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✓ 37 results · all verified · 26 also independently AI-judged

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Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini

1 · Prerequisites

2 · Summary

The page builds the tangent space from the cotangent space. For a point of a finite-type $k$ -scheme the intrinsic cotangent space is $m_{x} / m_{x}^{2}$ over the residue field $κ (x)$ , and the Zariski tangent space is its dual; at a $k$ -rational point this is the classical $m_{P} / m_{P}^{2}$ , and the tangent vectors are exactly the dual-number points $k [ϵ] / (ϵ^{2}) \to X$ lifting the point, equivalently the $k$ -derivations $O_{X, x} \to k$ . The Jacobian matrix of a finite generating list of the defining ideal computes the tangent space at a rational point, $T_{a} X ≅ \ker J (a)$ , independently of the chosen finite generating list, and the calculus continues with tangent points over square-zero vector extensions, functoriality for morphisms, with open immersions inducing isomorphisms, and the splitting of the tangent space of a product. No identification at a nonrational point is asserted.

A Noetherian local ring is regular when its embedding dimension equals its dimension, and the embedding dimension is always at least the local dimension. For a reduced finite-type scheme over an algebraically closed field a closed point is regular exactly when $\dim_{k} T_{x} X = \dim_{x} X$ , so regularity is readable from tangent dimensions. The Jacobian criterion turns this into an algebraic test: over a perfect field $A_{m}$ is regular if and only if $rank J (m) = n - \dim A_{m}$ , and at a rational point the same rank formula holds over an arbitrary field. The consequences assembled here are that the singular locus of a squarefree hypersurface is the common zero locus of its partial derivatives, that a regular point lies on exactly one irreducible component, that the regular locus is open, and that for a nonempty reduced finite-type scheme over a perfect field the regular locus is nonempty and dense in every irreducible component. For an irreducible classical variety the minimum of $\dim_{k} T_{x} X$ over the closed points equals $\dim X$ , so regularity is equivalent to constancy of the tangent-dimension function; a transitive group action likewise forces regularity.

Smoothness of a finite-type $k$ -scheme is the locally standard-smooth presentation, local on source and target. Over a perfect field it coincides with regularity, while over imperfect fields the two notions separate: for $k$ of characteristic $p$ and $a \notin k^{p}$ the scheme $Spec k [t] / (t^{p} - a)$ is regular but not smooth. Smoothness is stable under products and under base change of the standard-smooth presentation, the submersion criterion identifies smoothness at a point of a morphism between smooth classical varieties with surjectivity of the differential there, a hyperplane slice transverse to the tangent space is smooth, and every tangent direction is realized by a reduced curve that is smooth at the point and has the prescribed tangent line. In characteristic zero a dominant morphism of smooth classical varieties restricts to a smooth morphism over a nonempty open subset of the source: the locus where the differential has rank at most $r$ has image of dimension at most $r$ , so away from these images the differential is everywhere surjective.

The tangent cone at a rational point is the spectrum of the associated graded ring of the local ring, presented by the initial ideal with respect to a regular system of parameters, and its $k$ -linear span is the tangent space, in the scheme-theoretic sense that no proper linear closed subscheme of the tangent affine space contains the cone; the full, possibly nonreduced, cone is retained, while its reduction can span less. Multiplicity enters through the lowest nonvanishing homogeneous part of a hypersurface equation: a hypersurface point is smooth exactly at multiplicity one, over any field. The page closes with the Bertini package: the zero scheme of a section of a line bundle, base loci of linear systems, smoothness of the incidence correspondence over a smooth base, and the theorem that in characteristic zero the general member of a nonzero linear system is smooth on the complement of its base locus, the hyperplane case being recovered for an embedding. The corollary specialises to complete intersections: for a nonempty smooth projective variety of pure dimension $d$ over an algebraically closed field of characteristic zero and prescribed positive degrees, a general tuple of hypersurfaces meets in a nonempty smooth scheme of pure dimension $d - r$ when $r \leq d$ , and in the empty scheme when $r > d$ . A closing remark separates the intrinsic conventions of the page from those that depend on a chosen presentation. The Axiom of Choice is declared and inherited only through the cited suppliers; the Jacobian, differential and linear-algebra computations are choice-free.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-sol)audited 2026-09-30Open item page →

The intrinsic cotangent space

Definition

Let $X$ be a scheme and $x \in X$ . Write $A = O_{X, x}$ and let $m_{x}$ be the maximal ideal of this local ring. The intrinsic Zariski cotangent space of $X$ at $x$ is $C_{x} X : = m_{x} / m_{x}^{2} .$ It is a vector space over the residue field $κ (x) = O_{X, x} / m_{x}$ (The residue field at a point of an affine scheme).

The scalar action is induced by multiplication in $A$ : the class of $a \in A$ acts on the class of $b \in m_{x}$ by the class of $a b$ . This action depends only on the residue class of $a$ , since replacing $a$ by an element congruent modulo $m_{x}$ changes $a b$ by an element of $m_{x}^{2}$ . Thus the action factors through the field $κ (x)$ .

For a classical variety over an algebraically closed field at a closed $k$ -rational point, the residue field is $k$ and this is the usual cotangent space $m / m^{2}$ of the local ring. The definition above also applies to nonclosed points of arbitrary schemes.

This is the cotangent space of the underlying scheme at the point. No identification with a relative cotangent space over a chosen base is included in this definition.

For a direct check of the extreme dimensions, at the origin of $Spec k [t]$ the local ring is $k [t]_{(t)}$ and its maximal ideal is generated by $t$ , so $C_{0} X = (t) / (t^{2}) ≅ k$ with basis the class of $t$ . At the generic point $η$ of $Spec k [t]$ , the local ring is the field $k (t)$ , so its maximal ideal is zero and $C_{η} X = 0$ .

DefinitionDefinition: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

The intrinsic Zariski tangent space

Definition

Let $X$ be a scheme and $x \in X$ . Write $C_{x} X = m_{x} / m_{x}^{2}$ for the intrinsic cotangent space from The intrinsic cotangent space. The intrinsic Zariski tangent space of $X$ at $x$ is its linear dual over the residue field: $T_{x} X : = {Hom}_{κ (x)} (C_{x} X, κ (x)) .$ If $m_{x} = 0$ , then $C_{x} X = 0$ and $T_{x} X = 0$ .

For a $k$ -scheme $f : X \to Spec k$ (Schemes and morphisms over a base), the relative tangent space $T_{X / k, x}$ is separately defined as the $κ (x)$ -dual of $Ω_{X / k, x} \otimes_{O_{X, x}} κ (x)$ (Relative cotangent and tangent spaces). This definition makes no identification between $T_{x} X$ and $T_{X / k, x}$ at a nonrational point. At a $k$ -rational point they agree by the cotangent-space isomorphism Cotangent space at a rational point.

If $f : X \to Spec k$ is locally of finite type, then $T_{x} X$ is finite-dimensional. Indeed, an affine neighborhood has coordinate algebra finite type over $k$ , hence Noetherian; its local ring is a localization and is Noetherian. Its maximal ideal is therefore finitely generated, so the images of a finite generating set span $m_{x} / m_{x}^{2}$ . The dual of a finite-dimensional vector space is finite-dimensional. This argument uses no Axiom of Choice.

For example, at the closed origin of $X = Spec k [t]$ , the local ring is $k [t]_{(t)}$ and its maximal ideal is generated by $t$ . Hence $C_{0} X = (t) / (t^{2}) ≅ k$ with basis the class of $t$ , so $T_{0} X ≅ k$ . At the generic point $η = (0)$ , the local ring is the field $k (t)$ and its maximal ideal is zero, so $C_{η} X = 0$ and $T_{η} X = 0$ . In contrast, the relative tangent space over $k$ at $η$ is one-dimensional: the generic stalk of $Ω_{X / k}$ is $k (t) d t$ , so $T_{X / k, η} ≅ k (t)$ . Thus the two notions can differ at a nonrational point.

The definition also applies without a reducedness hypothesis. For the closed point of $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ (The affine scheme of dual numbers), every $a + b ϵ$ with $a \neq 0$ has inverse $a^{- 1} - a^{- 2} b ϵ$ . Thus its local ring is $k [ϵ] / (ϵ^{2})$ with maximal ideal $(ϵ)$ and square zero. Hence $C_{x} D_{k} ≅ k ϵ$ and $T_{x} D_{k} ≅ k$ .

Facts & Assumptions

Given: A scheme $X$ , a point $x \in X$ , and, for the finiteness assertion, a field $k$ and a morphism $X \to Spec k$ locally of finite type.

[F1]

The intrinsic cotangent space: $C_{x} X = m_{x} / m_{x}^{2}$ is a $κ (x)$ -vector space. At the origin of $Spec k [t]$ it is $k$ with basis the class of $t$ , while at the generic point it is zero.

[F2]

Locally finite type and finite type morphisms: locally of finite type means every point has an affine open neighborhood $U = Spec B$ over an affine base $Spec A$ , with $A \to B$ of finite type.

[F3]

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras: a commutative $R$ -algebra of finite type has the form $R [a_{1}, \dots, a_{n}]$ for some finite list, so the evaluation map $R [x_{1}, \dots, x_{n}] \to R [a_{1}, \dots, a_{n}]$ is surjective.

[F4]

A field has only the zero ideal and itself, hence is Noetherian: every field is a Noetherian ring.

[F5]

If $R$ is Noetherian then $R [x_{1}, \dots, x_{n}]$ is Noetherian for every $n \in N$ : if $R$ is Noetherian, then $R [x_{1}, \dots, x_{n}]$ is Noetherian for every $n \in N$ .

[F6]

The stalk of the affine structure sheaf at a prime is A_p: for $p \in Spec B$ , $O_{Spec B, p} ≅ B_{p}$ .

[F7]

Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ : if $p$ is prime in $B$ , then $B_{p}$ consists of fractions $b / s$ with $s \notin p$ .

[F8]

Noetherian commutative rings and modules: a commutative ring is Noetherian exactly when every ideal is finitely generated.

[F9]

Relative cotangent and tangent spaces: $T_{X / S, x} = {Hom}_{κ (x)} (Ω_{X / S, x} \otimes_{O_{X, x}} κ (x), κ (x))$ .

[F10]

Affine charts recover the algebraic module of differentials: on an affine scheme, $Ω_{X / S} (D (g)) ≅ Ω_{B_{g} / A}$ , compatibly with the localization maps.

[F11]

Polynomial differentials are free: $Ω_{A [x] / A}$ is free on $d x$ when there is one polynomial variable.

[F12]

Kähler differentials commute with localization: Kähler differentials commute with localization.

[F13]

Cotangent space at a rational point: at a $k$ -rational point, $m_{x} / m_{x}^{2} ≅ Ω_{X / k, x} \otimes_{O_{X, x}} κ (x)$ .

Proof

technique · direct

1.1F1F2F3F4F5F6F7F8algebra

Finite-dimensionality in the locally finite-type case. Choose an affine neighborhood $Spec B$ of $x$ provided by [F2], with $B$ a finite-type $k$ -algebra. By [F3], for some finite list $b_{1}, \dots, b_{n}$ the evaluation map $P = k [x_{1}, \dots, x_{n}] \to B$ is surjective. By [F4] and [F5], $k$ and then $P$ are Noetherian. The preimage in $P$ of any ideal of $B$ is an ideal of $P$ ; it is finitely generated by [F8], and its generators map to generators of the ideal in $B$ . Thus $B$ is Noetherian. If $x$ corresponds to $p \subset B$ , [F6] identifies $O_{X, x}$ with $B_{p}$ , and [F7] describes this localization by fractions. For any ideal $J \subseteq B_{p}$ , its contraction $I = {b \in B : b / 1 \in J}$ is an ideal of $B$ , hence is generated by finitely many $b_{1}^{'}, \dots, b_{r}^{'}$ by [F8]. If $b / s \in J$ , then $b / 1 = (s / 1) (b / s) \in J$ , so $b \in I$ and $b = \sum_{i} c_{i} b_{i}^{'}$ ; consequently $b / s = \sum_{i} (c_{i} / s) (b_{i}^{'} / 1)$ . Conversely each $b_{i}^{'} / 1$ lies in $J$ , so they generate $J$ . Thus $B_{p}$ is Noetherian. Its maximal ideal $m_{x}$ is finitely generated by [F8], say by $a_{1}, \dots, a_{r}$ . Modulo $m_{x}^{2}$ , every element $\sum_{i} b_{i} a_{i}$ is the $κ (x)$ -linear combination $\sum_{i} {\overline{b}}_{i} [a_{i}]$ , so $C_{x} X$ is finite-dimensional by [F1]. The dual is finite-dimensional as well: a surjection $κ (x)^{r} ↠ C_{x} X$ induces an injection ${Hom}_{κ (x)} (C_{x} X, κ (x)) ↪ {Hom}_{κ (x)} (κ (x)^{r}, κ (x))$ . This uses only finite generating lists and ordinary induction on the polynomial degree; no dependent choice or Axiom of Choice is invoked.

1.2F1F6F7F9F10F11F12

Difference at the generic point of the affine line. Let $X = Spec k [t]$ and $η = (0)$ . By [F6] and [F7], its local ring is the field $k (t)$ with maximal ideal zero, so [F1] gives $C_{η} X = 0$ and hence $T_{η} X = 0$ . On this affine chart [F11] gives $Ω_{k [t] / k} = k [t] d t$ ; using [F10] and [F12] to pass to the generic stalk gives $Ω_{X / k, η} = k (t) d t$ . Its residue-field fibre is the one-dimensional $k (t)$ -vector space $k (t) d t$ , so [F9] gives $T_{X / k, η} ≅ k (t)$ . This proves that the intrinsic and relative tangent spaces need not agree at a nonrational point.

2.1F9F13∎

Rational-point comparison. If $x$ is $k$ -rational, [F13] identifies the intrinsic cotangent space with the relative cotangent space. Taking $k$ -linear duals and using [F9] identifies $T_{x} X$ with $T_{X / k, x}$ . No such identification is asserted for nonrational points.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Cotangent spaces commute with localization at a rational point

Statement

Let $k$ be a field, let $A$ be a commutative $k$ -algebra, and let $m \subset A$ be a maximal ideal whose residue field is $A / m = k$ via the structure map. Set $S = A ∖ m$ , $A_{m} = S^{- 1} A$ , and $n = m A_{m}$ . Use the conventions $m^{0} = A$ and $m^{r + 1} = m^{r} m$ , and similarly for $n$ . For every $r \in N$ , the canonical map

$θ_{r} : m^{r} / m^{r + 1} ⟶ n^{r} / n^{r + 1}, [a] ⟼ a / 1 + n^{r + 1}$

is an isomorphism. At $r = 1$ this is the localization comparison for the intrinsic cotangent space The intrinsic cotangent space.

Facts & Assumptions

Given: A field $k$ , a commutative $k$ -algebra $A$ , and a maximal ideal $m$ such that $A / m = k$ as a $k$ -algebra.

[F1]

The intrinsic cotangent space: the intrinsic cotangent space at a point is the maximal ideal of its local ring modulo its square, over the residue field.

[F2]

Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ : for a prime ideal $p$ , $A_{p} = (A ∖ p)^{- 1} A$ and its elements are fractions $a / s$ with $s \notin p$ .

[F3]

$R_{p}$ is local with unique maximal ideal $p R_{p}$ : $A_{p}$ is local with unique maximal ideal $p A_{p} = {a / s : a \in p, s \notin p}$ .

[F4]

Ideals of $S^{- 1} R$ correspond to $S$ -saturated ideals of $R$ , and prime ideals correspond to primes disjoint from $S$ : for an ideal $I$ of $A$ , its extension is $S^{- 1} I = {a / s : a \in I, s \in S}$ .

[F5]

Equality, vanishing, and the kernel of the localisation map: $a / s = b / t$ in $S^{- 1} A$ exactly when $u (a t - b s) = 0$ for some $u \in S$ .

[F6]

The sum $I + J$ and product $I J$ of two-sided ideals: the product $I J$ of ideals consists of finite sums of products $i j$ with $i \in I$ and $j \in J$ .

Proof

technique · direct

1.1F2F3F4F6givenalgebra

Localization of the powers. If $a b \in m$ , then $\overset{ˉ}{a} \overset{ˉ}{b} = 0$ in the field $A / m$ , so $\overset{ˉ}{a} = 0$ or $\overset{ˉ}{b} = 0$ ; hence $m$ is prime and $S = A ∖ m$ is multiplicative. By [F2] and [F3], $A_{m} = S^{- 1} A$ and its maximal ideal is $n$ . By [F4], $n = S^{- 1} m$ and each extension $S^{- 1} I$ consists of fractions with numerator in $I$ . With the stated recursive convention for powers, [F6] gives $n^{r} = S^{- 1} (m^{r})$ for every $r \geq 0$ : it holds for $r = 0$ . If it holds at $r$ , every element of $n^{r + 1} = n^{r} n$ is a finite sum of products $(a_{i} / s_{i}) (b_{i} / t_{i}) = a_{i} b_{i} / (s_{i} t_{i})$ with $a_{i} \in m^{r}$ and $b_{i} \in m$ , so it lies in $S^{- 1} (m^{r + 1})$ . Conversely, writing a numerator in $m^{r + 1} = m^{r} m$ as a finite sum of such products expresses every fraction in $S^{- 1} (m^{r + 1})$ as an element of $n^{r + 1}$ . Therefore $θ_{r}$ is well-defined.

2.1step 1.1F4F5F6givenalgebra

Injectivity. Suppose $a \in m^{r}$ and $θ_{r} ([a]) = 0$ . By step 1.1 and [F4], $a / 1 = b / s$ for some $b \in m^{r + 1}$ and $s \in S$ . By [F5], there is $u \in S$ with $u (a s - b) = 0$ , so $u s a = u b \in m^{r + 1}$ . The residue of $u s$ in the field $A / m$ is nonzero; choose $t \in A$ whose residue is its inverse. Then $v = t u s - 1 \in m$ and $a = t u s a - v a \in m^{r + 1} + m m^{r} = m^{r + 1}$ . Thus $[a] = 0$ and $θ_{r}$ is injective.

3.1step 1.1F1F4F6givenalgebra∎

Surjectivity and boundary instances. Let a class in $n^{r} / n^{r + 1}$ be represented, by step 1.1 and [F4], by $a / s$ with $a \in m^{r}$ and $s \in S$ . Choose $t \in A$ whose residue is the inverse of the nonzero residue of $s$ . Then $1 - t s \in m$ , and $(a / s) - (t a / 1) = ((1 - t s) a) / s \in S^{- 1} (m^{r + 1}) = n^{r + 1}$ . Hence the class is $θ_{r} ([t a])$ , proving surjectivity. This covers $r = 0$ , where the map is $A / m \to A_{m} / n$ , and $r = 1$ , the cotangent-space map of [F1]. If $m = 0$ , then $A = k$ : $θ_{0}$ is the identity of $k$ and for every $r \geq 1$ both sides are zero. The lifts $t$ above are chosen separately for each displayed fraction, so no choice principle is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Tangent vectors at rational points are dual-number points

Statement

Let $X$ be any $k$ -scheme and let $x \in X (k)$ be a $k$ -rational point. The intrinsic Zariski tangent space $T_{x} X$ is naturally isomorphic, as a $k$ -vector space, to the fibre over $x$ of ${Hom}_{k} (Spec (k [ϵ] / (ϵ^{2})), X) ⟶ X (k),$ where the map is induced by $ϵ \mapsto 0$ . Equivalently, $T_{x} X ≅ {Der}_{k} (O_{X, x}, k),$ where $O_{X, x}$ acts on $k$ through evaluation at $x$ . The bijection is induced by writing a local $k$ -algebra map as $a \mapsto a (x) + ϵ D (a)$ . No identification at a nonrational point is asserted.

Facts & Assumptions

Given: A field $k$ , a $k$ -scheme $X$ , and a $k$ -rational point $x \in X (k)$ . Put $D = k [ϵ] / (ϵ^{2})$ and let $ρ : D \to k$ send $ϵ$ to zero.

[F1]

The intrinsic Zariski tangent space: $T_{x} X$ is the $k$ -linear dual of $m_{x} / m_{x}^{2}$ when $κ (x) = k$ .

[F2]

The affine scheme of dual numbers: $Spec D$ is the dual-numbers scheme, with $ϵ^{2} = 0$ .

[F3]

Affine schemes are contravariantly equivalent to commutative rings: for commutative rings $A, B$ , ring maps $A \to B$ correspond contravariantly to morphisms $Spec B \to Spec A$ .

[F4]

Cotangent spaces commute with localization at a rational point: if $A / m = k$ , localization induces an isomorphism $m / m^{2} \to (m A_{m}) / (m A_{m})^{2}$ .

[F5]

Schemes: every point of a scheme has an affine open neighborhood.

[F6]

Affine open subschemes: an open subscheme has the restricted structure sheaf; an affine open is affine with this structure.

[F7]

Open immersions of schemes: the inclusion of an open subscheme is an open immersion.

[F8]

The underlying space of an affine spectrum: the points of $Spec A$ are the prime ideals of $A$ .

[F9]

The stalk of the affine structure sheaf at a prime is A_p: at $p \in Spec A$ , $O_{Spec A, p} ≅ A_{p}$ .

[F10]

Schemes and morphisms over a base: a $k$ -morphism commutes with the structure maps to $Spec k$ .

[F11]

Prime ideals and maximal ideals in a commutative ring: a proper ideal $P$ is prime when $a b \in P$ implies $a \in P$ or $b \in P$ ; a maximal ideal has no proper ideal strictly between it and the ring.

[F12]

The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ : $R / I$ is formed from cosets with $(r + I) (s + I) = r s + I$ .

[F13]

$R / M$ is a field if and only if $M$ is a maximal ideal: for a commutative ring $R$ , $R / M$ is a field exactly when $M$ is maximal.

Proof

technique · direct

1.1F2F11F12F13givenalgebra

The dual-numbers scheme has one point. If $p$ is a prime ideal of $D$ , then $ϵ^{2} = 0 \in p$ implies $ϵ \in p$ by [F11]. The quotient $D / (ϵ)$ is $k$ , so $(ϵ)$ is maximal by [F12, F13]. Every prime containing this maximal ideal equals it. Thus $Spec D$ has the single point defined by $(ϵ)$ , and the map $Spec k \to Spec D$ induced by $ρ$ selects that point.

2.1F3F5F6F7F8F10step 1.1givenalgebra

Based morphisms can be computed in an affine neighborhood. Choose an affine open $U = Spec A$ containing $x$ by [F5, F6]. Its inclusion into $X$ is an open immersion by [F7]. Since $Spec D$ has only one point, every morphism in the fibre over $x$ factors uniquely through $U$ . The affine anti-equivalence [F3], together with the $k$ -morphism condition [F10], identifies such maps with $k$ -algebra homomorphisms $φ : A \to D$ whose reduction $ρ \circ φ : A \to k$ is the point $x$ . Conversely every such homomorphism gives a based morphism. If $m = \ker (A \to k)$ , then $m$ is the point of $U$ by [F8].

3.1F2F10step 2.1givenalgebra

These homomorphisms are exactly derivations. Each $a \in A$ has a unique image $φ (a) = \overset{ˉ}{a} + ϵ δ (a)$ , where $\overset{ˉ}{a} = x^{#} (a)$ . Since $φ$ is a $k$ -algebra homomorphism, $δ$ is $k$ -linear and vanishes on $k$ . Comparing the $ϵ$ -coefficients of $φ (a b) = φ (a) φ (b)$ gives $δ (a b) = \overset{ˉ}{a} δ (b) + \overset{ˉ}{b} δ (a)$ , so $δ$ is a derivation for the $A$ -module structure on $k$ given by evaluation at $x$ . Conversely each such derivation defines a homomorphism by this formula, since $ϵ^{2} = 0$ . These constructions are inverse.

4.1step 3.1givenalgebra

Derivations on $A$ are the dual of its cotangent space at $x$ . The structure map $k \to A$ splits evaluation $A \to k$ , so $A = k \oplus m$ as $k$ -vector spaces. The derivation identity makes $δ$ vanish on $m^{2}$ , and restriction gives a linear form on $m / m^{2}$ . Conversely, for $ℓ \in {Hom}_{k} (m / m^{2}, k)$ , define $δ (c + u) = ℓ (u + m^{2})$ for $c \in k$ , $u \in m$ . For $c + u, c^{'} + u^{'} \in A$ , the product has $m$ -part $c u^{'} + c^{'} u + u u^{'}$ , and $u u^{'} \in m^{2}$ ; hence this formula satisfies the Leibniz rule. It is inverse to restriction.

5.1F1F2F4F9F10step 2.1step 3.1step 4.1algebra∎

Passing to the stalk gives the claimed intrinsic tangent and local derivation formulation. By [F9], $O_{X, x} ≅ A_{m}$ with maximal ideal $m_{x} = m A_{m}$ . The rational-point localization isomorphism [F4] identifies $m / m^{2}$ with $m_{x} / m_{x}^{2}$ , so their $k$ -linear duals agree; [F1] identifies the latter dual with $T_{x} X$ . Also every $s \in A ∖ m$ has $φ (s) = \overset{ˉ}{s} + ϵ δ (s)$ with $\overset{ˉ}{s} \neq 0$ , which is a unit in $D$ with inverse ${\overset{ˉ}{s}}^{- 1} - {\overset{ˉ}{s}}^{- 2} δ (s) ϵ$ . Thus $φ$ extends uniquely to a local $k$ -algebra map $O_{X, x} = A_{m} \to D$ . Conversely, every local $k$ -algebra map $α : O_{X, x} \to D$ has a unique form $α (u) = u (x) + ϵ d (u)$ , where multiplicativity makes $d$ a $k$ -derivation through the residue action. Every such derivation defines a local map by this formula, since units have nonzero residue. Applying the decomposition $O_{X, x} = k \oplus m_{x}$ as in step 4.1 gives ${Der}_{k} (O_{X, x}, k) ≅ {Hom}_{k} (m_{x} / m_{x}^{2}, k)$ . The localization, extension, and restriction maps commute on smaller affine neighborhoods, so the identifications are independent of $U$ and natural. Scaling $ϵ$ by $c \in k$ scales the derivation and tangent vector by $c$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Square-zero vector extensions encode tangent vectors with coefficients

Statement

Let $k$ be algebraically closed, let $X$ be a classical affine variety over $k$ , and write $A = k [X]$ . Regard $X$ with its associated affine scheme $Spec A$ when forming $T_{x} X$ . For a finite-dimensional $k$ -vector space $W$ , give $R_{W} = k \oplus W$ the square-zero $k$ -algebra structure $(a, w) (b, v) = (a b, a v + b w), (a, w), (b, v) \in k \oplus W .$ Then reduction by the augmentation $π : R_{W} \to k$ , $π (a, w) = a$ , defines a natural bijection ${Hom}_{k -alg} (A, R_{W}) ≅ {(x, t) : x \in X (k), t \in W \otimes_{k} T_{x} X} .$

Facts & Assumptions

Given: An algebraically closed field $k$ , a classical affine variety $X \subseteq k^{n}$ , its coordinate ring $A = k [X]$ , and a finite-dimensional $k$ -vector space $W$ . The product on $R_{W}$ is the one displayed above.

[F1]

A classical affine variety over an algebraically closed field is a nonempty irreducible affine algebraic set $X \subseteq k^{n}$ (A classical affine variety).

[F2]

$A = k [t_{1}, \dots, t_{n}] / I (X)$ , and the coordinate classes generate $A$ as a $k$ -algebra. The zero algebra is allowed, so $k [\emptyset] = 0$ (The coordinate ring of a classical affine algebraic set).

[F3]

$I (X)$ consists exactly of the polynomials vanishing at every point of $X$ (The classical vanishing ideal).

[F4]

For a commutative ring $R$ , a homomorphism $R [t] \to S$ is uniquely determined by its coefficient map and the image of $t$ ; iteration gives evaluation on $k [t_{1}, \dots, t_{n}]$ (Universal property of $R [x]$ : a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism).

[F5]

An affine scheme is a locally ringed space isomorphic to $(Spec A, O_{Spec A})$ (Affine schemes and their coordinate rings).

[F6]

The underlying topological spectrum has the prime ideals of $A$ as its points (The underlying space of an affine spectrum).

[F7]

A proper ideal $P$ is prime when $a b \in P$ implies $a \in P$ or $b \in P$ (Prime ideals and maximal ideals in a commutative ring).

[F8]

For a prime ideal $p$ , $A_{p}$ is the localization using denominators outside $p$ (Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ ).

[F9]

$A_{p}$ is local with unique maximal ideal $p A_{p}$ ( $R_{p}$ is local with unique maximal ideal $p R_{p}$ ).

[F10]

The stalk of the affine structure sheaf at $p$ is canonically $A_{p}$ (The stalk of the affine structure sheaf at a prime is A_p).

[F11]

If $A / m = k$ via the structure map, localization canonically identifies $m / m^{2}$ with $m A_{m} / (m A_{m})^{2}$ (Cotangent spaces commute with localization at a rational point).

[F12]

The intrinsic cotangent space at $x$ is the maximal ideal of $O_{X, x}$ modulo its square (The intrinsic cotangent space).

[F13]

$T_{x} X = {Hom}_{k} (C_{x} X, k)$ at a $k$ -rational point (The intrinsic Zariski tangent space).

[F14]

If $V$ is finite-dimensional, the canonical map $V^{*} \otimes_{k} W \to {Hom}_{k} (V, W)$ is a natural isomorphism (For finite-dimensional $V$ , the canonical map $V^{*} \otimes_{F} W \to {Hom}_{F} (V, W)$ is an isomorphism).

[F15]

A balanced bilinear map out of two modules induces a unique map from their tensor product (Universal property of the tensor product for balanced maps into abelian groups).

[F16]

At a rational point, tangent vectors are naturally the based points of the dual-numbers scheme (Tangent vectors at rational points are dual-number points).

[F17]

The coordinate ring convention allows the zero algebra and gives $k [\emptyset] = 0$ (The coordinate ring of a classical affine algebraic set).

Proof

technique · direct

1.1F1F2F3F4givenalgebra

For $ϕ : A \to R_{W}$ , compose with $π$ and put $a_{i} = (π ϕ) ({\overset{ˉ}{t}}_{i})$ . Every $f \in I (X)$ satisfies $f (a_{1}, \dots, a_{n}) = (π ϕ) (\overset{ˉ}{f}) = 0$ , so $a = (a_{i}) \in V (I (X)) = X$ by [F1, F2, F3, F4]; conversely evaluation at each $x \in X (k)$ is a $k$ -algebra map $e_{x} : A \to k$ , and the coordinate classes generate $A$ , so this identifies ${Hom}_{k -alg} (A, k)$ with $X (k)$ .

1.2F2F3F4F5F6F7F8F9F10F11givenalgebra

Fix $x = (a_{1}, \dots, a_{n})$ and put $m = \ker e_{x}$ ; evaluation is surjective on constants, so $A / m = k$ , which makes $m$ proper, maximal, and prime. Since $I (X)$ is contained in the polynomial evaluation kernel at $a$ by [F3], and that kernel is generated by $t_{1} - a_{1}, \dots, t_{n} - a_{n}$ by telescoping each monomial's factors using [F4], their classes generate $m$ and $C_{A} : = m / m^{2}$ is finite-dimensional. By [F5, F6, F8, F9, F10, F11], $O_{X, x} = A_{m}$ has maximal ideal $m A_{m}$ and localization induces a canonical isomorphism $θ : C_{A} \tilde{\to} C_{x} X$ .

2.1step 1.1step 1.2givenalgebra

Among maps whose reduction is $e_{x}$ , write uniquely $ϕ (a) = e_{x} (a) + D (a)$ with $D (a) \in W$ ; comparing products in $R_{W}$ shows $D$ is a $k$ -derivation for the $A$ -module structure on $W$ through $e_{x}$ , with $D (a b) = e_{x} (a) D (b) + e_{x} (b) D (a)$ , and conversely every such derivation gives a map because $W^{2} = 0$ . It kills $m^{2}$ and restricts to a linear map $C_{A} \to W$ ; conversely, for $h : C_{A} \to W$ , $D (a) = h ([a - e_{x} (a) 1])$ defines the inverse derivation, since writing $a = e_{x} (a) 1 + u$ , $b = e_{x} (b) 1 + v$ with $u, v \in m$ leaves only the terms $e_{x} (a) v + e_{x} (b) u$ modulo $m^{2}$ .

3.1F12F13F14F15F16step 1.1step 1.2step 2.1algebra

The canonical tensor-Hom map sends $w \otimes λ$ to $[c \mapsto λ (c) w]$ ; by [F14] and the canonical tensor symmetry obtained from [F15], it identifies $W \otimes_{k} C_{A}^{*}$ with ${Hom}_{k} (C_{A}, W)$ . Dualizing $θ$ identifies this with $W \otimes_{k} T_{x} X$ by [F12, F13], so $ϕ$ maps to $(x, t)$ where $x$ is its reduction and $t$ encodes its induced map $C_{A} \to W$ , and the inverse sends $(x, t)$ to evaluation plus the corresponding derivation from step 1.2. These constructions are canonical and natural in $W$ ; when $W = k$ , $k [ϵ] / (ϵ^{2}) \to R_{k}$ , $ϵ \mapsto (0, 1)$ , identifies this with [F16].

4.1F1F17F14step 1.1step 1.2step 2.1step 3.1givenalgebra∎

If $W = 0$ , then $R_{W} = k$ and the bijection is $ϕ = e_{x} \leftrightarrow (x, 0)$ ; if $C_{x} X = 0$ , then $T_{x} X = 0$ and every map over $x$ is evaluation. For the empty algebraic set, outside the variety hypothesis, $k [X] = 0$ and no unital map $k [X] \to R_{W}$ exists, matching the absence of pairs. The construction works for every tensor $t$ , including $t = 0$ , and has no reverse implication. Only the finite coordinate presentation of this fixed $X$ is used to prove finite-dimensionality; the tensor-Hom map is canonical, no family of bases or points is selected, and no Axiom of Choice is used.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Equation rows and coordinate columns in an affine Jacobian

Definition

Let $k$ be a field, let $I \subseteq k [t_{1}, \dots, t_{n}]$ be an ideal with a specified finite generating list $f_{1}, \dots, f_{r}$ , and let $a = (a_{1}, \dots, a_{n}) \in k^{n}$ satisfy $f (a) = 0$ for every $f \in I$ . The Jacobian matrix at $a$ , with the equation-row convention, is the $r \times n$ matrix $J_{(f_{1}, \dots, f_{r})} (a) = {(\frac{\partial f_{i}}{\partial t_{j}} (a))}_{1 \leq i \leq r, 1 \leq j \leq n} .$ Thus row $i$ is the differential of equation $f_{i}$ , and column $j$ corresponds to coordinate $t_{j}$ . Formal derivatives are computed on monomials by $\partial_{t_{j}} (t_{1}^{e_{1}} \dots t_{n}^{e_{n}}) = {\begin{cases} e_{j} t_{1}^{e_{1}} \dots t_{j}^{e_{j} - 1} \dots t_{n}^{e_{n}}, & e_{j} > 0, \\ 0, & e_{j} = 0. \end{cases}$ with the integer coefficient read in $k$ , and are extended $k$ -linearly. The definition uses the actual scheme ideal $I$ ; it does not assume that $I$ is radical or that $k$ is perfect. For a reduced classical algebraic set over an algebraically closed field, this specializes to its coordinate ring $k [X] = k [t_{1}, \dots, t_{n}] / I (X)$ (The coordinate ring of an affine algebraic set).

For a scheme-theoretic affine zero locus, an equation list for the same underlying point set is not substituted for the actual ideal: nilpotent structure changes the Jacobian problem.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

The Jacobian kernel computes the tangent space

Statement

Let $k$ be any field, let $n \geq 0$ be finite, let $I$ be an ideal of $k [t_{1}, \dots, t_{n}]$ , and put $X = Spec (k [t_{1}, \dots, t_{n}] / I) .$ Let $a = (a_{1}, \dots, a_{n}) \in X (k)$ , and let $f_{1}, \dots, f_{r}$ be any finite generating list of the actual ideal $I$ . Then the coordinate-velocity map gives a canonical $k$ -linear isomorphism $T_{a} X ≅ \ker ⁣ (J_{(f_{1}, \dots, f_{r})} (a) : k^{n} ⟶ k^{r}) .$ The kernel is independent of the chosen finite generating list of $I$ . No reducedness, perfectness, or characteristic hypothesis is needed. Finiteness of the list is available for every finite $n$ by the finite-variable polynomial Noetherian result cited below.

Facts & Assumptions

Given: A field $k$ , finite $n \geq 0$ , an ideal $I \subseteq k [t_{1}, \dots, t_{n}]$ , the affine $k$ -scheme $X = Spec (A)$ with $A = k [t_{1}, \dots, t_{n}] / I$ , and a rational point $a \in X (k)$ , represented by its coordinate tuple $(a_{1}, \dots, a_{n})$ . Set $D = k [ϵ] / (ϵ^{2})$ .

[F1]

Equation rows and coordinate columns in an affine Jacobian: the equation-row Jacobian matrix uses formal monomial derivatives at a rational point and the actual scheme ideal.

[F2]

Tangent vectors at rational points are dual-number points: $T_{a} X$ is naturally isomorphic as a $k$ -vector space to the fibre of based dual-number maps over $a$ ; equivalently, its vectors are the coefficient derivations of those maps.

[F3]

The affine scheme of dual numbers: $D = k [ϵ] / (ϵ^{2})$ , so every element is uniquely $c + ϵ d$ with $c, d \in k$ and $ϵ^{2} = 0$ .

[F4]

Schemes and morphisms over a base: a $k$ -morphism commutes with the structure maps to $Spec k$ .

[F5]

Affine schemes are contravariantly equivalent to commutative rings: ring maps $A \to B$ correspond contravariantly to morphisms $Spec B \to Spec A$ ; together with [F4], the maps over $k$ are the $k$ -algebra maps.

[F6]

Universal property of a polynomial ring on an arbitrary family of indeterminates: a coefficient map and assigned images of the variables determine a unique polynomial-ring homomorphism.

[F7]

A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring: a ring map from $k [t_{1}, \dots, t_{n}]$ that kills $I$ factors uniquely through $k [t_{1}, \dots, t_{n}] / I$ .

[F8]

Finite-variable polynomial algebras over fields are Noetherian by finite generators: for every field and finite $n$ , every ideal of $k [t_{1}, \dots, t_{n}]$ has a finite generating list; the result is choice-free.

Proof

technique · direct

1.1F4F5F6F8givenalgebra

By [F8], fix a finite list $f_{1}, \dots, f_{r}$ generating $I$ . Since $a$ is a $k$ -rational point, the affine anti-equivalence [F5] and the base condition [F4] give a $k$ -algebra map $A \to k$ . Its composite with the quotient map is evaluation at $a$ : the polynomial universal property [F6] identifies the composite as the unique map sending $t_{j}$ to $a_{j}$ . It kills $I$ , so $f_{i} (a) = 0$ for each $i$ .

1.2F1F3F6givenalgebra

For any $v = (v_{1}, \dots, v_{n}) \in k^{n}$ , [F6] gives a unique $k$ -algebra map $ϕ_{v} : k [t_{1}, \dots, t_{n}] \to D$ with $t_{j} \mapsto a_{j} + ϵ v_{j}$ . For a monomial $t^{e} = \prod_{j} t_{j}^{e_{j}}$ , expansion and $ϵ^{2} = 0$ give $t^{e} (a + ϵ v) = a^{e} + ϵ \sum_{j : e_{j} > 0} e_{j} a_{1}^{e_{1}} \dots a_{j}^{e_{j} - 1} \dots a_{n}^{e_{n}} v_{j}$ . Extending over its finitely many monomials yields $p (a + ϵ v) = p (a) + ϵ \sum_{j = 1}^{n} (\partial p / \partial t_{j}) (a) v_{j}$ ; the integer $e_{j}$ is read in $k$ , including in positive characteristic, and the empty sum and product conventions cover $n = 0$ .

2.1F2F3F4F5F6F7step 1.1step 1.2givenalgebra

The map $ϕ_{v}$ kills $I$ exactly when it kills every generator $f_{i}$ . By steps 1.1 and 1.2, $ϕ_{v} (f_{i}) = ϵ \sum_{j = 1}^{n} (\partial f_{i} / \partial t_{j}) (a) v_{j}$ , which is zero exactly when row $i$ of $J_{(f_{1}, \dots, f_{r})} (a)$ annihilates $v$ . Thus $ϕ_{v}$ factors uniquely through $A$ by [F7] exactly when $J (a) v = 0$ , and its reduction modulo $ϵ$ is $a$ . Conversely, any based $k$ -morphism $Spec D \to X$ corresponds by [F4, F5] to a $k$ -algebra map $A \to D$ reducing to evaluation at $a$ ; the images of the coordinates have unique form $a_{j} + ϵ v_{j}$ . By [F6] its composite from the polynomial ring is $ϕ_{v}$ , and the same calculation forces $J (a) v = 0$ . The two constructions are inverse. Their coefficient derivations depend $k$ -linearly on $v$ , and [F2] identifies them with $T_{a} X$ , proving the canonical linear isomorphism in the statement and both membership implications.

2.2F1step 1.1step 1.2algebra

Let $g_{1}, \dots, g_{s}$ be another finite generating list of $I$ , and write each $g_{ℓ} = \sum_{i} h_{ℓ i} f_{i}$ . The coefficient-of- $ϵ$ formula in step 1.2 is a derivation because each $ϕ_{v}$ is a ring homomorphism. Its product rule in each coordinate direction, together with $f_{i} (a) = 0$ , gives $d g_{ℓ} (a) = \sum_{i} h_{ℓ i} (a) d f_{i} (a)$ , so each row of $J_{g} (a)$ lies in the row span of $J_{f} (a)$ . Reversing the lists gives equality of row spans and hence equality of their annihilators, which are the kernels in $k^{n}$ . For $n = 0$ all rows are empty and both row spans are zero.

3.1F2F3F8step 1.1step 1.2step 2.1step 2.2givenalgebra∎

The boundary cases are explicit. If $X$ is empty there is no rational point, so the pointwise statement has no instance. If $r = 0$ , then $I = (0)$ , the matrix has no rows, and the result says $T_{a} A_{k}^{n} = k^{n}$ . If $n = 0$ and a rational point exists, its $k$ -algebra map $k / I \to k$ composed with $k \to k / I$ is the identity, so $I = (0)$ ; hence $T_{a} X = k^{0} = 0$ . In one coordinate, $X = Spec k [t] / (t^{2})$ at $0$ has Jacobian row $2 t ∣_{0} = 0$ (also in characteristic $2$ ), so its tangent space is all of $k$ , as the scheme-theoretic nilpotent structure requires. The zero vector corresponds to the constant based map $t_{j} \mapsto a_{j}$ . No AC or DC is used: the finite generating tuple is chosen for this single ideal, and no basis or family of choices is made. Steps 2.1 and 2.2 prove both directions of the kernel characterization and generator independence.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Differentials, open restriction, and the chain rule

Statement

Let $k$ be a field, let $X, Y$ be $k$ -schemes, and let $f : X \to Y$ be a $k$ -morphism. For points $x \in X$ and $y = f (x) \in Y$ whose structure maps $k \to κ (x)$ and $k \to κ (y)$ are isomorphisms (that is, the points are $k$ -rational), write $C_{x} X = m_{x} / m_{x}^{2}$ and $C_{y} Y = m_{y} / m_{y}^{2}$ . The local map $f_{x}^{♯} : O_{Y, y} ⟶ O_{X, x}$ induces a $k$ -linear map ${\overset{ˉ}{f}}_{x}^{♯} : C_{y} Y \to C_{x} X$ . Its dual is the differential $d_{x} f = ({\overset{ˉ}{f}}_{x}^{♯})^{*} : T_{x} X ⟶ T_{y} Y .$ It agrees with post-composition by $f$ on based dual-number points. For the identity, $d_{x} ({id}_{X}) = {id}_{T_{x} X}$ ; for composable $k$ -morphisms $X \overset{f}{\to} Y \overset{g}{\to} Z$ and rational points $x \in X (k)$ , $y = f (x)$ , one has $d_{x} (g \circ f) = d_{y} g \circ d_{x} f .$ Every $k$ -open immersion induces an isomorphism on tangent spaces at each rational point.

No finite-type, reducedness, or smoothness hypothesis is needed. No Axiom of Choice is assumed or used.

Facts & Assumptions

Given: A field $k$ , $k$ -schemes, a $k$ -morphism, and points $x, y = f (x)$ whose residue fields are identified with $k$ by their structure maps. For the last assertion, the morphism is an open immersion over $k$ and the source point has residue field $k$ .

[F1]

Morphisms of schemes: a scheme morphism is a morphism of locally ringed spaces, and its induced maps on stalks are local homomorphisms.

[F2]

Morphisms of locally ringed spaces: a local stalk homomorphism sends the maximal ideal at the image point into the maximal ideal at the source.

[F3]

Schemes and morphisms over a base: a $k$ -morphism commutes with the structure maps to $Spec k$ .

[F4]

The intrinsic cotangent space: $C_{x} X = m_{x} / m_{x}^{2}$ is a vector space over the residue field. At a $k$ -rational point this residue field is identified with $k$ by the structure map.

[F5]

The intrinsic Zariski tangent space: $T_{x} X$ is the linear dual of $C_{x} X$ over the residue field; at a rational point it is ${Hom}_{k} (C_{x} X, k)$ .

[F6]

Tangent vectors at rational points are dual-number points: at a rational point of a $k$ -scheme, tangent vectors are naturally the fibre of based morphisms from $Spec (k [ϵ] / (ϵ^{2}))$ .

[F7]

Open immersions of schemes: an open immersion identifies its source isomorphically with an open subscheme of its target.

[F8]

Affine open subschemes: an open subscheme $U \subseteq X$ has structure sheaf $O_{X} ∣_{U}$ .

Proof

technique · direct

1.1F1F2F3F4F5givenalgebra

Put $B = O_{Y, y}$ , $A = O_{X, x}$ , $n = m_{y}$ , and $m = m_{x}$ . By [F1], $f_{x}^{♯} : B \to A$ is local; [F2] means that $f_{x}^{♯} (n) \subseteq m$ and $f_{x}^{♯} (n^{2}) \subseteq m^{2}$ . It therefore induces a map $n / n^{2} \to m / m^{2}$ . Since $f$ is a $k$ -morphism, [F3] says that its stalk map commutes with the two structure maps from $k$ ; because $x$ and $y$ are rational, these maps identify both residue fields with $k$ . The quotient map is thus $k$ -linear by [F4]. Dualizing it over $k$ gives the stated map $d_{x} f : T_{x} X \to T_{y} Y$ by [F5].

1.2F1F2F4F5F7F8givenalgebra

Let $j : U \to X$ be a $k$ -open immersion and let $u \in U (k)$ map to $x \in X (k)$ . By [F7], $j$ identifies $U$ with an open subscheme of $X$ ; by [F8] that open subscheme carries the restricted structure sheaf. Hence the induced stalk map $j_{u}^{♯} : O_{X, x} \to O_{U, u}$ is an isomorphism. It identifies maximal ideals and their squares, so the induced cotangent map $C_{x} X \to C_{u} U$ is an isomorphism. Its dual $d_{u} j$ is therefore an isomorphism $T_{u} U \to T_{x} X$ .

2.1F1F3F4F5step 1.1givenalgebra

For the identity morphism, the local-ring and cotangent maps are identities, so their dual is the identity. If $g : Y \to Z$ is another $k$ -morphism and $z = g (y)$ , contravariance on stalks gives $(g \circ f)_{x}^{♯} = f_{x}^{♯} \circ g_{y}^{♯}$ . Passing to maximal ideals modulo squares gives ${\overline{(g \circ f)}}_{x}^{♯} = {\overset{ˉ}{f}}_{x}^{♯} \circ {\overset{ˉ}{g}}_{y}^{♯}$ . Dualizing reverses this order, so $d_{x} (g \circ f) = ({\overset{ˉ}{g}}_{y}^{♯})^{*} \circ ({\overset{ˉ}{f}}_{x}^{♯})^{*} = d_{y} g \circ d_{x} f$ . This proves identity and chain rules without choosing coordinates or bases.

3.1F4F5F6step 1.1step 2.1givenalgebra

Under [F6], a tangent vector $t \in T_{x} X$ is represented by a based map $γ_{t} : Spec (k [ϵ] / (ϵ^{2})) \to X$ . For $b \in n$ , the coefficient of $ϵ$ in the pullback of $b$ by $f \circ γ_{t}$ is the value of $t$ on $f_{x}^{♯} (b) m o d m^{2}$ , namely $t ({\overset{ˉ}{f}}_{x}^{♯} (b m o d n^{2}))$ . This is exactly the functional $d_{x} f (t)$ on $C_{y} Y$ . Constants have zero $ϵ$ -coefficient, so the agreement holds on the whole local ring. Thus the dualized construction is the map on based dual-number points induced by post-composition with $f$ ; the identity and composition laws also agree with composition of these maps.

4.1F4F5F6step 1.1step 2.1step 3.1givenalgebra∎

If a source or target cotangent space is zero, the induced cotangent map still has the displayed source and target, and its dual is the unique corresponding linear map; in particular zero tangent vectors map to zero. If both cotangent spaces are one-dimensional and the cotangent map sends a chosen target generator to $c$ times a chosen source generator, its dual sends a source functional with value $a$ on the source generator to the target functional with value $c a$ on the target generator. This is precisely the same formula as step 1.1 and introduces no exceptional one-dimensional case. The construction is defined for every local map, including zero, noninjective, or nonsurjective cotangent maps. If $X$ is empty there is no source rational point and the pointwise assertions are vacuous. The zero tangent vector is the based map factoring through $Spec k$ and is preserved by post-composition. All maps used are canonical, so no choice of bases or other choices, and no Axiom of Choice, is used. The statement contains no iff claim.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Tangent spaces of products over a field

Statement

Let $k$ be a field, let $X, Y$ be $k$ -schemes, and let $x \in X$ , $y \in Y$ be $k$ -rational points. Write $p_{X} : X \times_{k} Y \to X$ and $p_{Y} : X \times_{k} Y \to Y$ for the projections. The canonical map $T_{(x, y)} (X \times_{k} Y) ⟶ T_{x} X \oplus T_{y} Y$ that sends a tangent vector, represented by a based map $γ : Spec (k [ϵ] / (ϵ^{2})) \to X \times_{k} Y$ , to $(p_{X} \circ γ, p_{Y} \circ γ)$ is a $k$ -linear isomorphism. No finite-type, reducedness, or smoothness hypothesis is needed.

Facts & Assumptions

Given: A field $k$ , $k$ -schemes $X, Y$ , and points $x, y$ whose residue fields are identified with $k$ by their structure maps.

[F1]

Schemes: each point of a scheme has an open neighbourhood that is an affine scheme with the restricted structure sheaf.

[F2]

Schemes and morphisms over a base: a $k$ -scheme and its morphisms to other $k$ -schemes have structure maps to $Spec k$ and commute with those maps.

[F3]

The affine scheme of dual numbers: the dual-numbers scheme is $Spec (k [ϵ] / (ϵ^{2}))$ .

[F4]

Affine schemes are contravariantly equivalent to commutative rings: a map between affine schemes corresponds contravariantly to a ring map; in particular, based maps from the dual-numbers scheme into an affine chart correspond to $k$ -algebra maps from its coordinate ring to $k [ϵ] / (ϵ^{2})$ .

[F5]

Existence of all scheme fibre products: for affine covers of $k$ -schemes $X, Y$ , the product $X \times_{k} Y$ has an open affine cover with charts $Spec (A \otimes_{k} B)$ for charts $Spec A \subseteq X$ and $Spec B \subseteq Y$ .

[F6]

Universal mapping property of the tensor product of commutative algebras: given $k$ -algebra maps $A \to C$ and $B \to C$ , there is a unique $k$ -algebra map $A \otimes_{k} B \to C$ whose restrictions to $A$ and $B$ are the given maps; it sends $a \otimes b$ to the product of their images.

[F7]

Tangent vectors at rational points are dual-number points: for any $k$ -scheme at a $k$ -rational point, its tangent vectors are naturally the based dual-number maps, as a $k$ -vector space.

[F8]

Tangent vectors at rational points are dual-number points: under the same identification, a based local map $a \mapsto a (x) + ϵ D (a)$ is the $k$ -derivation $D$ representing the tangent vector.

Proof

technique · direct

1.1F1F2F5givenalgebra

By [F1], choose affine open neighbourhoods $U = Spec A$ of $x$ and $V = Spec B$ of $y$ . Their structure maps make $A$ and $B$ $k$ -algebras by [F2]. The fibre-product theorem [F5] gives an open affine neighbourhood of $(x, y)$ in $X \times_{k} Y$ with coordinate ring $R = A \otimes_{k} B$ ; the two projections correspond to its canonical $k$ -algebra maps from $A$ and $B$ .

2.1F3F4F5F6F7step 1.1givenalgebra

Let $e_{x} : A \to k$ and $e_{y} : B \to k$ be the maps of the rational points, and put $D = k [ϵ] / (ϵ^{2})$ . By [F7] and [F4], a tangent vector at $x$ or $y$ is represented in these charts by a $k$ -algebra map $α : A \to D$ or $β : B \to D$ , with reductions $e_{x}$ and $e_{y}$ . Conversely, any such pair determines by [F6] a unique $k$ -algebra map $φ : R \to D$ satisfying $φ (a \otimes b) = α (a) β (b)$ . Its reduction is $a \otimes b \mapsto e_{x} (a) e_{y} (b)$ , so it is based at $(x, y)$ . Restriction along the two projection maps recovers $α$ and $β$ ; therefore post-composition by the projections gives a bijection between the based dual-number maps of the product and pairs of based dual-number maps of the factors.

3.1F6F7F8step 1.1step 2.1givenalgebra

Write $α (a) = e_{x} (a) + ϵ D_{x} (a)$ and $β (b) = e_{y} (b) + ϵ D_{y} (b)$ ; [F8] says $D_{x}, D_{y}$ are the derivations representing the two tangent vectors. Since $ϵ^{2} = 0$ , the map of step 2.1 satisfies $φ (a \otimes b) = e_{x} (a) e_{y} (b) + ϵ (e_{y} (b) D_{x} (a) + e_{x} (a) D_{y} (b)) .$ Thus its coefficient derivation is linear in $(D_{x}, D_{y})$ . Conversely, restriction of the coefficient derivation of $φ$ along the two projection maps returns $D_{x}, D_{y}$ . The bijection in step 2.1 and its inverse are therefore $k$ -linear, proving the asserted natural vector-space isomorphism. Its construction uses only the projections, so it is independent of the chosen affine neighbourhoods.

4.1F1F3F7F8step 1.1step 2.1step 3.1givenalgebra∎

If either factor has zero tangent space, its based maps consist only of the constant map at that point, and step 2.1 pairs it with the based maps of the other factor; if both tangent spaces are zero, the product tangent space is zero as well. For a one-dimensional tangent factor with generator derivation $D_{x}$ , step 3.1 sends $(D_{x}, 0)$ to the coefficient derivation $a \otimes b \mapsto e_{y} (b) D_{x} (a)$ ; a generator $D_{y}$ in the other factor is sent to $a \otimes b \mapsto e_{x} (a) D_{y} (b)$ . Each scalar multiple is sent to the same scalar multiple; no one-dimensional exception occurs. The formula also covers nonsmooth and nonreduced schemes because it uses only their based dual-number maps. The zero vector is the constant based map, and the zero pair corresponds to the constant map at $(x, y)$ . If either scheme is empty, there is no point pair and the assertion has no instance. Only one affine neighbourhood for each of the two fixed points is used, no bases are chosen, and no Axiom of Choice is needed. The statement contains no iff claim.

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Regular points of locally Noetherian schemes

Statement

Let $X$ be a locally Noetherian scheme and $x \in X$ . Write $R = O_{X, x}$ , let $m_{x}$ be its maximal ideal, and set $κ (x) = R / m_{x}$ . The point $x$ is regular when $R$ is a regular local ring, with regularity defined by $edim R = \dim R$ . Then the intrinsic tangent space $T_{x} X$ is finite-dimensional over $κ (x)$ , and $x is regular ⟺ \dim_{κ (x)} T_{x} X = \dim O_{X, x} .$ This is absolute regularity of the local ring; it asserts no smoothness over a base field.

Facts & Assumptions

Given: A locally Noetherian scheme $X$ and a point $x \in X$ .

[F1]

Locally Noetherian and Noetherian schemes: a locally Noetherian scheme has an affine open cover by spectra of Noetherian rings.

[F2]

Affine open subschemes: an open subscheme carries the restricted structure sheaf, and it is affine when that restricted locally ringed space is affine.

[F3]

The stalk of a presheaf at a point: the stalk $F_{x}$ is the filtered colimit of $F (U)$ over open neighborhoods $U$ of $x$ .

[F4]

The stalk of the affine structure sheaf at a prime is A_p: for a point $p \in Spec B$ , the affine structure-sheaf stalk is canonically $O_{Spec B, p} ≅ B_{p}$ .

[F5]

Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ : $B_{p}$ consists of fractions $b / s$ with $s \notin p$ .

[F6]

$R_{p}$ is local with unique maximal ideal $p R_{p}$ : $B_{p}$ is a nonzero local ring with maximal ideal $p B_{p}$ .

[F7]

Left and right Noetherian rings: a ring is left Noetherian when its left regular module is Noetherian; here $B$ is commutative, so its ideals are submodules of that regular module.

[F8]

Noetherian modules: every submodule is finitely generated: every submodule of a Noetherian module is finitely generated.

[F9]

The intrinsic Zariski tangent space: $T_{x} X = {Hom}_{κ (x)} (m_{x} / m_{x}^{2}, κ (x))$ .

[F10]

embedding dimension and regular local ring: for a nonzero Noetherian local ring, $edim R = \dim_{κ (x)} (m_{x} / m_{x}^{2})$ , and $R$ is regular local exactly when $edim R = \dim R$ .

Proof

technique · direct

1.1F1F2F3F4F5F6F7F8givenalgebra

Noetherian local stalk. Fix $x$ . By [F1], there is an affine open neighborhood $U = Spec B$ of $x$ with $B$ Noetherian. Because the sheaf on $U$ is the restriction from $X$ [F2], neighborhoods of $x$ contained in $U$ are cofinal among its neighborhoods in $X$ ; the stalk-colimit description [F3] therefore identifies $O_{X, x}$ with $O_{U, x}$ . Let $p \subset B$ be the prime corresponding to $x$ . By [F4], $R ≅ B_{p}$ , and [F6] makes this a nonzero local ring with maximal ideal $p B_{p}$ . We verify Noetherianity directly. Let $J$ be any ideal of $B_{p}$ and contract it to $I = {b \in B : b / 1 \in J}$ . This is an ideal of $B$ , hence a submodule of its regular module [F7]; by [F8], take a finite generating list $b_{1}, \dots, b_{q}$ of $I$ . If $a / s \in J$ , then [F5] and the ideal property give $a / 1 = (s / 1) (a / s) \in J$ , so $a \in I$ and $a = \sum_{i} c_{i} b_{i}$ . It follows that $a / s = \sum_{i} (c_{i} / s) (b_{i} / 1)$ . Conversely every $b_{i} / 1$ lies in $J$ , so these images generate $J$ . Thus every ideal of $B_{p}$ is finitely generated and $R$ is Noetherian. This uses a chart for the fixed point and a finite list for the fixed ideal, not a simultaneous choice over all points or ideals.

2.1step 1.1F7F8F9F10givenalgebra∎

Intrinsic tangent dimension and regularity. By step 1.1, $R$ is Noetherian local, so its maximal ideal is finitely generated by [F7, F8]. The images of a finite generating list span $V = m_{x} / m_{x}^{2}$ over $κ (x)$ ; hence $V$ is finite-dimensional. A finite basis of $V$ gives the same number of dual basis vectors, so [F9] yields $\dim_{κ (x)} T_{x} X = \dim_{κ (x)} V = edim R$ by [F10]. Therefore $R$ is regular local if and only if $\dim_{κ (x)} T_{x} X = \dim R$ , proving both directions. If $\dim R = 0$ or $1$ , this is respectively the equality $edim R = 0$ or $1$ ; if $m_{x} = 0$ , both criteria reduce to $0 = \dim R$ . Since $T_{x} X$ is finite-dimensional, an infinite value of $\dim R$ cannot satisfy either criterion. If $X$ is empty, there is no point to test. The argument uses only finite generation and finite-dimensional linear algebra, so neither AC nor DC is used.

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Local dimension for a reducible classical algebraic set

Statement

Assume the Axiom of Choice. Let $X$ be a reduced classical finite-type space over an algebraically closed field $k$ , and let $x \in X$ be a closed point. If $X_{i}$ are the irreducible components of $X$ , then $\dim O_{X, x} = \max_{x \in X_{i}} \dim X_{i} .$

Facts & Assumptions

Given: AC, an algebraically closed field $k$ , a reduced classical finite-type space $X$ over $k$ , and a closed point $x \in X$ .

[F1]

A classical variety is Noetherian with finitely many irreducible components; every open or closed subvariety has a finite affine cover (Classical varieties have finite irreducible decompositions).

[F2]

For an affine algebraic set $U \subseteq A_{k}^{n}$ , its coordinate ring is $k [U] = k [t_{1}, \dots, t_{n}] / I (U)$ (The coordinate ring of an affine algebraic set).

[F3]

Over algebraically closed $k$ and under AC, the Nullstellensatz correspondence identifies radical ideals of $k [U]$ with closed subsets of $U$ ; nonempty irreducible closed subsets correspond to proper prime ideals (Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals).

[F4]

For a classical affine variety $U$ and $x \in U$ , the local ring is canonically $O_{U, x} ≅ k [U]_{m_{x}}$ , where $m_{x}$ is the ideal of functions vanishing at $x$ (The local ring at a point of an affine variety is the localization at its maximal ideal).

[F5]

For a ring $A$ and multiplicative set $S$ , prime ideals of $S^{- 1} A$ correspond by an inclusion-preserving bijection to the prime ideals of $A$ disjoint from $S$ (Prime ideals of a localization are exactly the primes disjoint from the denominator set).

[F6]

If $Y$ is an irreducible classical variety and $x$ is a closed point of $Y$ , then $\dim O_{Y, x} = \dim Y$ (Closed-point local dimension equals ambient irreducible dimension).

[F7]

AC says that every family of nonempty sets has a choice function (The Axiom of Choice).

Proof

1.1F1F2F4F7givenchoose

Choose an affine open neighborhood $U \subseteq X$ of $x$ , put $A = k [U]$ , let $m \subset A$ be the maximal ideal of functions vanishing at $x$ , and write $R = O_{X, x}$ . By [F4], $R ≅ A_{m}$ . The finite component decomposition of $X$ restricts to a finite decomposition of $U$ by its irreducible components $U_{j} = X_{j} \cap U$ ; precisely those $U_{j}$ containing $x$ come from the global components $X_{j}$ containing $x$ .

2.1F3F5step 1.1

For each component $U_{j}$ , let $p_{j} = I_{U} (U_{j}) \subseteq A$ . The Nullstellensatz correspondence makes $p_{j}$ prime and reverses inclusions of closed subsets. The localization correspondence identifies the primes of $R = A_{m}$ with the primes $p \subseteq m$ of $A$ , preserving strict chains. In particular, if $x \in U_{j}$ , then $q_{j} : = p_{j} A_{m}$ is a prime of $R$ .

3.1F1F3F5step 2.1algebra

Consider any strict prime chain $q_{0} ⊊ q_{1} ⊊ \dots ⊊ q_{r}$ in $R$ , and contract it to $p_{0} ⊊ p_{1} ⊊ \dots ⊊ p_{r}$ in $A$ using [F5]. The irreducible closed subset $V_{U} (p_{0})$ contains $x$ , since $p_{0} \subseteq m$ . Because $U$ is a finite union of its irreducible components, irreducibility forces $V_{U} (p_{0}) \subseteq U_{j}$ for some $j$ . Thus $x \in U_{j}$ and $p_{j} \subseteq p_{0}$ , so $q_{j} \subseteq q_{0}$ . The chain therefore gives a chain of length $r$ in $R / q_{j}$ .

4.1F2F4F6step 3.1algebra

The quotient-localization isomorphism gives $R / q_{j} ≅ (A / p_{j})_{m / p_{j}}$ , which is the local ring $O_{U_{j}, x} = O_{X_{j}, x}$ because $U_{j} = X_{j} \cap U$ is an open neighborhood of $x$ in $X_{j}$ . Hence [F6] gives $\dim (R / q_{j}) = \dim X_{j}$ . Step 3.1 now bounds every chain length in $R$ by $\max_{x \in X_{i}} \dim X_{i}$ .

5.1F6step 2.1step 4.1algebra

Conversely, for every global component $X_{i}$ containing $x$ , its $q_{i}$ is prime in $R$ by step 2.1 and $\dim (R / q_{i}) = \dim X_{i}$ by step 4.1. Every prime chain in $R / q_{i}$ lifts to a prime chain in $R$ , so $\dim R \geq \dim X_{i}$ . Taking the maximum gives the reverse inequality.

6.1F7step 3.1step 4.1step 5.1∎

Steps 3.1–5.1 prove the asserted equality. The argument uses AC only through the explicitly AC-dependent component, affine-correspondence, local-ring, and irreducible local-dimension suppliers; after their finite component and prime correspondences are in hand, the chain comparison makes no further choice.

Source note

Milne’s §3c notes 3.13–3.14 identify local primes with irreducible closed subsets through a point and identify the components through that point with minimal local primes. The proof of Corollary 4.45 in §4i uses this local component description. The dimension of each irreducible component at a closed point is supplied here by Closed-point local dimension equals ambient irreducible dimension; Milne’s Chapter 10 supplement, 10.54–10.56, gives the corresponding irreducible-scheme dimension conventions. The finite reducible case above is proved by the displayed prime-chain comparison.

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Tangent dimension bounds local dimension

Statement

Assume the Axiom of Choice. For every point $x$ of a locally Noetherian scheme $X$ , $\dim_{κ (x)} T_{x} X \geq \dim O_{X, x} .$ For a reduced classical finite-type variety $X$ over an algebraically closed field and a closed point $x$ , this gives $\dim T_{x} X \geq \dim_{x} X,$ where $\dim_{x} X : = \max_{x \in X_{i}} \dim X_{i}$ over the irreducible components $X_{i}$ containing $x$ .

Facts & Assumptions

Given: AC, a locally Noetherian scheme $X$ , and a point $x \in X$ . The classical specialization additionally assumes that $X$ is a reduced finite-type variety over an algebraically closed field and that $x$ is closed.

[F1]

The Axiom of Choice: Every family of nonempty sets has a choice function.

[F2]

Schemes: A scheme is a locally ringed space $(X, O_{X})$ such that every point has an open neighbourhood which, with the restricted structure sheaf, is an affine scheme.

[F3]

Locally Noetherian and Noetherian schemes: if it has an affine open cover by spectra of Noetherian rings.

[F4]

Affine open subschemes: For a scheme $X$ and an open set $U \subseteq X$ , the open subscheme $U$ means $(U, O_{X} ∣_{U})$ .

[F5]

The underlying space of an affine spectrum: whose points are the prime ideals of $A$ .

[F6]

The stalk of the affine structure sheaf at a prime is A_p: there is a canonical isomorphism $O_{Spec A, p} ≅ A_{p}$ .

[F7]

Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ : $R_{p} = (R ∖ p)^{- 1} R$ .

[F8]

$R_{p}$ is local with unique maximal ideal $p R_{p}$ : $R_{p}$ is a nonzero local ring. Its unique maximal ideal is

[F9]

The residue field at a point of an affine scheme: $κ (x) = O_{X, x} / m_{x}$ .

[F10]

Noetherian commutative rings and modules: Equivalently, every ideal of $R$ is finitely generated.

[F11]

The intrinsic cotangent space: $C_{x} X : = m_{x} / m_{x}^{2}$ .

[F12]

The intrinsic Zariski tangent space: $T_{x} X : = {Hom}_{κ (x)} (C_{x} X, κ (x))$ .

[F13]

embedding dimension and regular local ring: define $edim R = \dim_{k} (m / m^{2})$ .

[F14]

dimension at most embedding dimension: every nonzero commutative Noetherian local ring $R$ satisfies $\dim R \leq edim R < \infty$ .

[F15]

Local dimension for a reducible classical algebraic set: $\dim O_{X, x} = \max_{x \in X_{i}} \dim X_{i}$ .

[F16]

The affine scheme of dual numbers: $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ .

[F17]

Prime ideals and maximal ideals in a commutative ring: A proper ideal $P ⊊ R$ is prime when $a b \in P$ implies $a \in P$ or $b \in P$ .

[F18]

Prime ideals and maximal ideals in a commutative ring: there is no proper ideal strictly between $M$ and $R$ .

[F19]

Krull dimension of a nonzero ring: the Krull dimension of $R$ is the supremum of all integers $n \geq 0$ for which such a chain exists.

Proof

technique · direct

1.1F2F3F4F5F6F7F8F9givenchoose

Choose an affine open neighborhood $U = Spec A$ of $x$ with $A$ Noetherian by [F2, F3]. Since $U$ is an open subscheme with the restricted structure sheaf [F4], neighborhoods contained in $U$ are cofinal among neighborhoods of $x$ , so $O_{X, x} = O_{U, x}$ . The point $x$ corresponds to a prime $p \subset A$ by [F5], and [F6, F7] identify $R : = O_{X, x}$ with $A_{p}$ . By [F8], $R$ is a nonzero local ring with maximal ideal $m = p A_{p}$ ; [F9] identifies its residue field with $κ (x)$ .

2.1F7F10step 1.1algebra

The local ring $R = A_{p}$ is Noetherian. Let $J$ be any ideal of $A_{p}$ and contract it to $I : = {a \in A : a / 1 \in J}$ . Since $A$ is Noetherian, [F10] gives generators $a_{1}, \dots, a_{n}$ of $I$ . If $a / s \in J$ with $s \notin p$ , then $a / 1 = (s / 1) (a / s) \in J$ , so $a \in I$ and $a = \sum_{i} c_{i} a_{i}$ . Therefore $a / s = \sum_{i} (c_{i} / s) (a_{i} / 1)$ , while each $a_{i} / 1$ lies in $J$ . Thus the images $a_{i} / 1$ generate $J$ . As this holds for every $J$ , [F10] implies that $R$ is Noetherian.

3.1F11F12F13step 2.1algebra

The tangent dimension equals the embedding dimension of $R$ . The maximal ideal $m$ is finitely generated because $R$ is Noetherian, so $m / m^{2}$ is a finite-dimensional vector space over $κ (x)$ . By [F11] this quotient is $C_{x} X$ , and by [F12] $T_{x} X$ is its $κ (x)$ -linear dual; a finite-dimensional vector space and its dual have equal dimension. By [F13], this common dimension is $edim R$ .

4.1F1F8F14step 2.1step 3.1

Now [F8] and step 2.1 make $R = O_{X, x}$ a nonzero Noetherian local ring, so the AC-dependent bound [F14] applies. Together with step 3.1 it gives $\dim O_{X, x} \leq edim R = \dim_{κ (x)} T_{x} X$ . AC is used here through [F14], whose height-theorem input requires it; it is an explicit assumption, not a consequence of finite choice.

5.1F1F15step 4.1algebra

In the stated classical closed-point specialization, [F15] gives $\dim O_{X, x} = \max_{x \in X_{i}} \dim X_{i} = \dim_{x} X$ . Substituting this equality into step 4.1 proves $\dim T_{x} X \geq \dim_{x} X$ . This local-dimension supplier also assumes AC, already declared in the statement.

6.1F5F6F7F8F9F11F12F16F17F18F19step 4.1algebra∎

At $X = Spec k$ , the only prime is $(0)$ , so the unique local ring is $k$ , its maximal ideal is zero, and [F19] gives local dimension zero; [F11, F12] give tangent dimension zero. The nonreduced dual-numbers scheme $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ shows that the inequality may be strict. Its ring is a two-dimensional $k$ -vector space, so every ideal, as a subspace, has a finite basis that generates it as an ideal; hence $D_{k}$ is Noetherian. Every prime contains $ϵ$ because $ϵ^{2} = 0$ by [F17]. The ideal $(ϵ)$ is proper because its elements are multiples of $ϵ$ and cannot equal $1$ . Any proper ideal strictly containing $(ϵ)$ would contain $a + b ϵ$ with $a \neq 0$ , a unit with inverse $a^{- 1} - a^{- 2} b ϵ$ . Thus $(ϵ)$ is maximal by [F18] and, since every prime contains it, it is the unique prime. By [F5], $D_{k}$ has one point. Its local ring $D_{k, (ϵ)}$ is $D_{k}$ since every denominator outside $(ϵ)$ is a unit; by [F6, F7, F8, F19] its local dimension is zero. The residue field is $D_{k} / (ϵ) ≅ k$ by [F9], while the maximal ideal squares to zero, so $(ϵ) / (ϵ^{2})$ is one-dimensional over the residue field. By [F11, F12], the tangent dimension is one.

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Regular and singular loci

Definition

Let $X$ be a locally Noetherian scheme. Define subsets of its underlying point set by

X_{r e g} = {x \in ∣ X ∣ : O_{X, x} is a regular local ring}, X_{s i n g} = ∣ X ∣ ∖ X_{r e g} .

These are the regular locus and singular locus of $X$ . This definition alone asserts no openness or closedness property and no smoothness over a chosen base.

For the classical dimension test, assume the Axiom of Choice and suppose that $X$ is a reduced classical finite-type space over an algebraically closed field $k$ . If $x \in X$ is closed, define

\dim_{x} X : = \max_{x \in X_{i}} \dim X_{i},

where $X_{i}$ range over the irreducible components containing $x$ . Then

x \in X_{r e g} ⟺ \dim_{κ (x)} T_{x} X = \dim_{x} X .

The Axiom of Choice is used for this classical component-dimension identification through Local dimension for a reducible classical algebraic set; it is not needed to define either locus. At reducible points, $\dim_{x} X$ uses only components through $x$ , not a single global dimension for all of $X$ .

Facts & Assumptions

Given: A locally Noetherian scheme $X$ ; for the numerical specialization, also AC and a reduced classical finite-type $X$ over an algebraically closed field with a closed point $x$ .

[F1]

Regular points of locally Noetherian schemes: for any point of a locally Noetherian scheme, regularity is equivalent to $\dim_{κ (x)} T_{x} X = \dim O_{X, x}$ .

[F2]

Local dimension for a reducible classical algebraic set: under AC, for a reduced classical finite-type space over an algebraically closed field and a closed point $x$ , $\dim O_{X, x}$ is the maximum of $\dim X_{i}$ over components containing $x$ .

[F3]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function; its use here is inherited only through [F2].

Proof

technique · direct

1.1F1givenalgebra

Use the regular-point predicate of [F1] to define $X_{r e g}$ as the points whose local rings are regular local, and take its set-theoretic complement in $∣ X ∣$ for $X_{s i n g}$ . These definitions apply to every locally Noetherian scheme, including nonreduced schemes; they do not assert that either set is open or closed.

2.1F1F2F3givenalgebra

Under the classical hypotheses, [F2] gives $\dim O_{X, x} = \dim_{x} X$ . By [F1], $x \in X_{r e g}$ exactly when $\dim_{κ (x)} T_{x} X = \dim O_{X, x}$ . Substituting the equality from [F2] proves $x \in X_{r e g}$ if and only if $\dim_{κ (x)} T_{x} X = \dim_{x} X$ . This argument uses AC only through [F2], not for the locus definitions in step 1.1.

3.1F2step 2.1givenalgebra

When $X$ is reducible, the right side uses the maximum dimension of components containing this particular $x$ by the definition of $\dim_{x} X$ and [F2]. Components not containing $x$ do not enter the local dimension, so replacing $\dim_{x} X$ by the global $\dim X$ is not justified in general. If $\dim_{x} X = 0$ or $1$ , the same equivalence specializes respectively to equality of tangent and local dimension zero or one; it does not require all components of $X$ to have the same dimension.

4.1F1F2step 1.1step 2.1step 3.1givenalgebra∎

If $X = Spec k$ for a field $k$ , its only local ring is the field $k$ , its maximal ideal is zero, and its tangent and local dimensions are both zero, so its point belongs to $X_{r e g}$ . For an empty scheme, both loci are empty by step 1.1. Nilpotents do not affect the definition in step 1.1, but the numerical component formula is stated only for reduced classical spaces, exactly as required by [F2]. The proof makes no choices beyond AC's stated use through [F2], and the displayed criterion has both directions by step 2.1.

Source note

Milne's book-wide field convention is algebraically closed. In §4h, Definition 4.35, printed pp. 93–94, a point on an affine algebraic variety is called nonsingular when it lies on a single irreducible component $W$ and $\dim T_{x} X = \dim W$ ; otherwise it is singular. In §4i, Theorem 4.44 and Corollary 4.45, printed pp. 96–97, Milne identifies that classical notion with regularity of the local ring; the corollary's proof uses that a regular local ring is a domain to exclude points on multiple components. Those passages support the classical terminology, not a general scheme definition or any openness assertion here. The scheme-theoretic locus definition and the reducible local-dimension test are supplied and proved through [F1] and [F2].

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Jacobian rank detects regularity at closed points

Statement

Let $k$ be a field, let $n \geq 0$ be finite, put $P = k [t_{1}, \dots, t_{n}]$ , and let $A = P / I$ with a specified finite generating list $I = (f_{1}, \dots, f_{r})$ for the actual ideal defining the affine scheme. For a maximal ideal $m \subset A$ , write $L = A / m$ . Let $J (m)$ be the $r \times n$ matrix over $L$ obtained by mapping the formal partial derivatives $\partial f_{i} / \partial t_{j}$ through $P \to A \to L$ .

If $k$ is perfect, then $L / k$ is finite separable and

{rank}_{L} J (m) = n - \dim A_{m}

if and only if $A_{m}$ is a regular local ring. For any field $k$ , the same equivalence holds at a $k$ -rational point, where $L = k$ , without a perfectness assumption. If $I = I (X)$ for a reduced classical affine algebraic set $X$ over an algebraically closed field and $m$ corresponds to a closed point $x$ , then, assuming AC,

\dim A_{m} = \dim_{x} X = \max_{x \in X_{i}} \dim X_{i},

where $X_{i}$ ranges over the irreducible components through $x$ . The finite generating list need not be minimal, and $I$ need not be radical in the first two assertions.

Facts & Assumptions

Given: A field $k$ , a finite $n$ , the polynomial ring $P = k [t_{1}, \dots, t_{n}]$ , an ideal $I \subseteq P$ with a specified finite generating list $f_{1}, \dots, f_{r}$ , the quotient $A = P / I$ , and a maximal ideal $m \subset A$ with residue field $L = A / m$ . For the rational case, $L = k$ . For the classical dimension clause, $k$ is algebraically closed, $I = I (X)$ for a reduced classical affine algebraic set $X$ , and AC is assumed.

[F1]

Finite-variable polynomial algebras over fields are Noetherian by finite generators: every finite-variable polynomial ring over a field is Noetherian, so its quotients and localizations are Noetherian.

[F2]

A maximal ideal of an affine algebra has finite residue field over the base field: if $A$ is a finite-type $k$ -algebra and $m$ is maximal, then $A / m$ is finite over $k$ .

[F3]

Every algebraic extension of a perfect field is separable: every algebraic extension of a perfect field is separable.

[F4]

Separable residue and the cotangent sequence of a local algebra: for a Noetherian local $k$ -algebra with finite separable residue field $L$ , the map $n / n^{2} \to Ω_{R / k} \otimes_{R} L$ is an isomorphism, where $n$ is the maximal ideal of $R$ .

[F5]

Localization, base change and functoriality of differentials: localization of the source algebra localizes its module of Kähler differentials, so $Ω_{A_{m} / k} ≅ (Ω_{A / k})_{m}$ .

[F6]

Localisation of modules is extension of scalars: for a multiplicative set $S$ , $S^{- 1} M ≅ S^{- 1} A \otimes_{A} M$ ; after tensoring with the residue field of $A_{m}$ this identifies $(Ω_{A / k})_{m} \otimes_{A_{m}} L$ with $Ω_{A / k} \otimes_{A} L$ .

[F7]

Differentials of a polynomial quotient and the Jacobian cokernel: if $A = P / I$ and $I = (f_{1}, \dots, f_{r})$ , then $Ω_{A / k}$ is the cokernel of the map $A^{r} \to A^{n}$ whose columns are the formal derivative vectors of the $f_{i}$ .

[F8]

Tensoring is right exact: tensoring a cokernel presentation with $L$ gives the cokernel of the base-changed map.

[F9]

The intrinsic Zariski tangent space: at a point with residue field $L$ , $T_{x} X = {Hom}_{L} (m_{x} / m_{x}^{2}, L)$ .

[F10]

Regular points of locally Noetherian schemes: for a locally Noetherian scheme, $x$ is regular exactly when $\dim_{κ (x)} T_{x} X = \dim O_{X, x}$ .

[F11]

The Jacobian kernel computes the tangent space: at a rational point of the affine scheme defined by the actual ideal $I$ , the coordinate-velocity tangent space is canonically $\ker J (a)$ .

[F12]

The coordinate ring of a classical affine algebraic set: the coordinate ring of an affine algebraic set $X \subseteq k^{n}$ is $k [X] = k [t_{1}, \dots, t_{n}] / I (X)$ .

[F13]

Local dimension for a reducible classical algebraic set: for a reduced classical finite-type variety and closed point $x$ , $\dim O_{X, x} = \max_{x \in X_{i}} \dim X_{i}$ .

[F14]

Global and local dimension of classical varieties: at a closed point, $\dim_{x} X = \max_{x \in X_{i}} \dim X_{i}$ over the components containing $x$ .

[F15]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function; only the classical component-dimension clause below uses it, through [F13] and the convention in [F14].

Proof

technique · direct

1.1F1F2F3F4given

Since $A$ is a quotient of the finite-variable polynomial ring $P$ , [F1] makes $A_{m}$ a Noetherian local ring. The algebra $A$ is finite type over $k$ , so [F2] makes $L / k$ finite. If $k$ is perfect, [F3] then makes $L / k$ separable; this verifies the residue-field hypothesis in [F4] without assuming that $m$ is rational.

1.2F9F10F11algebra

Now let $k$ be any field and let $m$ be $k$ -rational. By [F11], $T_{x} (Spec A) ≅ \ker J (m)$ , so rank-nullity gives $\dim_{k} T_{x} (Spec A) = n - {rank}_{k} J (m)$ . Applying [F10] proves the same equivalence without a perfectness assumption. This argument uses the rational-point theorem only in the case $L = k$ .

1.3F12F13F14F15given

In the reduced classical case, [F12] identifies $A$ with the coordinate ring $k [X]$ . Under the stated AC assumption, [F13] gives $\dim A_{m} = \max_{x \in X_{i}} \dim X_{i}$ , and [F14] identifies this maximum with $\dim_{x} X$ . This is the claimed classical dimension formula; AC enters this clause through the local-dimension lemma [F13] and the fixed-field convention in [F14].

2.1F4F5F6step 1.1

In the perfect-field case put $R = A_{m}$ and $n = m R$ . By [F4], $n / n^{2} ≅ Ω_{R / k} \otimes_{R} L$ . Applying [F5] and then [F6] identifies this with $Ω_{A / k} \otimes_{A} L$ .

3.1F7F8F9step 2.1algebra

By [F7], $Ω_{A / k}$ is the cokernel of $A^{r} \to A^{n}$ represented by the derivative vectors of $f_{1}, \dots, f_{r}$ . Right exactness [F8] identifies $Ω_{A / k} \otimes_{A} L$ with the cokernel of $L^{r} \to L^{n}$ represented by those same vectors after mapping their entries to $L$ . This is the transpose presentation of the equation-row matrix $J (m)$ , so the map has rank ${rank}_{L} J (m)$ . Hence $\dim_{L} (n / n^{2}) = n - {rank}_{L} J (m)$ . Since this cokernel is finite-dimensional, [F9] gives $\dim_{L} T_{x} (Spec A) = n - {rank}_{L} J (m)$ .

4.1F10step 3.1algebra

The local-ring definition [F10] says $A_{m}$ is regular exactly when its tangent dimension equals $\dim A_{m}$ . Substituting the dimension computed in step 3.1 gives $A_{m}$ regular iff $n - {rank}_{L} J (m) = \dim A_{m}$ , equivalently iff ${rank}_{L} J (m) = n - \dim A_{m}$ . This proves both directions for every closed point over a perfect field.

5.1F7F9F10F13F14F15step 1.2step 1.3step 3.1step 4.1algebra∎

The degenerate cases fit the same calculations. If $I = P$ , then $A = 0$ has no maximal ideal and the pointwise assertions are vacuous. If $n = 0$ and a maximal ideal exists, then $A = k$ , $m = 0$ , the local ring is a field of dimension zero, and the empty-column Jacobian has rank zero; if $r = 0$ , the map $L^{0} \to L^{n}$ has rank zero and the cokernel calculation in step 3.1 still applies. For one equation in one variable, $A = k [t] / (t)$ at $(t)$ has local ring $k$ , Jacobian $[1]$ , and rank $1 = n - 0$ , so it is regular. In contrast, $A = k [t] / (t^{2})$ at $(t)$ has a unique prime $(t)$ , local dimension zero, one-dimensional cotangent space $(t) / (t^{2})$ , and Jacobian entry $2 t = 0$ in the residue field (including characteristic two); it is not regular and its rank $0$ does not equal $n - \dim A_{(t)} = 1$ . The equivalence in steps 1.2 and 4.1 handles both iff directions. At local dimension zero the regularity equality requires full Jacobian rank; when tangent dimension is the ambient dimension $n$ , it requires rank zero. No separate dimension-range assertion is used. No minimality of the generator list or reducedness of $I$ entered [F7], and the cokernel's dimension is independent of the chosen list. The general-field rational proof and the perfect-field proof use no choice or DC; AC enters only the classical clause through [F13] and [F14].

Source qualification

Milne, Algebraic Geometry v6.10, §4d, Definition 4.23 and the Jacobian tangent-rank discussion (printed pp. 87–88 / PDF pp. 86–87; web lines 4636–4672), computes $\dim T_{a} X = n - rank J (a)$ and gives the classical nonsingularity criterion for algebraic sets over an algebraically closed field. §4i, Corollary 4.45 (printed p. 97 / PDF p. 96; web lines 5224–5229), identifies nonsingularity with regularity under its classical variety conventions. Those passages do not establish the arbitrary scheme-ideal or nonrational perfect-field clauses here. Milne, Algebraic Geometry, Chapter 10 supplement, §f, 10.58 and 10.60–10.64 (web lines 892–985), gives the cotangent-dimension/regularity comparison, rational-point tangent description, Jacobian-minor construction, and regularity/smoothness comparison in the stated classical settings; in particular 10.62 is for an irreducible closed subscheme and does not prove the arbitrary quotient statement here. The proof above instead uses the complete separable-residue cotangent sequence, differential localization, polynomial-quotient differential presentation, and tensor right exactness recorded in [F4]–[F8]. Stacks Lemma 10.140.4 (tag 00TU), full statement and proof, proves the separable-residue injection by constructing a section modulo $m^{2}$ after lifting a separating transcendence basis and correcting a lift using the derivative of its separable minimal polynomial. Stacks Lemma 10.140.5 (tag 00TV), full statement and proof, corroborates the regularity comparison for finite-type algebras with separable residue field but is not used as a logical input here.

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The gradient test for a reduced hypersurface

Statement

Assume the Axiom of Choice. Let $k$ be an algebraically closed field, let $n \geq 1$ , and let $f \in k [t_{1}, \dots, t_{n}]$ be nonconstant and squarefree, meaning that no irreducible factor occurs more than once. Put $X = V (f)$ with its reduced classical variety structure. For every $a \in X (k)$ , the point $a$ is singular exactly when every formal first partial derivative of $f$ vanishes at $a$ . Equivalently, $X_{s i n g} = V (f, \partial_{1} f, \dots, \partial_{n} f) as subsets of k^{n} .$

The affine scheme $Spec (k [t_{1}, \dots, t_{n}] / (f))$ uses the actual ideal $(f)$ , which is already radical for squarefree $f$ . For a non-squarefree equation, passing from its principal ideal to its radical can change the scheme and its tangent space; for example, $t^{2}$ and its radical $t$ have different tangent spaces at $0$ .

Facts & Assumptions

Given: AC, an algebraically closed field $k$ , a finite integer $n \geq 1$ , a nonconstant squarefree polynomial $f \in R = k [t_{1}, \dots, t_{n}]$ , the classical zero set $X = V (f)$ , and a point $a \in X (k)$ . The word squarefree means that the finite factorization of $f$ in the polynomial-ring UFD has no repeated irreducible factor.

[F1]

The Jacobian kernel computes the tangent space: for an affine scheme over any field, its tangent space at a rational point is the kernel of the Jacobian matrix of any finite generating list for the actual scheme ideal.

[F2]

Regular and singular loci: for a reduced classical finite-type space over an algebraically closed field and a closed point, the singular locus is the complement of the regular locus, and regularity is characterized by tangent dimension equalling the maximum dimension of the irreducible components through the point.

[F3]

Local dimension for a reducible classical algebraic set: under AC, the local dimension at a closed point of a reduced classical finite-type space is the maximum dimension of its irreducible components through that point.

[F4]

A nontrivial principal section has pure codimension one: under AC, the zero locus of a nonzero nonunit on an irreducible affine variety is nonempty and each irreducible component has dimension one less than the ambient variety.

[F5]

Affine geometric dimension equals ring dimension: under AC, the dimension of a nonempty affine algebraic set is the Krull dimension of its coordinate ring.

[F6]

A polynomial ring in n variables over a field has dimension n: for a field $k$ and finite $n$ , $\dim k [t_{1}, \dots, t_{n}] = n$ .

[F7]

Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes: the finite variable polynomial ring over a field is a UFD, and its irreducible elements are prime.

[F8]

Strong Nullstellensatz: I(V(I)) equals the radical of I: under AC and for an algebraically closed field, $I (V (J)) = \sqrt{J}$ for every polynomial ideal $J$ .

[F9]

The Axiom of Choice: AC says every family of nonempty sets has a choice function; its uses here are inherited through [F2]–[F5] and [F8].

Proof

technique · direct

1.1F7F8F9givenalgebra

The polynomial ring $R$ is a UFD by [F7], so write $f = u q_{1} \dots q_{m}$ with $u \in k^{\times}$ and pairwise nonassociate irreducibles $q_{i}$ ; each $q_{i}$ is prime. If $g^{r} \in (f)$ for some $r \geq 1$ , every $q_{i}$ divides $g^{r}$ and hence divides $g$ . Since the $q_{i}$ are distinct prime factors, their product divides $g$ , so $g \in (f)$ and $(f)$ is radical. By [F8], $I (X) = I (V (f)) = \sqrt{(f)} = (f)$ , so $X$ is reduced and its affine scheme is $Spec (R / (f))$ with the actual equation ideal. This finite factorization argument makes no choice; AC is used here only for the Nullstellensatz identification.

1.2F3F4F5F6F7F9givenalgebra

The affine space $A_{k}^{n}$ is irreducible because $R$ is a domain by [F7], and [F5] and [F6] give $\dim A_{k}^{n} = n$ . The polynomial $f$ is a nonzero nonunit of its coordinate ring, so [F4] gives that $X$ is nonempty and each irreducible component $X_{i}$ has dimension $n - 1$ . For the fixed closed point $a$ , [F3] therefore gives $\dim O_{X, a} = \max_{a \in X_{i}} \dim X_{i} = n - 1$ . Component dimensions come from [F4], and AC identifies their maximum with local dimension through [F3].

1.3F1givenalgebra

By [F1] applied to the actual ideal $(f)$ and its one-element generating list, the intrinsic tangent space at $a$ is the kernel of the single row $d f (a) = (\partial_{1} f (a), \dots, \partial_{n} f (a)) : k^{n} ⟶ k$ . If some coefficient $c_{j} = \partial_{j} f (a)$ is nonzero, the equation $\sum_{i} \partial_{i} f (a) v_{i} = 0$ determines $v_{j} = - c_{j}^{- 1} \sum_{i \neq j} \partial_{i} f (a) v_{i}$ , so the other $n - 1$ coordinates vary freely and $\dim_{k} T_{a} X = n - 1$ . If every partial vanishes, the kernel is all of $k^{n}$ and has dimension $n$ . These alternatives include every characteristic because [F1] uses formal polynomial derivatives without a characteristic restriction.

2.1F2F3F4F9step 1.2step 1.3givenalgebra

The point $a$ is closed in the reduced classical finite-type space $X$ . By [F2], it is regular exactly when $\dim_{k} T_{a} X = \max_{a \in X_{i}} \dim X_{i}$ . Step 1.2 identifies this maximum as $n - 1$ , and step 1.3 shows that equality holds exactly when some partial derivative of $f$ is nonzero. Since the singular locus is the complement of the regular locus by [F2], $a$ is singular exactly when all partial derivatives vanish. Since $f (a) = 0$ , this proves both inclusions in the displayed equality. The choice use is inherited through [F2]–[F4], as recorded in [F9].

3.1F1F4F9step 1.1step 1.2step 1.3step 2.1givenalgebra∎

The non-squarefree distinction is visible in one variable: in $k [t] / (t^{2})$ at $0$ the actual Jacobian row is $(2 t) ∣_{0} = 0$ in every characteristic, so [F1] gives tangent space $k$ , whereas for the radical ideal $(t)$ the row is $(1)$ and the tangent space is zero. Thus radicalizing a non-squarefree equation changes its tangent computation; for the squarefree $f$ here, step 1.1 proves radicalization is redundant. For $n = 1$ and $f = t$ , the unique point has nonzero derivative, tangent dimension zero and local dimension zero, so it is regular. For the reducible squarefree example $f = x y$ in $k [x, y]$ , at the origin the gradient $(y, x)$ vanishes, the tangent dimension is two and the local dimension is one; away from the origin on either axis the gradient is nonzero and tangent dimension is one, so those points are regular. The zero tangent vector lies in every Jacobian kernel. Here $n \geq 1$ and nonconstant nonzero $f$ make the principal-subvariety theorem applicable; [F4] makes the hypersurface nonempty, so the empty case has no instance. Steps 1.3–2.1 establish both implications, and no arbitrary choice is made beyond the declared AC uses.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

A regular point lies on one irreducible component

Statement

Assume the Axiom of Choice (The Axiom of Choice). A regular point of a reduced Noetherian scheme lies on exactly one irreducible component.

Facts & Assumptions

Given: A reduced Noetherian scheme $X$ and a point $x \in X$ whose local ring is regular.

[F1]

AC says every family of nonempty sets has a choice function (The Axiom of Choice).

[F2]

A Noetherian scheme has a finite affine open cover by spectra of Noetherian rings (Locally Noetherian and Noetherian schemes).

[F3]

An open subscheme has the restricted structure sheaf, and an affine open subscheme is affine with that structure sheaf (Affine open subschemes).

[F4]

For $U = Spec A$ and $x \leftrightarrow p \in Spec A$ , the stalk is $O_{U, x} ≅ A_{p}$ (The stalk of the affine structure sheaf at a prime is A_p).

[F5]

A point is regular when its local ring is regular local (Regular points of locally Noetherian schemes).

[F6]

Under AC, every regular local ring is a domain (regular local rings are domains and cohen macaulay).

[F7]

An irreducible component of a scheme is a maximal irreducible closed subset of its underlying space (Irreducible components as schemes).

[F8]

In an irreducible space, every nonempty open subset is dense (Irreducibility via nonempty open subsets, connectedness and open subspaces).

[F9]

Under AC, the closure of an irreducible subset is irreducible (Existence and basic properties of irreducible components).

[F10]

Under AC, the irreducible components of $Spec A$ are exactly $V (q)$ for minimal prime ideals $q$ of $A$ (Irreducible components of the spectrum correspond to minimal prime ideals).

[F11]

For a multiplicative set $S \subseteq A$ , primes of $S^{- 1} A$ correspond bijectively and in an inclusion-preserving way to primes of $A$ disjoint from $S$ ; the inverse is extension (Prime ideals of a localization are exactly the primes disjoint from the denominator set).

[F12]

$V (q)$ consists of the primes containing $q$ (The prime spectrum and vanishing sets).

[F13]

A nonempty open subset of an irreducible space is irreducible (Irreducibility via nonempty open subsets, connectedness and open subspaces).

[F14]

Under AC, every point lies in an irreducible component (Existence and basic properties of irreducible components).

[F15]

$A_{p}$ means $(A ∖ p)^{- 1} A$ (Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ ).

Proof

1.1F1F2F3F4F5F6F15given

Fix $x$ . By the AC assumption [F1] and [F2], choose an affine open neighbourhood $U = Spec A$ of $x$ with $A$ Noetherian. Write $x$ as the prime $p \subset A$ . The open-scheme structure in [F3] and the affine stalk calculation [F4] identify $O_{X, x}$ with $A_{p}$ . Since $x$ is regular, this is a regular local ring by [F5], and therefore a domain by [F6]. Put $S = A ∖ p$ ; by [F15], $A_{p} = S^{- 1} A$ . These are pointwise choices of one chart and its corresponding prime; no family of charts is chosen.

1.2F7F8F9F10F12F13given

Let $C$ be any irreducible component of $X$ containing $x$ . By [F7], $C$ is closed and irreducible. The subset $C \cap U$ is a nonempty open subset of $C$ , so [F8] makes it dense in $C$ and [F13] makes it irreducible. It is closed in $U$ because $C$ is closed in $X$ . To see it is maximal irreducible in $U$ , let $Z$ be an irreducible closed subset of $U$ containing $C \cap U$ . Its closure $\overline{Z}$ in $X$ is irreducible by [F9]. Since $C \cap U \subseteq Z \subseteq \overline{Z}$ and $C \cap U$ is dense in $C$ , we have $C \subseteq \overline{Z}$ . The maximality of $C$ then gives $C = \overline{Z}$ . As $Z$ is closed in $U$ , $\overline{Z} \cap U = Z$ , so $C \cap U = Z$ . Thus $C \cap U$ is an irreducible component of $U$ . By [F10], there is a unique minimal prime $q_{C}$ of $A$ with $C \cap U = V (q_{C})$ . Since $x$ corresponds to $p$ and lies in this vanishing set, [F12] gives $q_{C} \subseteq p$ .

2.1F8F11F15step 1.1step 1.2

The prime correspondence [F11] identifies the primes of $A_{p}$ with primes of $A$ contained in $p$ . Since $q_{C}$ is minimal in $A$ , its extension $q_{C} A_{p}$ is minimal in $A_{p}$ : a prime properly below it would contract to a prime properly below $q_{C}$ . But $A_{p}$ is a domain by step 1.1, so its only minimal prime is $(0)$ . Hence $q_{C} A_{p} = (0)$ . The localization correspondence is one-to-one, so all components $C$ through $x$ have the same prime $q_{C}$ . Their intersections with $U$ are therefore the same; each such intersection is dense in its component by [F8], so taking its closure in $X$ recovers that component. Thus there is at most one component through $x$ .

3.1F1F6F10F14step 1.1step 2.1given∎

Under the assumed AC [F1], [F14] gives at least one irreducible component through $x$ . Together with step 2.1 this proves there is exactly one. If $X$ is empty, there is no point $x$ and the assertion is vacuous. If the local dimension is zero, $A_{p}$ is a zero-dimensional local domain and hence a field; the same minimal-prime argument still gives one component. No dimension restriction was used in steps 1.1–2.1. AC is also used through [F6] and [F10] for the local-domain theorem and the affine minimal-prime correspondence. For the fixed point $x$ , the proof chooses one chart and makes no simultaneous choices. The statement is a uniqueness-and-existence claim, not an iff criterion; no endpoint parameter is present.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Openness of the regular locus over a perfect field

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a perfect field (Perfect fields: every irreducible polynomial is separable) and let $X$ be a $k$ -scheme of finite type over $k$ . Then the regular locus $X_{r e g} = {x \in ∣ X ∣ : O_{X, x} is a regular local ring}$ of Regular and singular loci is open in $X$ . No reducedness, irreducibility, equidimensionality, or separatedness hypothesis is imposed, and $X$ may be empty.

Facts & Assumptions

Given: AC; a perfect field $k$ ; a $k$ -scheme $X$ of finite type over $k$ .

[F1]

The Axiom of Choice: every family of nonempty sets has a choice function.

[F2]

Perfect fields: every irreducible polynomial is separable: a field $F$ is perfect when every nonconstant irreducible polynomial in $F [x]$ is separable.

[F3]

Regular and singular loci: for a locally Noetherian scheme $X$ one defines $X_{r e g} = {x \in ∣ X ∣ : O_{X, x} is a regular local ring}$ and $X_{s i n g} = ∣ X ∣ ∖ X_{r e g}$ ; these definitions assert no openness or closedness property and apply to nonreduced schemes as well.

[F4]

Regular points of locally Noetherian schemes: for a point $x$ of a locally Noetherian scheme, $x$ is regular exactly when $O_{X, x}$ is a regular local ring, and then $\dim_{κ (x)} T_{x} X = \dim O_{X, x}$ ; this is absolute regularity and asserts no smoothness over a base field.

[F5]

Locally finite type and finite type morphisms: a morphism $f : X \to S$ is locally of finite type when every point of $X$ has an affine open neighbourhood $U$ whose image lies in an affine open $V = Spec A$ of $S$ with $U = Spec B$ and $A \to B$ of finite type; it is of finite type when it is locally of finite type and quasi-compact.

[F6]

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras: a commutative $R$ -algebra $A$ is of finite type over $R$ exactly when $A$ is isomorphic as an $R$ -algebra to a quotient $R [x_{1}, \dots, x_{n}] / a$ for some $n \in N$ and some ideal $a$ ; the case $n = 0$ gives the quotients of $R$ itself.

[F7]

Finite-variable polynomial algebras over fields are Noetherian by finite generators: for every field $K$ and every finite $d \geq 0$ the ring $K [x_{1}, \dots, x_{d}]$ is Noetherian, each ideal of it having a finite generating list; the proof is choice-free.

[F8]

Locally Noetherian and Noetherian schemes: a scheme is locally Noetherian when it has an affine open cover by spectra of Noetherian rings.

[F9]

Affine open subschemes: for a scheme $X$ and an open set $U \subseteq X$ , the open subscheme $U$ means $(U, O_{X} ∣_{U})$ , so its structure sheaf is the restriction of the structure sheaf of $X$ ; it is affine when this restricted ringed space is affine.

[F10]

The stalk of a presheaf at a point: the stalk of a presheaf $F$ at a point $x$ is the filtered colimit of the sections $F (U)$ over the open neighbourhoods $U$ of $x$ , concretely equivalence classes of pairs $(U, s)$ with $s \in F (U)$ .

[F11]

The stalk of the affine structure sheaf at a prime is A_p: for $p \in Spec A$ there is a canonical isomorphism $O_{Spec A, p} ≅ A_{p}$ .

[F12]

Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison: a topology satisfies (T1) $\emptyset \in T$ and $X \in T$ and (T2) $⋃ S \in T$ for every family $S \subseteq T$ of open sets, so arbitrary unions of open sets are open.

[F13]

Jacobian criterion and openness of the regular locus over a perfect field: under AC, for a perfect field $k$ , a polynomial ring $P = k [x_{1}, \dots, x_{n}]$ with $n \geq 0$ , an ideal $I \subseteq P$ and $A : = P / I$ , clause 3 states that the regular locus ${q \in Spec A : A_{q} regular}$ is open in $Spec A$ .

Proof

technique · direct

1.1F3F4F5F6F7F8given

Setup. Since $X \to Spec k$ is of finite type it is locally of finite type [F5], so every point of $X$ has an affine open neighbourhood $U = Spec B$ with the structure map $k \to B$ of finite type, and by [F6] such a $B$ is isomorphic as a $k$ -algebra to $k [x_{1}, \dots, x_{n}] / a$ for some $n \geq 0$ and some ideal $a$ . The polynomial ring $k [x_{1}, \dots, x_{n}]$ is Noetherian by [F7], hence so is its quotient $B$ ; as the point was arbitrary, $X$ has an affine open cover by spectra of Noetherian rings and is locally Noetherian by [F8]. Consequently the regular locus $X_{r e g}$ is defined by [F3] and the regular-point predicate of [F4] applies to $X$ .

1.2F12given

Locality of openness. It suffices to prove that every point of $X_{r e g}$ has an open neighbourhood contained in $X_{r e g}$ : if that holds, then $X_{r e g}$ is the union of the family of all open subsets of $X$ contained in $X_{r e g}$ , and this union is open by (T2) [F12]. The family is specified by a property of its members rather than by a selection, so no choice is used here.

2.1F5F6step 1.1given

The chart at a regular point. Fix $x \in X_{r e g}$ . By [F5] the point $x$ has an affine open neighbourhood $U = Spec B$ with the structure map $k \to B$ of finite type, and [F6] gives an isomorphism $B ≅ k [x_{1}, \dots, x_{n}] / a$ of $k$ -algebras for some $n \geq 0$ and some ideal $a \subseteq k [x_{1}, \dots, x_{n}]$ ; this chart is chosen for the single fixed point $x$ .

3.1F3F4F9F10F11step 2.1algebra

Stalks on the chart and the equivalence. Let $y \in U$ correspond to the prime $p \subseteq B$ . The open neighbourhoods of $y$ contained in $U$ are cofinal among all open neighbourhoods of $y$ in $X$ , because the intersection of any open neighbourhood with the open set $U$ is again an open neighbourhood of $y$ inside $U$ ; since the structure sheaf of the open subscheme $U$ is the restriction $O_{X} ∣_{U}$ [F9], the stalk colimits of [F10] agree on these cofinal systems and give $O_{X, y} ≅ O_{U, y}$ , while [F11] gives $O_{U, y} ≅ B_{p}$ . It follows that for $y \in U$ one has $y \in X_{r e g}$ if and only if $B_{p}$ is a regular local ring, both directions being the definition of $X_{r e g}$ in [F3] together with the regular-point criterion [F4]; hence $X_{r e g} \cap U = {p \in Spec B : B_{p} is a regular local ring}$ .

4.1F2F13step 2.1step 3.1given

The supplier. The field $k$ is perfect [F2], and $B ≅ k [x_{1}, \dots, x_{n}] / a$ with $n \geq 0$ by step 2.1, so clause 3 of [F13] applies with $P = k [x_{1}, \dots, x_{n}]$ and $I = a$ : the set ${p \in Spec B : B_{p} is a regular local ring}$ is open in $Spec B$ . By step 3.1 this set is $X_{r e g} \cap U$ , so $X_{r e g} \cap U$ is open in $U$ ; since $U$ is open in $X$ , such an open subset of $U$ is open in $X$ , and $X_{r e g} \cap U$ is an open neighbourhood of $x$ contained in $X_{r e g}$ .

5.1F1F2F3F4F12F13step 1.2step 2.1step 3.1step 4.1given∎

Conclusion and boundaries. The point $x \in X_{r e g}$ of step 2.1 was arbitrary, so step 4.1 shows that every point of $X_{r e g}$ has an open neighbourhood contained in $X_{r e g}$ , and step 1.2 then makes $X_{r e g}$ open in $X$ , which is the assertion. Boundaries. If $X$ is empty then $X_{r e g} = \emptyset$ is open by (T1) [F12] and the argument is vacuous. If the chart of step 2.1 has $n = 0$ then $B ≅ k / a$ is a quotient of the field $k$ and clause 3 of [F13] still applies, covering $X = Spec k$ (one point, whose local ring is the field $k$ and is regular) and $X = Spec (k [ϵ] / (ϵ^{2}))$ (no regular point). The scheme $X$ may be reducible or nonequidimensional: no purity, irreducibility, or dimension-uniformity input occurs, the supplier being applied chart by chart, and its clause 3 speaks about every point of the spectrum and not only the closed ones, so nonclosed points are covered as well. Nilpotents are retained and no reduction is performed, so the argument does not use reducedness and in fact proves the statement for arbitrary finite-type $k$ -schemes over a perfect field. AC is declared as [F1] and enters only through the supplier [F13], which assumes it; the chart of step 2.1 is chosen for one fixed point, and the union of step 1.2 is defined by a property, so no further choice occurs. The statement is not an if-and-only-if assertion; the one equivalence used, the characterization of $X_{r e g} \cap U$ in step 3.1, is proved in both directions.

Source qualification

Milne, Algebraic Geometry v6.10, §4h, Theorem 4.37 (printed p. 95; PDF page 94) proves that the set of nonsingular points of an affine algebraic variety over an algebraically closed field is dense and open, arguing that the singular locus is the zero set of the $(n - d) \times (n - d)$ minors of the Jacobian matrix and then that it is proper on each irreducible component; Milne works with closed points of classical varieties, and his density half is not asserted here, being the subject of the next theorem on this page. The Stacks Project, Varieties Lemma 33.25.8 (tag 0B8X) states the scheme-level result over a perfect field in the reduced case, where the regular locus equals the smooth locus and is dense open. The proof above instead applies the affine clause 3 of Jacobian criterion and openness of the regular locus over a perfect field, which is stated for every quotient $P / I$ of a polynomial ring over a perfect field and therefore also covers nonreduced and nonequidimensional charts; reducedness is consequently not used, and the statement is phrased without it. The scheme-theoretic locus is $X_{r e g}$ in the sense of Regular and singular loci, and the argument is deliberately local: it compares the stalk of an affine chart with the stalk of $X$ and quotes the supplier on that chart rather than re-proving the Jacobian rank criterion.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

A dense hypersurface chart with a nonzero partial derivative

Statement

Assume the Axiom of Choice. Let $k$ be algebraically closed and let $X$ be an irreducible classical variety over $k$ of dimension $d$ . There is a nonconstant irreducible polynomial $P (T_{1}, \dots, T_{d}, Z) \in k [T_{1}, \dots, T_{d}, Z]$ such that the hypersurface $H = V (P) \subseteq A_{k}^{d + 1}$ is irreducible, $\partial P / \partial Z \neq 0$ , and $X$ and $H$ contain isomorphic nonempty open subvarieties. In particular, $\dim H = d$ .

Facts & Assumptions

Given: The Axiom of Choice; an algebraically closed field $k$ ; and an irreducible classical variety $X$ of dimension $d$ over $k$ .

[F1]

The Axiom of Choice says every family of nonempty sets has a choice function (The Axiom of Choice).

[F2]

Every algebraically closed field is perfect (Fields of characteristic zero, finite fields, and algebraically closed fields are perfect); every nonconstant polynomial over it has a root (An algebraically closed field: every nonconstant polynomial has a root in the field).

[F3]

A classical variety has a finite affine-model cover; for an irreducible variety its compatible affine atlas makes it an integral classical variety. Thus a nonempty affine chart $U$ exists (Classical algebraic prevarieties, regular maps, and varieties, Integral classical varieties in the compatible affine-atlas register).

[F4]

For an affine algebraic set $U$ , its coordinate ring is the quotient of a finite-variable polynomial ring by its vanishing ideal (The coordinate ring of an affine algebraic set). In particular it is a finitely generated $k$ -algebra.

[F5]

The fraction fields of the nonempty affine charts of an integral classical variety identify canonically as $k (X)$ (Function fields and dominant pullbacks on general varieties).

[F6]

If $X$ is irreducible classical, then $\dim X = {trdeg}_{k} k (X) < \infty$ (Dimension equals transcendence degree).

[F7]

A finitely generated field extension of a perfect field has a separating transcendence basis: for some algebraically independent $t_{1}, \dots, t_{r}$ , the extension over $k (t_{1}, \dots, t_{r})$ is finite separable (Finitely generated extensions of a perfect field are separably generated).

[F8]

Every finite separable field extension is generated by one element (A finite extension generated by elements all but possibly one of which are separable is simple).

[F9]

The minimal polynomial of an algebraic element is monic irreducible and generates the kernel of its evaluation map (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).

[F10]

An element of a field extension is separable when its minimal polynomial is separable; an extension is separable when every element is (Separable algebraic elements and separable extensions).

[F11]

For an irreducible polynomial $m$ over a field $L$ , the quotient $L [Z] / (m)$ is a field (For a nonconstant $p$ in $F [x]$ , the ideal $(p)$ is maximal and $F [x] / (p)$ is a field exactly when $p$ is irreducible).

[F12]

An irreducible polynomial over a field is separable exactly when its formal derivative is nonzero (An irreducible polynomial over a field is separable exactly when its derivative is nonzero).

[F13]

Every finite-variable polynomial ring over a field is a UFD, and its irreducible elements are prime (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes).

[F14]

Over a UFD, a primitive positive-degree polynomial is irreducible exactly when it is irreducible over the fraction field (Gauss lemma over a UFD).

[F15]

Over algebraically closed $k$ and under AC, irreducible affine algebraic sets correspond to proper prime ideals, and $I (V (J)) = \sqrt{J}$ (Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals).

[F16]

For an affine classical variety $Y$ , its function field is $k (Y) = Frac (k [Y])$ (The function field of an irreducible classical affine variety).

[F17]

Under AC, two integral classical varieties over algebraically closed $k$ are birationally equivalent exactly when their function fields are $k$ -isomorphic (Classical integral varieties are birational exactly when their function fields are isomorphic over $k$ ).

[F18]

For integral classical varieties with compatible affine atlases, birational equivalence means that they have isomorphic nonempty open subvarieties (Birational maps and birational equivalence of classical varieties).

Proof

1.1F3F4F5F6choose

By [F3], choose a nonempty affine chart $U \subseteq X$ . Its coordinate ring is finite type by [F4], so its fraction field is finitely generated over $k$ ; by [F5] this field is $K : = k (X)$ . Equation [F6] gives ${trdeg}_{k} K = d$ .

2.1F2F6F7F8F9step 1.1choose

By [F2] and [F7], choose a separating transcendence basis $t_{1}, \dots, t_{d}$ of length $d$ for $K / k$ ; its residual extension $K / L$ , for $L = k (t_{1}, \dots, t_{d})$ , is finite separable. Here $A = k [T_{1}, \dots, T_{d}]$ has fraction field $L$ under $T_{i} \mapsto t_{i}$ . If $d = 0$ , then $K / k$ is finite and every element has an irreducible minimal polynomial over the algebraically closed field $k$ ; [F2] and [F9] force each such polynomial to be linear, so $K = k$ and we may take $γ = 0$ . In general, [F8] gives $γ \in K$ with $K = L (γ)$ ; when $K = L$ , again take $γ = 0$ .

3.1F9F10F11F12step 2.1

Let $m (Z) \in L [Z]$ be the monic minimal polynomial of $γ$ . It is irreducible by [F9] and separable by [F10], since $K / L$ is separable. Hence $m^{'} (Z) \neq 0$ by [F12]. For the trivial extension this is $m (Z) = Z$ , so the same derivative conclusion holds. Also [F9] and [F11] identify $L [Z] / (m)$ with the field $L (γ) = K$ .

4.1F13F14step 3.1algebra

Clear the finitely many coefficient denominators of $m$ with a nonzero $q \in A$ , obtaining $Q = q m \in A [Z]$ . In the UFD $A$ , factor the common irreducible divisors of the finitely many coefficients of $Q$ to write $Q = c P$ , where $c \in A ∖ {0}$ is their common content and $P$ is primitive. The polynomial $P$ has positive $Z$ -degree and is an associate of $m$ over $L$ . Thus it is irreducible in $L [Z]$ , and [F14] makes it irreducible in $A [Z]$ . Since $q / c \in L^{\times}$ is independent of $Z$ , $P = (q / c) m$ in $L [Z]$ and $P_{Z} = (q / c) m^{'} \neq 0$ .

5.1F4F11F13F15F16step 4.1algebra

Put $H = V (P) \subseteq A_{k}^{d + 1}$ . By [F13] the irreducible polynomial $P$ is prime in $A [Z]$ ; it is a nonunit because it has positive $Z$ -degree. Therefore $(P)$ is a proper prime ideal. By [F15], $H$ is nonempty and irreducible and $I (H) = \sqrt{(P)} = (P)$ . Thus [F4] gives $k [H] = A [Z] / (P)$ , a domain, and [F16] gives $k (H) = Frac (A [Z] / (P))$ . Localizing this coordinate ring at the nonzero elements of $A$ yields $L [Z] / (P) ≅ L [Z] / (m) ≅ L (γ) = K$ . Here $A$ embeds in $A [Z] / (P)$ because $P$ has positive $Z$ -degree, and the localized quotient is a field by [F11]; hence it is the fraction field of $A [Z] / (P)$ . Consequently $k (H) ≅ k (X)$ over $k$ .

6.1F1F3F5F6F15F16F17F18step 4.1step 5.1∎

Both $X$ and $H$ are integral classical varieties by their hypotheses and step 5.1. Their function fields are $k$ -isomorphic, so [F17] and [F18] supply isomorphic nonempty open subvarieties. By [F6], $\dim H = {trdeg}_{k} k (H) = d$ . Step 4.1 gives the required nonzero last-coordinate partial derivative, and the construction makes $H$ an irreducible hypersurface in $A_{k}^{d + 1}$ . AC is propagated through the integral-atlas, chart-function-field, dimension, Nullstellensatz, affine-function-field and birational interfaces [F3, F5, F6, F15, F16, F17]; once these apply, the construction uses only the finite chart, basis, generator and denominator/content selections above.

Source note

Milne, Algebraic Geometry v6.10, §3k Proposition 3.36 and Theorem 3.37, printed p. 74, reduces birationality of affine varieties to isomorphic nonempty affine opens and constructs a birational hypersurface from a $d$ -dimensional function field generated by $d + 1$ elements. Proposition 3.38, printed pp. 74–75, supplies the separable last generator over a perfect base; §4h, proof of Theorem 4.37, printed p. 95, uses the resulting hypersurface chart and a nonvanishing partial derivative. The present proof obtains the specific last-coordinate derivative directly: a primitive generator over a separating transcendence basis has separable minimal polynomial, and clearing denominators and removing content multiplies it only by a nonzero scalar in $k (T_{1}, \dots, T_{d})$ , so that derivative stays nonzero.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Dense regular loci on every component

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a perfect field (Perfect fields: every irreducible polynomial is separable) and let $X$ be a reduced $k$ -scheme of finite type over $k$ . Then:

the regular locus $X_{r e g} = {x \in ∣ X ∣ : O_{X, x} is a regular local ring}$ (Regular and singular loci) is open in $X$ ;
for every irreducible component $Z$ of $X$ (Irreducible components of a topological space) the intersection $X_{r e g} \cap Z$ is a dense open subset of $Z$ ; in particular every irreducible component contains a nonempty dense open subset of points regular on $X$ ;
if $X \neq \emptyset$ , then $X_{r e g} \neq \emptyset$ .

No separatedness, irreducibility or equidimensionality hypothesis is imposed, and $X$ may be empty, in which case the second clause is vacuous and the third is not asserted.

Facts & Assumptions

Given: AC; a perfect field $k$ ; a reduced $k$ -scheme $X$ of finite type over $k$ .

[F1]

The Axiom of Choice: every family of nonempty sets has a choice function.

[F2]

Perfect fields: every irreducible polynomial is separable: a field $F$ is perfect when every nonconstant irreducible polynomial in $F [x]$ is separable.

[F3]

The reduction of a scheme and Reduced affine schemes: the nilradical ideal sheaf $N_{X}$ has nilpotent germs and $X$ is reduced exactly when $N_{X} = 0$ ; on $Spec A$ the reduction is $Spec (A / \sqrt{(0)})$ , and an affine scheme is reduced exactly when its coordinate ring is reduced.

[F4]

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Locally finite type and finite type morphisms and Every algebra of finite type over a Noetherian ring is a Noetherian ring: a $k$ -algebra of finite type is a quotient of a polynomial ring in finitely many variables; a morphism of finite type is locally of finite type, so an affine chart $U = Spec A$ of a finite-type $k$ -scheme has $A$ of finite type over $k$ ; an algebra of finite type over a Noetherian ring is Noetherian, and Locally Noetherian and Noetherian schemes makes $Spec A$ locally Noetherian for such an $A$ , so a finite-type $k$ -scheme is locally Noetherian.

[F5]

Regular and singular loci: for a locally Noetherian scheme, $X_{r e g} = {x \in ∣ X ∣ : O_{X, x} is a regular local ring}$ ; the definition alone asserts no openness.

[F6]

Schemes and Affine open subschemes: a scheme has an open cover by affine open subschemes, and for an open $U \subseteq X$ the open subscheme is $(U, O_{X} ∣_{U})$ ; The stalk of a presheaf at a point then gives $O_{U, x} = (O_{X} ∣_{U})_{x} = O_{X, x}$ for every $x \in U$ , the neighbourhood systems in $U$ and in $X$ being cofinal.

[F7]

Openness of the regular locus over a perfect field: under AC, for a perfect field $k$ and every finite-type $k$ -scheme, the regular locus is open; no reducedness is needed.

[F8]

Jacobian criterion and openness of the regular locus over a perfect field: under AC, let $k$ be perfect, $P = k [x_{1}, \dots, x_{n}]$ , $I \subseteq P$ an ideal and $A : = P / I$ . Then the regular locus ${q : A_{q} regular}$ is open in $Spec A$ ; if $p$ is a minimal prime of $A$ with $A_{p}$ reduced, then the regular locus contains a dense open subset of $V (p)$ ; when $A$ is reduced this holds for every irreducible component.

[F9]

Irreducible components of the spectrum correspond to minimal prime ideals: under AC, the irreducible components of $Spec A$ are exactly the closed sets $V (p)$ for minimal primes $p$ of $A$ , each minimal prime giving one component.

[F10]

Irreducible components of a topological space and Existence and basic properties of irreducible components: components are nonempty maximal irreducible subsets; under AC they are closed, every irreducible subset is contained in a component, every point of a nonempty space lies in a component, and the closure of an irreducible subset is irreducible.

[F11]

Irreducibility via nonempty open subsets, connectedness and open subspaces: a space is irreducible if and only if it is nonempty and every nonempty open subset is dense; a nonempty open subspace of an irreducible space is irreducible.

[F12]

Interior, closure, boundary, exterior, derived set and isolated point in a topological space and Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace: the closure of a subset is the smallest closed superset of it, so a subset of a closed set $Z$ has its closure contained in $Z$ , and a subset is dense in $Z$ exactly when its closure computed in $Z$ is $Z$ ; the closed subsets of a subspace are exactly the traces of the closed subsets of the ambient space, so the closure in a subspace $Z$ of a subset of $Z$ is contained in its closure in the ambient space.

[F13]

Dual numbers give a one-point nonreduced affine scheme: for a field $k$ and $R = k [ϵ] / (ϵ^{2})$ , the scheme $Spec R$ has exactly one point, the prime $(ϵ)$ , whose residue field is $k$ , and it is not reduced.

[F14]

The stalk of the affine structure sheaf at a prime is A_p: for a prime $p$ of a commutative ring $A$ there is a canonical isomorphism $O_{Spec A, p} ≅ A_{p}$ .

[F15]

regular local rings are domains and cohen macaulay: under AC, a regular local ring is a domain (and Cohen--Macaulay).

Proof

technique · direct

1.1F1F2F3F4F5F6F7givenalgebra

Setup. The field $k$ is perfect [F2] and AC is assumed [F1]. By [F4] the scheme $X$ is locally Noetherian, so the regular locus $X_{r e g}$ is defined [F5], and it is open in $X$ by [F7]. For an open subscheme $U \subseteq X$ , [F6] gives $O_{U, x} = O_{X, x}$ for every $x \in U$ , so $X_{r e g} \cap U = {x \in U : O_{U, x} is a regular local ring} = U_{r e g},$ and if $U = Spec A$ is affine then $A$ is reduced: $X$ reduced means $N_{X} = 0$ [F3], the restriction of the zero sheaf is zero, and on $Spec A$ the reduction is $Spec (A / \sqrt{(0)})$ , so $\sqrt{(0)} = 0$ and $A$ has no nonzero nilpotent, that is, $A$ is reduced [F3]. This proves clause 1 of the statement and records the two facts used below.

1.2F6F9F10F11F12givenalgebra

Comparing a component of $X$ with a chart component. Let $Z$ be an irreducible component of $X$ ; it is nonempty and closed in $X$ [F10]. Choose $z \in Z$ ; by [F6] there is an affine open subscheme $U = Spec A$ of $X$ with $z \in U$ . Then $U \cap Z$ is a nonempty open subspace of $Z$ , hence irreducible [F11], and dense in $Z$ [F11]; by [F10] it is contained in an irreducible component $W$ of $U$ , and by [F9] we have $W = V (p)$ for a minimal prime $p$ . The closure $\overline{W}$ of $W$ in $X$ is irreducible and closed [F10], and it contains $U \cap Z$ , whose closure in $X$ equals $Z$ : indeed $U \cap Z$ is dense in $Z$ , so $Z = {\overline{U \cap Z}}^{Z} \subseteq {\overline{U \cap Z}}^{X} \subseteq Z$ , the second inclusion because $U \cap Z \subseteq Z$ and $Z$ is closed in $X$ [F12]. Since $Z$ is a maximal irreducible subset of $X$ and $\overline{W} \supseteq Z$ is irreducible, $\overline{W} = Z$ ; finally $W = \overline{W} \cap U$ , because $W$ is closed in $U$ [F10] and any point of $\overline{W} \cap U$ outside $W$ would have the open neighbourhood $U ∖ W$ in $X$ disjoint from $W$ . Hence $W = Z \cap U .$

2.1F8F9F10step 1.1givenalgebra

The affine chart input. Let $U = Spec A$ be a nonempty affine open subscheme of $X$ , with $A$ reduced of finite type over the perfect field $k$ [step 1.1, F4]. Let $W$ be an irreducible component of $U$ ; by [F9] there is a minimal prime $p \subseteq A$ with $W = V (p)$ , and $W \neq \emptyset$ because $p \in V (p)$ [F10]. Since $A$ is reduced, clause 3 of [F8] applies and the regular locus of $Spec A$ contains a dense open subset $D$ of $W$ ; in particular $D \subseteq U_{r e g} \cap W$ [step 1.1], the set $U_{r e g} \cap W$ is open in $W$ because $U_{r e g}$ is open in $U$ , and it is dense in $W$ because it contains the dense subset $D$ ; also $D \neq \emptyset$ , because a dense subset of the nonempty space $W$ cannot be empty.

3.1F11step 1.1step 2.1step 1.2givenalgebra

Density of the regular locus on every component. With $Z$ , $U = Spec A$ and $W = Z \cap U$ as in step 1.2, step 2.1 applied to the component $W$ of $U$ produces the dense open subset $D \subseteq U_{r e g} \cap W$ with $D \neq \emptyset$ . Here $U_{r e g} \cap W = (X_{r e g} \cap U) \cap (Z \cap U) = X_{r e g} \cap (Z \cap U) \subseteq X_{r e g} \cap Z$ [step 1.1, step 1.2], so $D$ is a nonempty subset of $X_{r e g} \cap Z$ that is open in $W$ ; since $W = Z \cap U$ is open in $Z$ (as $U$ is open in $X$ ), $D$ is open in $Z$ . Thus $X_{r e g} \cap Z$ is a nonempty subset of $Z$ that is open in $Z$ (ostensibly open in $X$ by clause 1, hence open in $Z$ ), and therefore it is dense in $Z$ because $Z$ is irreducible [F11]. This proves clause 2 of the statement, including the assertion that each component contains a nonempty dense open set of regular points, namely $X_{r e g} \cap Z$ itself.

4.1F1F2F3F4F5F7F8F9F10F11F12F13F14F15step 2.1step 3.1givenalgebra∎

Nonemptiness and boundaries. If $X \neq \emptyset$ , pick a point $z \in X$ ; by [F10] it lies in some irreducible component $Z$ , and step 3.1 gives $\emptyset \neq X_{r e g} \cap Z \subseteq X_{r e g}$ , so the regular locus is nonempty; this proves clause 3. If $X = \emptyset$ then there are no irreducible components [F10] and clauses 2 and 3 are vacuous, while $X_{r e g} = \emptyset$ is open in $X$ . Reducedness cannot be dropped: let $k$ be a perfect field [F2], let $R = k [ϵ] / (ϵ^{2})$ and $X = Spec R$ . By [F13] the scheme $X$ has exactly one point, the prime $(ϵ)$ , whose residue field is $k$ , and $X$ is not reduced, while $X$ is of finite type over $k$ because $R$ is a quotient of the polynomial ring $k [ϵ]$ [F4]. Every element of $R$ has the form $a + b ϵ$ with $a, b \in k$ ; such an element with $a \neq 0$ is a unit, with inverse $a^{- 1} - a^{- 2} b ϵ$ , and the elements with $a = 0$ are exactly the multiples of $ϵ$ , so $(ϵ)$ is the unique maximal ideal and the localization at it is $R$ itself; the stalk at the unique point is therefore $O_{X, (ϵ)} ≅ R_{(ϵ)} = R$ [F14]. The nonreducedness of $X$ means by [F3] that the coordinate ring $R$ is not reduced, so $R$ has a nonzero nilpotent element and is not a domain, a domain having no nonzero nilpotent; since a regular local ring is a domain [F15], the local ring $O_{X, (ϵ)} ≅ R$ is not regular. Hence $X_{r e g} = \emptyset$ by [F5], as $(ϵ)$ is the only point of $X$ [F13] and its local ring is not regular, while $X \neq \emptyset$ is irreducible [F11] with sole irreducible component $X$ itself [F10] and $X_{r e g} \cap X = \emptyset$ is not dense in $X$ [F12], so clauses 2 and 3 fail for this finite-type $k$ -scheme, which is not reduced. Perfectness is used only through [F7] and [F8] and nothing is asserted for imperfect $k$ . The Axiom of Choice enters through the statement [F1] and through the suppliers that assume it, namely [F7], [F8], [F9], [F10] and [F15], each cited at the step that uses it; the remaining steps use only explicit set-theoretic and ring-theoretic operations.

Source qualification

Milne, Algebraic Geometry v6.10, §4h, Theorem 4.37 (printed p. 95; PDF p. 94) proves that over a perfect field the singular locus of a variety is closed and that the regular points are dense in every irreducible component, working with classical varieties over an algebraically closed field and asserting the density through the nonsingularity of a suitable hypersurface section; the item above instead derives the density clause for an arbitrary reduced finite-type $k$ -scheme from clause 3 of Jacobian criterion and openness of the regular locus over a perfect field, which packages the affine-adapted version of the same theorem, and makes the passage from affine charts to global components explicit through the closure of a chart component. The Stacks Project's treatment of the same statement (Varieties, Lemma 33.25.8, tag 0B8X, and the more general criterion for the smooth locus) agrees with the affine form used here; its reducedness hypothesis on the ambient scheme matches the hypothesis above, which is necessary as the dual-numbers example of step 4.1 records. No separatedness is imposed, no smoothness is concluded, and nothing is asserted over imperfect fields.

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

Minimal tangent dimension and homogeneous regularity

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field and let $X$ be an irreducible classical variety over $k$ (Classical algebraic prevarieties, regular maps, and varieties), with dimension $\dim X$ (Global and local dimension of classical varieties). Then $\dim X = \min_{x \in X} \dim_{k} T_{x} X,$ the minimum taken over the closed points of $X$ (The intrinsic Zariski tangent space), and $X$ is regular (Regular and singular loci) if and only if the function $x \mapsto \dim_{k} T_{x} X$ is constant on the closed points of $X$ .

More generally, a nonempty reduced classical finite-type space over $k$ whose automorphism group acts transitively on its point set is regular.

Facts & Assumptions

Given: AC; an algebraically closed field $k$ ; a classical variety $X$ over $k$ ; and the intrinsic tangent spaces $T_{x} X$ at its closed points.

[F1]

The Axiom of Choice: Every family of nonempty sets has a choice function.

[F2]

[F3]

Global and local dimension of classical varieties: for a classical variety $X$ , $\dim X$ is the chain dimension and $\dim_{x} X = \max_{x \in X_{i}} \dim X_{i}$ over the irreducible components containing the closed point $x$ .

[F4]

Local dimension for a reducible classical algebraic set: for a reduced classical finite-type space $X$ over an algebraically closed field and a closed point $x$ , $\dim O_{X, x} = \max_{x \in X_{i}} \dim X_{i}$ over the irreducible components containing $x$ .

[F5]

Regular and singular loci: the regular locus is $X_{r e g} = {x \in ∣ X ∣ : O_{X, x} is a regular local ring}$ , and for a reduced classical finite-type space over an algebraically closed field, a closed point $x$ lies in $X_{r e g}$ exactly when $\dim_{κ (x)} T_{x} X = \dim_{x} X$ .

[F6]

Regular points of locally Noetherian schemes: a point $x$ of a locally Noetherian scheme is regular when its local ring is a regular local ring, and then $x$ is regular if and only if $\dim_{κ (x)} T_{x} X = \dim O_{X, x}$ .

[F7]

Tangent dimension bounds local dimension: for every point $x$ of a locally Noetherian scheme, $\dim_{κ (x)} T_{x} X \geq \dim O_{X, x}$ ; for a reduced classical finite-type variety over an algebraically closed field and a closed point $x$ , $\dim T_{x} X \geq \dim_{x} X$ .

[F8]

Dense regular loci on every component: for a perfect field $k$ and a reduced $k$ -scheme $X$ of finite type, the regular locus is open, its trace on every irreducible component is a dense open subset of that component, and $X_{r e g} \neq \emptyset$ whenever $X \neq \emptyset$ .

[F9]

Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: every algebraically closed field is perfect.

[F10]

Existence and basic properties of irreducible components: every irreducible subset is contained in an irreducible component, and a nonempty irreducible space is its own unique irreducible component.

[F11]

Differentials, open restriction, and the chain rule: for a $k$ -morphism $f$ of $k$ -schemes, the differential $d_{x} f : T_{x} X \to T_{f (x)} Y$ is defined at $k$ -rational points, is compatible with composition, and $d_{x} ({id}_{X}) = {id}_{T_{x} X}$ .

[F12]

Morphisms of locally ringed spaces: a morphism of locally ringed spaces induces at every point $x$ a local ring homomorphism $f_{x}^{♯} : O_{Y, f (x)} \to O_{X, x}$ on stalks.

[F13]

The intrinsic Zariski tangent space: the intrinsic tangent space $T_{x} X$ is the $κ (x)$ -dual of $m_{x} / m_{x}^{2}$ , and differentials of $k$ -morphisms act on it by the dual of the induced cotangent map.

[F14]

The coordinate ring of a classical affine algebraic set: the coordinate ring $k [X] = k [x_{1}, \dots, x_{n}] / I (X)$ of an affine algebraic set over an algebraically closed field is reduced, and the finite coordinate classes generate it as a $k$ -algebra.

[F15]

In a finite-type algebra over a field, closed points are dense in every closed subset of the spectrum: for every finite-type $k$ -algebra $A$ , each nonempty open subset of a closed subset of $Spec A$ contains a closed point of $Spec A$ . On a reduced affine model over algebraically closed $k$ , these points are exactly the classical $k$ -points by Classical k-points give closed points over an algebraically closed field. A $k$ -point is closed in the whole finite-type model: its intersection with any affine chart containing it is a maximal ideal there, while its intersection with a chart not containing it is empty.

[F16]

localisations of regular local rings are regular: assuming AC, every prime localization of a regular local ring is regular. If $p \subseteq m$ in a finite-type affine coordinate ring $A$ and $A_{m}$ is regular, then $A_{p} = (A_{m})_{p A_{m}}$ is regular.

Proof

technique · direct

1.1F2F3F4F5F6F7F8F9F10F13F14givenalgebra

Since $X$ is an irreducible classical variety over the algebraically closed field $k$ , it is nonempty, because irreducible means nonempty [F3]. Its affine models have reduced coordinate rings [F14], so $X$ is a reduced classical finite-type space over $k$ and the classical suppliers [F4], [F5], [F7] apply to it, while [F8] applies to the reduced finite-type spectra of its affine coordinate rings; in particular $X$ is its own unique irreducible component [F10], and the field $k$ is perfect [F9]. Fix a closed point $x$ of $X$ . Because the only irreducible component of $X$ is $X$ itself [F10], [F3] and [F4] give $\dim_{x} X = \dim X$ , and then [F7] gives $\dim_{k} T_{x} X \geq \dim_{x} X = \dim X$ . Apply [F8] to the spectrum of any nonempty affine model chart. Its regular locus is open and nonempty, so [F15] supplies a closed, hence classical, point there whose local ring is regular by [F5]. At such a point [F6] gives $\dim_{k} T_{x} X = \dim O_{X, x}$ , while [F4] with [F3] gives $\dim O_{X, x} = \dim_{x} X = \dim X$ ; hence $\dim_{k} T_{x} X = \dim X$ for every closed point $x \in X_{r e g}$ .

1.2F2F5F6F11F12F13givenalgebra

Let $σ$ be an automorphism of the classical variety $X$ , that is, an isomorphism of locally ringed spaces over $k$ with inverse $σ^{- 1}$ . At every closed point $x$ the induced stalk map of [F12], $σ_{x}^{♯} : O_{X, σ (x)} \to O_{X, x}$ , is a local ring homomorphism, and the stalk maps of $σ$ and $σ^{- 1}$ are mutually inverse isomorphisms of local rings, so $O_{X, σ (x)}$ is a regular local ring if and only if $O_{X, x}$ is; hence $σ (X_{r e g}) = X_{r e g}$ . Likewise, since $σ^{- 1} \circ σ = {id}_{X}$ and $σ \circ σ^{- 1} = {id}_{X}$ , the functoriality of the differential [F11] applied to these two composites gives $d_{σ (x)} (σ^{- 1}) \circ d_{x} σ = d_{x} ({id}_{X}) = {id}_{T_{x} X}$ and $d_{x} σ \circ d_{σ (x)} (σ^{- 1}) = {id}_{T_{σ (x)} X}$ , so $d_{x} σ$ is an isomorphism and $\dim_{k} T_{σ (x)} X = \dim_{k} T_{x} X$ for every closed point $x$ .

2.1F2F3F5F6F8step 1.1algebra

By the affine application of [F8] and [F15] in step 1.1 there is a classical closed point $x_{0} \in X_{r e g}$ ; by step 1.1 its tangent dimension equals $\dim X$ , and by step 1.1 again every closed point has tangent dimension at least $\dim X$ . Hence the minimum of $\dim_{k} T_{x} X$ over the closed points of $X$ is attained and $\dim X = \min_{x \in X} \dim_{k} T_{x} X$ . Comparing step 1.1 with the criterion of [F5] and the definition of $\dim_{x} X$ in [F3] shows in addition that a closed point $x$ attains the minimum exactly when $\dim_{k} T_{x} X = \dim_{x} X$ , that is, exactly when $x \in X_{r e g}$ .

2.2F2F5F8F9F15F16step 1.2givenalgebra

Now let $X$ be a nonempty reduced classical finite-type space over $k$ whose automorphism group $G$ acts transitively on its classical point set. Take a nonempty affine model with reduced finite-type coordinate ring $A$ [F2, F14]. By [F8] and [F9] the regular locus of $Spec A$ is a nonempty open subset, so [F15] gives a classical closed point $x_{0}$ there with a regular local ring. By step 1.2, for every classical point $y$ an automorphism taking $x_{0}$ to $y$ identifies their local rings; thus every classical closed point is regular. Now take any point $z$ of the scheme model, represented by a prime $p$ in an affine chart $Spec A$ . Applying [F15] to the nonempty closed subset $V (p)$ gives a maximal ideal $m \supseteq p$ , hence a classical closed point. Its local ring $A_{m}$ is regular; [F16] then makes $A_{p} = (A_{m})_{p A_{m}}$ regular. Since $z$ was arbitrary, every scheme point is regular, so the classical space and its scheme model are regular in the sense of [F5].

3.1F2F3F5F6step 1.1step 2.1algebra

By definition [F5] the variety $X$ is regular when every point of it is regular, that is, when $X_{r e g} = X$ ; every point of a classical variety is a closed point [F2]. If $X$ is regular, step 1.1 applies at every closed point and gives $\dim_{k} T_{x} X = \dim X$ , so the function $x \mapsto \dim_{k} T_{x} X$ is constant. Conversely, suppose $\dim_{k} T_{x} X = c$ for every closed point; then $c$ is the minimum computed in step 2.1, so $c = \dim X$ , and step 1.1 with [F3] gives $\dim_{k} T_{x} X = \dim X = \dim_{x} X$ for every closed point $x$ ; by [F5] each such $x$ lies in $X_{r e g}$ , so $X_{r e g} = X$ and $X$ is regular. This proves both directions of the equivalence.

4.1F1F2F4F5F7F8F10F15F16step 1.1step 1.2step 2.1step 2.2step 3.1givenalgebra

Boundary and scope dispositions. Empty: an irreducible classical variety is nonempty by convention [F3], so the minimum of step 2.1 is taken over a nonempty set, and for the general claim the empty reduced space is excluded by hypothesis, the assertion being vacuous for it. Zero and one: at a point with $\dim_{x} X = 0$ the criterion [F5] reads " $x$ regular if and only if $\dim_{k} T_{x} X = 0$ ", so the zero-dimensional case is covered by the criterion without modification, and in the one-dimensional case the minimum of step 2.1 has the value one, attained at the regular points. Degenerate: reducedness is genuinely needed for the transitive claim, since a nonreduced local ring is not a regular local ring while the one-point nonreduced space $Spec k [ϵ] / (ϵ^{2})$ has a transitive automorphism group on its single point; for such a space the regular locus can be empty, so the supplier [F8] cannot be applied. Endpoints: the minimum of step 2.1 is attained exactly at the regular points, and the constant value of step 3.1 is exactly $\dim X$ . Choice: AC is declared in [F1] and is used only through the AC-assuming suppliers [F4], [F5], [F7], [F8], [F10], [F15] and [F16], each cited at the step that uses it, while the automorphism arguments of steps 1.2 and 2.2 make no choice. Biconditional directions: step 3.1 proves both directions of the regularity-constancy equivalence, using step 1.1 in the forward direction and the minimum of step 2.1 in the reverse direction, and the criterion [F5] is instantiated in step 3.1 in the direction "tangent dimension equal to local dimension implies regular" while its defining content, regularity of the local ring, is what defines $X_{r e g}$ in step 1.1.

∎

Source qualification

J. S. Milne, Algebraic Geometry v6.10, §4h, Corollaries 4.38-4.40 (printed p. 95), records for a variety over an algebraically closed field that the dimension is the minimum of the tangent-space dimensions, that nonsingularity is equivalent to constancy of the tangent dimension, and that homogeneous spaces are nonsingular; Milne's book-wide conventions (classical varieties, algebraically closed field) are narrower than the scheme-level inputs used here, so the statement is derived from the library's openness/density supplier for the regular locus and the embedding-dimension bound rather than quoted from the source. Donu Arapura, Notes on Basic Algebraic Geometry §5.2 Corollary 5.2.4, states the homogeneous regularity conclusion in the same classical setting. Neither source is used as a substitute for the proof, which is given above from the cited library items.

DefinitionDefinition: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Smoothness over a field by geometric regularity

Definition

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and $X$ a finite-type $k$ -scheme. Smoothness of $X \to Spec k$ is defined by local standard smooth presentations in Smooth morphisms via local standard smooth presentations. The following is an equivalent characterization: $X \to Spec k$ is smooth if and only if, for every field extension $K / k$ , every local ring of the scheme-theoretic base change $X_{K}$ is regular. Here $X_{K}$ is formed by tensoring affine coordinate rings with $K$ and gluing as in Extension of scalars of a scheme along a field extension. We call this condition geometric regularity of $X$ over $k$ . It retains nilpotents in every field change.

Facts & Assumptions

Given: A field $k$ , a finite-type $k$ -scheme $X$ , the earlier local-standard-smooth definition, and the Axiom of Choice.

[F1]

Locally finite type and finite type morphisms: a finite-type morphism is locally of finite type; hence every point has an affine neighbourhood $U = Spec A$ on which the structure algebra is of finite type.

[F2]

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras: an $R$ -algebra is of finite type when it is a finitely generated $R$ -algebra.

[F3]

Smooth morphisms via local standard smooth presentations and Standard smooth presentations and locally standard smooth maps: a morphism is smooth when every source point has affine neighbourhoods on which the induced ring map is standard smooth at the corresponding prime; standard smoothness at a prime holds after a further principal shrinking, and a standard smooth presentation is a finitely presented algebra.

[F4]

Geometrically regular algebras and geometrically regular fibres: a finite-type $k$ -algebra $A$ is geometrically regular over $k$ when $A \otimes_{k} K$ is a regular Noetherian ring for every finitely generated field extension $K / k$ .

[F5]

Locally standard smooth iff flat with geometrically regular fibres: for a finite-type $k$ -algebra $A$ , geometric regularity over $k$ is equivalent to the structure map $k \to A$ being locally standard smooth.

[F6]

Field tests for geometric regularity: if $A$ is geometrically regular over $k$ , then $A \otimes_{k} K$ is a regular ring for every field extension $K / k$ .

[F7]

regular noetherian ring: a commutative Noetherian ring is regular when its localization at every prime is a regular local ring.

[F8]

Extension of scalars of a scheme along a field extension: for every affine open $U = Spec A \subseteq X$ , its restriction in $X_{K}$ is canonically $Spec (A \otimes_{k} K)$ , open in $X_{K}$ ; the theorem's AC use is only to index affine opens by points, and it notes that the set of all affine opens gives a choice-free construction.

[F9]

Regular points of locally Noetherian schemes: a point $x$ of a locally Noetherian scheme is regular when $O_{X, x}$ is a regular local ring.

[F10]

embedding dimension and regular local ring: a nonzero Noetherian local ring is regular local exactly when its embedding dimension equals its Krull dimension.

[F11]

The affine scheme of dual numbers: for a field $k$ , $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ ; its class $ϵ$ is nilpotent.

[F12]

The stalk of the affine structure sheaf at a prime is A_p: for a point $p \in Spec A$ , the stalk is canonically $A_{p}$ .

[F13]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function.

Proof

1.1F1F2F5F6F8F13given

Finite-type affine charts and the assumption. The structural morphism $X \to Spec k$ is of finite type, so around each $x \in X$ there is an affine open $U = Spec A$ with $A$ of finite type over $k$ [F1, F2]. The field-change scheme $X_{K}$ and its affine restrictions are those of [F8]. AC is carried in the hypotheses of the standard-smooth equivalence [F5], the all-field field-test [F6], and the field-change construction [F8], so it is declared here [F13]. The field-change theorem says its AC use is only to index a pointwise affine cover and that the set of all affine opens also gives a choice-free construction [F8]. The proof below uses one chart at a time and makes no simultaneous choice of charts or local generators.

2.1F3F5F6F7F8F9F12step 1.1given

Smoothness implies regularity after every field change. Assume $X \to Spec k$ is smooth, fix any extension $K / k$ , and let $z \in X_{K}$ map to $x \in X$ . By [F3], there is an affine neighbourhood $V = Spec C$ of $x$ on which $k \to C$ is standard smooth at the prime for $x$ ; shrinking further by a principal open gives $W = Spec B \subseteq V$ containing $x$ with $B$ standard smooth over $k$ . Thus $k \to B$ is locally standard smooth, so [F5] makes $B$ geometrically regular. By [F6], $B \otimes_{k} K$ is a regular ring. By [F8], $W_{K} = Spec (B \otimes_{k} K)$ is an open neighbourhood of $z$ in $X_{K}$ . If $q$ is the prime for $z$ in this chart, [F12] identifies its local ring with $(B \otimes_{k} K)_{q}$ ; this is regular by [F7], and hence the local ring of $z$ is regular in the sense of [F9]. Since $K$ and $z$ were arbitrary, every local ring of every $X_{K}$ is regular.

2.2F3F4F5F7F8F12step 1.1given

Regularity after every field change implies smoothness. Suppose every local ring of $X_{K}$ is regular for every extension $K / k$ . Fix $x \in X$ and choose a finite-type affine neighbourhood $U = Spec A$ as in step 1.1. For every finitely generated extension $K / k$ , [F8] identifies $U_{K}$ with the open subscheme $Spec (A \otimes_{k} K) \subseteq X_{K}$ . All its local rings are regular by hypothesis and [F12], and the ring is Noetherian because it is a finite-type algebra over the field $K$ [F4]. It is therefore regular by [F7]. This holds for every finitely generated $K / k$ , hence $A$ is geometrically regular by [F4]. The equivalence [F5] gives that $k \to A$ is locally standard smooth, in particular standard smooth at the prime for $x$ . As this applies at every $x$ , [F3] says that $X \to Spec k$ is smooth. The empty scheme satisfies both conditions vacuously.

3.1F10F11F12step 2.1step 2.2givenalgebra∎

Boundary calculations. For $X = Spec k$ , every field change is $Spec K$ and its only local ring is the field $K$ , so the zero-dimensional one-point case is smooth. For the nonreduced point $D_{k}$ of [F11], the field change has coordinate ring $K [ϵ] / (ϵ^{2})$ : the map $(a + b ϵ) \otimes λ \mapsto a λ + b λ ϵ$ is a ring isomorphism with inverse $c + d ϵ \mapsto 1 \otimes c + ϵ \otimes d$ . Every prime contains the nilpotent $ϵ$ , and every element with nonzero constant term is a unit, so $(ϵ)$ is the unique prime and maximal ideal. The ring is a two-dimensional $K$ -vector space, so its ideals are finite-dimensional $K$ -subspaces and it is Noetherian. Its unique local ring has dimension zero, while its maximal ideal has square zero and one-dimensional quotient by its square; it is not regular local by [F10]. Thus the criterion detects the nilpotent structure and correctly says $D_{k}$ is not smooth. Taking $K = k$ shows the original scheme itself is included among the required field changes; no interval or endpoint parameter occurs.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Regular equals smooth over a perfect field

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a perfect field and let $X$ be a finite-type $k$ -scheme. Here regular means that $X$ is locally Noetherian and every local ring $O_{X, x}$ is regular local; smooth over $k$ means that $X \to Spec k$ is smooth under the local-standard-smooth convention of Smooth morphisms via local standard smooth presentations. Then $X is regular ⟺ X \to Spec k is smooth .$ No reducedness, irreducibility, or closed-point restriction is imposed.

Facts & Assumptions

Given: A perfect field $k$ , a finite-type $k$ -scheme $X$ , and the Axiom of Choice.

[F1]

Locally finite type and finite type morphisms: a finite-type morphism is locally of finite type, so every point of $X$ has an affine open neighbourhood $U = Spec A$ on which $A$ is of finite type over $k$ .

[F2]

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras: an algebra of finite type over $k$ is a quotient of a finite-variable polynomial $k$ -algebra.

[F3]

Finite-variable polynomial algebras over fields are Noetherian by finite generators: for every field $K$ and finite $d \geq 0$ , $K [x_{1}, \dots, x_{d}]$ is Noetherian, meaning each ideal has a finite generating list.

[F4]

Left and right Noetherian rings: a ring is left Noetherian when its left regular module is Noetherian.

[F5]

Noetherian modules: every submodule is finitely generated: a module is Noetherian when each submodule is finitely generated.

[F6]

Left, right and two-sided ideals: in a commutative ring, its ideals are exactly the submodules of its left regular module.

[F7]

Locally Noetherian and Noetherian schemes: a scheme is locally Noetherian when it has an affine open cover by spectra of Noetherian rings.

[F8]

Affine open subschemes: an open subscheme has the restricted structure sheaf $O_{X} ∣_{U}$ .

[F9]

The stalk of the affine structure sheaf at a prime is A_p: for $p \in Spec A$ , $O_{Spec A, p} ≅ A_{p}$ .

[F10]

Regular points of locally Noetherian schemes: on a locally Noetherian scheme, a point is regular exactly when its local ring is regular local.

[F11]

regular noetherian ring: a commutative Noetherian ring is regular when every prime localization is regular local; the zero ring is regular vacuously.

[F12]

Geometrically regular algebras and geometrically regular fibres: a finite-type $k$ -algebra $A$ is geometrically regular when $A \otimes_{k} K$ is a regular Noetherian ring for every finitely generated field extension $K / k$ .

[F13]

Regular algebras over a perfect field are geometrically regular: assuming AC, a regular finite-type algebra over a perfect field remains a regular ring after tensoring with every field extension.

[F14]

Locally standard smooth iff flat with geometrically regular fibres: under AC, for a finite-type $k$ -algebra $A$ , geometric regularity over $k$ is equivalent to the structure map $k \to A$ being locally standard smooth.

[F15]

Smooth morphisms via local standard smooth presentations: under AC, a finite-type scheme morphism is smooth when every source point has affine neighbourhoods on which the induced ring map is standard smooth at that point.

[F16]

Standard smooth presentations and locally standard smooth maps: locally standard smooth means that the map is standard smooth at every prime, where standard smoothness at a prime is checked after a further principal shrinking.

[F17]

Standard smooth presentations and locally standard smooth maps: the presentation definition allows $c = 0$ , and the case $n = c = 0$ with localization element $g = 1$ presents $R$ over itself.

[F18]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function. It is declared here because [F13], [F14], and [F15] carry that assumption; no additional simultaneous choice or DC is used.

[F19]

embedding dimension and regular local ring: a nonzero Noetherian local ring is regular local exactly when its embedding dimension equals its Krull dimension.

Proof

technique · direct

1.1F1F2given

Finite-type affine charts. If $X$ is empty, its local regularity and smoothness conditions are vacuous. Otherwise, by [F1], every point has an affine open neighbourhood $U = Spec A$ with $A$ of finite type over $k$ . By [F2], $A ≅ k [t_{1}, \dots, t_{n}] / I$ for some finite $n \geq 0$ and ideal $I$ . We consider all such affine charts, so no simultaneous choice of a chart at every point is made.

2.1F3F4F5F6F7F11step 1.1algebra

Noetherianity of the chart rings. Fix any chart from step 1.1 and write $P = k [t_{1}, \dots, t_{n}]$ . Let $J \subseteq A$ be an ideal and let $\tilde{J}$ be its preimage under the quotient map $P \to A$ . By [F3], $\tilde{J}$ has a finite generating list in $P$ ; the images of that list generate $J$ because $P \to A$ is surjective. Thus every ideal of $A$ is finitely generated. By [F4]–[F6], the left regular module of $A$ is Noetherian and $A$ is a Noetherian ring. The zero quotient $A = 0$ is also Noetherian (its only ideal is generated by the finite list $[0]$ ) and is regular vacuously by [F11]; its spectrum is empty. Since these charts cover $X$ , [F7] makes $X$ locally Noetherian.

3.1F8F9F10F11step 2.1

Regularity transfers to each affine chart ring. Assume $X$ is regular and fix $U = Spec A$ from step 1.1. For any $p \in Spec A$ , let $x$ be its point in $X$ . The restricted structure sheaf [F8] identifies the stalk on $U$ with $O_{X, x}$ , and [F9] identifies it with $A_{p}$ . Since $X$ is locally Noetherian by step 2.1, [F10] and the regularity hypothesis make $A_{p}$ regular local. We have already shown that $A$ is Noetherian, so [F11] gives that $A$ is a regular ring.

3.2F8F9F10F11F12F14F15F16step 2.1algebra

Smooth implies regular. Assume $X \to Spec k$ is smooth. Fix any affine chart $U = Spec A$ . By [F15], each point of $U$ has a neighborhood on which the structure map has a standard smooth presentation; restricting these neighborhoods within $U$ and using [F16] shows that $k \to A$ is locally standard smooth. By [F14], $A$ is geometrically regular over $k$ . The finitely generated extension $K = k$ is included in [F12], and the canonical isomorphism $A \otimes_{k} k ≅ A$ therefore makes $A$ a regular Noetherian ring. By [F11], every $A_{p}$ is regular local; [F8]–[F10] identify this with regularity of each corresponding point of $X$ . Since $X$ is locally Noetherian by step 2.1, $X$ is regular. This proves the reverse implication.

4.1F11F12F13step 3.1given

A regular chart is geometrically regular. Let $U = Spec A$ be any chart and assume $X$ is regular. Step 3.1 makes $A$ a regular finite-type $k$ -algebra. For every finitely generated field extension $K / k$ , [F13] gives that $A \otimes_{k} K$ is regular, and [F11] includes Noetherianity in the meaning of regular ring. Hence the defining condition [F12] holds and $A$ is geometrically regular over $k$ .

5.1F14step 4.1

Geometric regularity gives local standard smoothness. By [F14], the structure map $k \to A$ for each chart in step 4.1 is locally standard smooth.

6.1F15step 1.1step 5.1

Regular implies smooth. At each point of $X$ , take a chart from step 1.1. The local standard-smooth presentations supplied by step 5.1 make the morphism smooth at that point under [F15]. This proves the forward implication at all points, including nonclosed points.

7.1

Boundary and nilpotent checks. For $X = Spec k$ , the local ring is $k$ , with maximal ideal zero and both dimension and embedding dimension zero, so it is regular by [F19]. The map $k \to k$ has the standard smooth presentation with $n = c = 0$ and $g = 1$ by [F17], so $X$ is smooth. For $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ , the unique prime is $(ϵ)$ because every prime contains the nilpotent $ϵ$ and an element with nonzero constant term is a unit. The ring is two-dimensional over $k$ , hence Noetherian; its local dimension is zero, whereas $(ϵ) / (ϵ)^{2}$ is one-dimensional, so its embedding dimension is one and [F19] shows it is not regular. Consequently it is not geometrically regular, since [F12] includes the extension $K = k$ . By [F14] its structure map is not locally standard smooth, and [F15] says it is not smooth. Thus nilpotents are retained, and the theorem does not silently replace $D_{k}$ by its reduced point. Step 6.1 proves regular $\Rightarrow$ smooth, while step 3.2 proves smooth $\Rightarrow$ regular. AC is used only through the stated suppliers [F13]–[F15]; the proof treats one chart or point at a time, and no DC is invoked. [F12, F13, F14, F15, F17, F18, F19, step 6.1, step 3.2, given, algebra] $□$

Source qualification

Stacks Project Lemma 33.12.3 (tag 038V), lines 23–38, gives the affine-chart characterization of geometric regularity and its finite purely inseparable field tests. Lemma 33.12.6 (tag 038X), lines 23–28, proves that geometric regularity at a point is equivalent to smoothness there for a locally finite-type scheme. The latter statement does not assume perfectness; perfectness enters this item through the regular-algebra scalar-extension theorem. Stacks Algebra Lemma 10.166.1 (tag 0381), lines 22–36, gives the finitely-generated-field versus finite-purely-inseparable test. The stronger arbitrary-field scalar-extension assertion used here is supplied by the fully proved library item Regular algebras over a perfect field are geometrically regular via Field tests for geometric regularity.

Milne, Algebraic Geometry, Chapter 10 supplement, §f, item 10.64 (printed p. 18 / PDF page 18), states that a regular variety over a perfect field is smooth and that a smooth variety is regular. Milne's “variety” conventions are narrower than the present claim about arbitrary finite-type schemes and do not include this proof's nonreduced dual-number boundary; that citation is corroboration for the classical case, not a substitute for the scheme-level chart argument above.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

Purely inseparable field algebras separate regularity from smoothness

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field of characteristic $p > 0$ and put $k^{p} = {x^{p} : x \in k}$ . Let $a \in k$ satisfy $a \notin k^{p}$ , and put

$L = k [t] / (t^{p} - a), α = t + (t^{p} - a) \in L .$

Then $L$ is a field, the structural map $k \to L$ is injective, $α^{p} = a$ and $L = k [α]$ , so $L$ is a field extension of $k$ ; the affine $k$ -scheme $X = Spec L$ is of finite type over $k$ and regular; for every field extension $K / k$ and every $β \in K$ with $β^{p} = a$ there is a $K$ -algebra isomorphism

$L \otimes_{k} K ≅ K [u] / (u^{p}),$

where $K [u] / (u^{p})$ is a Noetherian local ring with unique prime $(u)$ , Krull dimension $0$ and embedding dimension $1$ , and is not regular; and consequently $X \to Spec k$ is not smooth, although $X$ is regular. The failure is witnessed already by $K = L$ and $β = α$ . No reduction, radicalization or Frobenius twist is applied: the displayed isomorphism is of the actual tensor product, and the nilpotent class $u$ is retained.

Facts & Assumptions

Given: A field $k$ of characteristic $p > 0$ , the set $k^{p} = {x^{p} : x \in k}$ , an element $a \in k$ with $a \notin k^{p}$ , the ring $L = k [t] / (t^{p} - a)$ with the class $α$ of $t$ , and the Axiom of Choice.

[F1]

If $a$ is not a $p$ th power in a characteristic- $p$ field, then $x^{p^{n}} - a$ is irreducible for every $n \geq 1$ : for a field $F$ of characteristic $p > 0$ , an element $c \in F$ that is not a $p$ th power, and $n \geq 1$ , the polynomial $x^{p^{n}} - c$ is irreducible in $F [x]$ .

[F2]

For a nonconstant $p$ in $F [x]$ , the ideal $(p)$ is maximal and $F [x] / (p)$ is a field exactly when $p$ is irreducible: for a field $F$ and a nonconstant $f \in F [x]$ , the quotient ring $F [x] / (f)$ is a field exactly when $f$ is irreducible.

[F3]

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras: a commutative $R$ -algebra $A$ is of finite type over $R$ when $A = R [a_{1}, \dots, a_{n}]$ for some finite list, equivalently when $A$ is isomorphic to a quotient $R [x_{1}, \dots, x_{n}] / a$ .

[F4]

Finite type is affine-local on source and target: a quasi-compact morphism locally of finite type is of finite type; equivalently, over each affine target open it may be tested on a finite affine source cover.

[F5]

Smoothness over a field by geometric regularity: under AC, for a finite-type $k$ -scheme $X$ , the morphism $X \to Spec k$ is smooth if and only if for every field extension $K / k$ every local ring of the scheme-theoretic base change $X_{K}$ is regular.

[F6]

Affine charts after extension of the ground field: for a field extension $K / k$ and a $k$ -scheme $X$ , the inverse image under $X_{K} \to X$ of every affine open $U = Spec A$ of $X$ is $Spec (A \otimes_{k} K)$ , and these affine charts cover $X_{K}$ .

[F7]

Presentations and localization under base extension: for a unital ring map $A \to C$ and an ideal $I \subseteq A [t_{i}]$ , there is a ring isomorphism $(A [t_{i}] / I) \otimes_{A} C ≅ C [t_{i}] / I C [t_{i}]$ ; no flatness, finite-generation or nonzero-ring hypothesis is required.

[F8]

Affine schemes are contravariantly equivalent to commutative rings: $Spec$ is a contravariant equivalence from commutative rings to affine schemes with quasi-inverse global sections, so a ring isomorphism induces an isomorphism of affine schemes.

[F9]

embedding dimension and regular local ring: for a nonzero commutative Noetherian local ring $(R, m, κ)$ , $edim R = \dim_{κ} (m / m^{2})$ , and $R$ is regular local exactly when $edim R = \dim R$ .

[F10]

An algebra that is finite dimensional as a vector space over a field is a Noetherian ring: a commutative algebra over a field whose underlying vector space is finite dimensional is a Noetherian ring.

[F11]

Fields and $Z$ are Noetherian, and so are their polynomial rings in finitely many variables: every field is a Noetherian ring.

[F12]

Locally Noetherian and Noetherian schemes: a scheme is locally Noetherian when it has an affine open cover by spectra of Noetherian rings.

[F13]

Regular points of locally Noetherian schemes: on a locally Noetherian scheme $X$ , a point $x$ is regular when the local ring $O_{X, x}$ is a regular local ring.

[F14]

The stalk of the affine structure sheaf at a prime is A_p: for a prime $p \in Spec A$ there is a canonical isomorphism $O_{Spec A, p} ≅ A_{p}$ .

[F15]

Krull dimension of a nonzero ring: for a nonzero commutative ring, the Krull dimension is the supremum of the lengths of strict chains of prime ideals.

[F16]

The binomial theorem over an arbitrary commutative ring: $(x + y)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{k} y^{n - k}$ in every commutative ring, with natural-number coefficients acting by repeated addition.

[F17]

A prime $p$ divides $(\binom{p}{k})$ for $0 < k < p$ : if $p$ is prime and $0 < k < p$ , then $p$ divides $(\binom{p}{k})$ .

[F18]

Field: a field has $0 \neq 1$ , and every nonzero element $x$ has a multiplicative inverse $x^{- 1}$ with $x x^{- 1} = 1$ .

[F19]

Field homomorphism and embedding: a field homomorphism $φ : F \to G$ satisfies $φ (x y) = φ (x) φ (y)$ and $φ (1_{F}) = 1_{G}$ , and an embedding is an injective field homomorphism.

[F20]

A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective: a field $F$ is perfect exactly when $char F = 0$ , or $char F = p > 0$ and the Frobenius map $a \mapsto a^{p}$ is surjective.

[F21]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function; it is declared here because [F5] carries that assumption, and no further simultaneous choice is used below.

Proof

technique · direct

1.1givenF1F2F3F18F19algebra

The class $α$ of $t$ satisfies $α^{p} = a$ , and $L = k [α]$ is a quotient of $k [t]$ , hence of finite type over $k$ by [F3]. The polynomial $t^{p} - a$ is nonconstant, and [F1] with $F = k$ , $c = a$ and $n = 1$ makes it irreducible in $k [t]$ ; by [F2] the quotient $L$ is therefore a field. The structural map $φ : k \to L$ is a field homomorphism by [F19] and $1_{L} \neq 0_{L}$ by [F18]; for $x \neq 0$ in $k$ the inverse relation $x x^{- 1} = 1$ of [F18] is preserved by [F19], giving $φ (x) φ (x^{- 1}) = φ (1) = 1 \neq 0$ , so $φ (x) \neq 0$ . Hence $φ$ is injective and exhibits $k$ as a subfield of $L$ .

1.2F10algebra

For any field $F$ , the quotient ring $R_{F} = F [u] / (u^{p})$ has the classes of $1, u, \dots, u^{p - 1}$ as an $F$ -basis, because division by the monic polynomial $u^{p}$ leaves unique remainders of degree less than $p$ . Hence $R_{F}$ is a commutative $F$ -algebra of dimension $p$ over $F$ , and it is a Noetherian ring by [F10]. Since $p \geq 2$ , the ring $R_{F}$ is nonzero and $u \neq 0$ in $R_{F}$ .

1.3F16F17algebra

Let $K / k$ be a field extension and let $β \in K$ satisfy $β^{p} = a$ . Then $K$ has characteristic $p$ , and [F16] with $x = z$ , $y = - β$ and $n = p$ gives $(z - β)^{p} = \sum_{k = 0}^{p} (\binom{p}{k}) z^{p - k} (- β)^{k}$ in the commutative ring $K [z]$ ; the intermediate coefficients $(\binom{p}{k})$ with $0 < k < p$ vanish by [F17], while $(\binom{p}{0}) = (\binom{p}{p}) = 1$ , so $(z - β)^{p} = z^{p} + (- β)^{p} = z^{p} - β^{p} = z^{p} - a$ , where $(- β)^{p} = - β^{p}$ holds in every characteristic.

2.1F4step 1.1

$X = Spec L$ is of finite type over $k$ : over the affine base $Spec k$ the source is covered by the single affine chart $Spec L$ , and the ring map $k \to L$ is of finite type by step 1.1, so [F4] applies.

2.2F9F11F12F13F14F15step 1.1

$X$ is locally Noetherian and regular. Since $L$ is a field, [F11] makes $L$ a Noetherian ring, and the one-chart cover ${Spec L}$ witnesses local Noetherianity by [F12]. The field $L$ has the single prime ideal $(0)$ , so $\dim L = 0$ by [F15]; its maximal ideal is $(0)$ , so $m / m^{2} = 0$ and $edim L = 0 = \dim L$ , which makes $L$ a regular local ring by [F9]. The single point $(0)$ of $X$ has local ring $O_{X, (0)} ≅ L_{(0)} = L$ by [F14], so it is regular by [F13]; being the only point, it makes $X$ regular.

2.3F6F7step 1.1

For a field extension $K / k$ , write $X_{K}$ for the base change of $X$ along $Spec K \to Spec k$ . The inverse image of the affine open $U = X$ under $X_{K} \to X$ is $Spec (L \otimes_{k} K)$ by [F6], and it is all of $X_{K}$ ; applying [F7] with $A = k$ , the variable $t$ , the ideal $I = (t^{p} - a) \subseteq k [t]$ and $C = K$ gives a ring isomorphism $L \otimes_{k} K ≅ K [z] / (z^{p} - a)$ .

2.4step 1.2algebra

For any field $F$ , the ring $R_{F} = F [u] / (u^{p})$ is local with unique prime $(u)$ . By the basis of step 1.2 every element of $R_{F}$ has a unique expression $c_{0} + c_{1} u + \dots + c_{p - 1} u^{p - 1}$ . If $c_{0} \neq 0$ , write the element as $c_{0} (1 + u h)$ ; then $(u h)^{p} = u^{p} h^{p} = 0$ , so $1 + u h$ has inverse $1 - u h + (u h)^{2} - \dots + (- u h)^{p - 1}$ and the element is a unit. If $c_{0} = 0$ , the element lies in $(u)$ and is not a unit, because $u \neq 0$ is nilpotent and a nilpotent element of a nonzero commutative ring cannot be a unit. Hence $(u)$ is the unique maximal ideal. Since $u^{p} = 0$ , every prime ideal contains $u$ and hence contains $(u)$ ; and $(u)$ is prime because $R_{F} / (u) ≅ F$ is a field. So $(u)$ is the only prime ideal.

3.1F8step 1.1step 1.2step 1.3step 2.3step 2.4algebra

Fix a field extension $K / k$ and $β \in K$ with $β^{p} = a$ . Combining steps 2.3 and 1.3 and substituting $u = z - β$ gives $K$ -algebra isomorphisms $L \otimes_{k} K ≅ K [z] / (z^{p} - a) = K [z] / ((z - β)^{p}) ≅ K [u] / (u^{p}) = R_{K}$ , with $R_{K}$ as in steps 1.2 and 2.4; in particular, taking $K = L$ and $β = α$ , which is legitimate by step 1.1, the base change $X_{L} = Spec (L \otimes_{k} L)$ is isomorphic to $Spec R_{L}$ by [F8].

3.2F9F15step 1.2step 2.4algebra

For any field $F$ , the ring $R_{F}$ is not a regular local ring. It is nonzero, Noetherian by step 1.2, and local with maximal ideal $(u)$ by step 2.4. As $(u)$ is the only prime ideal, [F15] gives $\dim R_{F} = 0$ . Every element of $(u)$ is congruent modulo $(u)^{2}$ to $c u$ for some $c \in F$ by the basis of step 1.2, and $u \notin (u)^{2}$ because $u$ has basis coefficient $1$ in degree $1$ while every element of $(u)^{2}$ has basis coefficients only in degrees at least $2$ ; hence $(u) / (u)^{2}$ is one-dimensional over $F$ with basis the class of $u$ , and $edim R_{F} = 1$ . Thus $edim R_{F} = 1 \neq 0 = \dim R_{F}$ , and [F9] shows that $R_{F}$ is not regular local.

4.1F14step 2.4step 3.1step 3.2

The scheme $X_{L}$ has a nonregular local ring. By step 3.1, $X_{L} ≅ Spec R_{L}$ , and by step 2.4 the ring $R_{L}$ is local with unique maximal ideal $(u)$ , so $Spec R_{L}$ consists of the single point $(u)$ . Its local ring is $O_{Spec R_{L}, (u)} ≅ (R_{L})_{(u)} = R_{L}$ by [F14], the last equality because every element outside $(u)$ is a unit by step 2.4. Step 3.2 says that $R_{L}$ is not a regular local ring, so this local ring of $X_{L}$ is not regular.

5.1F5F21step 2.1step 2.2step 4.1given

$X$ is not smooth over $k$ . By [F5], the AC-carrying geometric-regularity characterization, $X \to Spec k$ is smooth only if every local ring of every base change $X_{K}$ , with $K / k$ a field, is regular. The field extension $K = L$ of step 1.1 and the nonregular local ring of $X_{L}$ exhibited in step 4.1 contradict that condition, so $X \to Spec k$ is not smooth. Meanwhile $X$ is of finite type over $k$ by step 2.1 and regular by step 2.2, so an imperfect base field separates regularity from smoothness.

6.1F5F9F20F21givenstep 1.1step 1.2step 2.2step 2.3step 2.4step 3.1step 3.2step 5.1algebra∎

Boundary and hypothesis checks. (i) The hypothesis $a \notin k^{p}$ is exactly what step 1.1 needs, and by [F20] the existence of some such $a$ is equivalent to imperfection of $k$ in characteristic $p$ ; a perfect field of characteristic $p$ has no such $a$ , so the conclusion of step 5.1 cannot arise there. (ii) If instead $a = b^{p}$ lies in $k^{p}$ , then $t^{p} - a = (t - b)^{p}$ and $L ≅ k [u] / (u^{p})$ is the nonreduced ring of steps 1.2 and 3.2, which is not even regular; so the hypothesis is used, not decorative. (iii) The nilpotent class $u$ survives: steps 2.3 and 3.1 are isomorphisms of the actual tensor product, and no reduction or radical is taken, so the nonreduced base change is retained. (iv) The extension is genuinely needed: for $K = k$ the base change is $X$ itself, which is regular by step 2.2, and the witness $K = L$ is the field generated over $k$ by one $p$ th root of $a$ . (v) At $p = 2$ the ring $R_{F} = F [u] / (u^{2})$ is the classical dual-number ring of dimension $0$ and embedding dimension $1$ ; steps 1.2 through 3.2 divide by nothing except the monic polynomial $u^{p}$ , so characteristic $2$ is included. (vi) AC is declared in [F21] and used only through [F5]; the proof exhibits the single extension $K = L$ and one point, so it makes no simultaneous choice and invokes no dependent choice.

Source qualification

Stacks Project Example 33.12.7 (tag 038S) takes $k = F_{p} (t)$ and observes that $Spec (k [x] / (x^{p} - t))$ is a regular variety over $k$ that is not geometrically reduced, the base change to $k (t^{1 / p})$ becoming $k (t^{1 / p}) [ϵ] / (ϵ^{p})$ . That example is the case $a = t \notin k^{p}$ of the statement above. The example is used here as the literature source for the phenomenon only: the field, regularity, base-change and non-smoothness assertions are each proved from the library's own suppliers in steps 1.1--5.1, and the general statement over an arbitrary field $k$ of characteristic $p$ with an arbitrary $a \notin k^{p}$ is not asserted by that example. The equivalence between smoothness over a field and geometric regularity invoked in step 5.1 is the one proved in Smoothness over a field by geometric regularity, not an external citation. The dual-number case $p = 2$ is the published example of dual numbers not regular, which records the same dimension-zero, embedding-dimension-one computation for $u^{2} = 0$ .

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-sol)audited 2026-09-30Open item page →

The scheme-theoretic tangent cone at a point

Definition

Let $X$ be a locally Noetherian scheme and $x \in X$ . Write $A = O_{X, x}$ , let $m_{x}$ be its maximal ideal, and put $κ (x) = A / m_{x}$ , as in The intrinsic cotangent space. The associated graded ring for the maximal-ideal filtration is

{gr}_{m_{x}} (A) = ⨁_{n \geq 0} m_{x}^{n} / m_{x}^{n + 1}

with multiplication induced from $A$ (The associated graded ring and associated graded module of an ideal-adic filtration).

For every $n$ , multiplication by $m_{x}$ sends $m_{x}^{n}$ into $m_{x}^{n + 1}$ , so it acts trivially on the degree- $n$ quotient. Thus the scalar action on each graded piece factors canonically through $κ (x)$ , and the degree-zero piece is $A / m_{x} = κ (x)$ . This makes the associated graded ring a graded $κ (x)$ -algebra without choosing a coefficient field in $A$ .

The scheme-theoretic tangent cone of $X$ at $x$ is the affine $κ (x)$ -scheme

{Cone}_{x} (X) : = Spec ⁣ ({gr}_{m_{x}} (A)) ⟶ Spec κ (x) .

We use ${Cone}_{x} (X)$ to distinguish this scheme from the cotangent space denoted $C_{x} X$ in the preceding definition. Here Spec carries its affine scheme structure (The underlying space of an affine spectrum, Affine schemes and their coordinate rings), and the map to $Spec κ (x)$ is the structure morphism of a scheme over $κ (x)$ (Schemes and morphisms over a base) induced by the degree-zero inclusion. The grading is retained as part of the cone presentation. The reduction $({Cone}_{x} (X))_{r e d}$ is a closed subscheme that can differ from ${Cone}_{x} (X)$ ; the definition uses the full associated graded ring, without quotienting by its nilpotents (The reduction of a scheme).

If $A$ is already a field, then $m_{x} = 0$ , every positive graded piece vanishes, and ${Cone}_{x} (X) = Spec κ (x)$ .

For the closed point $x = (ϵ)$ of the dual-numbers scheme $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ (The affine scheme of dual numbers), every element $a + b ϵ$ with $a \neq 0$ is a unit, so $O_{D_{k}, x} = k [ϵ] / (ϵ^{2})$ and $m_{x} = (ϵ)$ . Its associated graded pieces are $k$ in degree $0$ , $k ϵ$ in degree $1$ , and zero in every degree $n \geq 2$ ; the degree-one class squares to zero. Hence

{gr}_{(ϵ)} O_{D_{k}, x} ≅ k [ϵ] / (ϵ^{2}), {Cone}_{x} (D_{k}) ≅ D_{k} .

The nilradical is $(ϵ)$ , so the reduction is $Spec k$ . Thus ${Cone}_{x} (D_{k})$ and its reduction have the same one-point topological space but different structure sheaves.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

All initial forms define the tangent cone

Statement

Let $k$ be any field, $n \in N$ , $P = k [t_{1}, \dots, t_{n}]$ , and $q = (t_{1}, \dots, t_{n})$ . Let $I \subseteq q$ , $A = P / I$ , $X = Spec A$ , and let $x$ be the rational origin, so $m = q / I$ and $A / m = k$ . For a nonzero polynomial $f = \sum_{d} f_{d}$ decomposed into total-degree homogeneous parts, let $f_{\min}$ be its lowest nonzero homogeneous part, and let ${in}_{q} (I)$ be the ideal generated by all $f_{\min}$ with $0 \neq f \in I$ . The canonical graded $k$ -algebra map $P / {in}_{q} (I) ⟶ {gr}_{m_{x}} O_{X, x}$ sending each variable to its degree-one initial class, is an isomorphism. Consequently ${Cone}_{x} (X) ≅ Spec (P / {in}_{q} (I)) .$ If $I = (f)$ with $f \neq 0$ , then ${in}_{q} (I) = (f_{\min})$ . The initial forms of a chosen generating set for $I$ need not generate ${in}_{q} (I)$ .

Facts & Assumptions

Given: A field $k$ , a finite $n \in N$ , and an ideal $I \subseteq q = (t_{1}, \dots, t_{n})$ of $P = k [t_{1}, \dots, t_{n}]$ .

[F1]

The scheme-theoretic tangent cone at a point: for a locally Noetherian scheme and a point $x$ , the scheme-theoretic tangent cone is $Spec ({gr}_{m_{x}} O_{X, x})$ over $κ (x)$ .

[F2]

Cotangent spaces commute with localization at a rational point: when $A / m = k$ , localization induces an isomorphism $m^{r} / m^{r + 1} \to (m A_{m})^{r} / (m A_{m})^{r + 1}$ for every $r \in N$ .

[F3]

The associated graded ring and associated graded module of an ideal-adic filtration: ${gr}_{J} (R) = ⨁_{r \geq 0} J^{r} / J^{r + 1}$ , with multiplication induced by multiplication in $R$ .

[F4]

Finite-variable polynomial algebras over fields are Noetherian by finite generators: for a field $k$ and finite $n$ , every ideal of $P = k [t_{1}, \dots, t_{n}]$ has a finite generating list.

[F5]

Left and right Noetherian rings: a ring is left Noetherian when its left regular module is Noetherian.

[F6]

Noetherian modules: every submodule is finitely generated: every submodule of a Noetherian module is finitely generated.

[F7]

Left, right and two-sided ideals: an additive subgroup of the left regular module is a left ideal exactly when it is closed under left multiplication; in a commutative ring the left, right and two-sided ideals agree.

[F8]

Submodule of a module: a subset of a module is a submodule when it is an additive subgroup closed under scalar multiplication.

[F9]

The sum $I + J$ and product $I J$ of two-sided ideals: products of ideals consist of finite sums of products of their elements.

[F10]

The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ : $P / I$ consists of additive cosets with $(a + I) (b + I) = a b + I$ .

[F11]

The canonical projection $R \to R / I$ is a surjective ring homomorphism with kernel $I$ : the canonical projection $P \to P / I$ is a surjective ring homomorphism with kernel $I$ .

[F12]

$R / M$ is a field if and only if $M$ is a maximal ideal: $R / M$ is a field if and only if $M$ is a maximal ideal.

[F13]

Every maximal ideal of a commutative ring is prime: every maximal ideal of a commutative ring is prime.

[F14]

The stalk of the affine structure sheaf at a prime is A_p: for $p \in Spec R$ , $O_{Spec R, p} ≅ R_{p}$ .

[F15]

$R_{p}$ is local with unique maximal ideal $p R_{p}$ : $R_{p}$ is a nonzero local ring with unique maximal ideal $p R_{p}$ .

[F16]

Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ : each polynomial in finitely many variables has a unique finite monomial expansion, and each monomial has its total degree.

[F17]

homogeneous polynomial and homogeneous ideal: a polynomial is homogeneous of degree $d$ when every occurring monomial has total degree $d$ .

[F18]

The ideal generated by a subset and principal ideals: the ideal generated by a subset is the smallest ideal containing that subset; the ideal generated by $f$ is written $(f)$ .

[F19]

In a commutative ring, $(S)$ consists of finite sums $\sum r_{i} s_{i}$ , and $(a) = R a$ : in a commutative ring, an ideal generated by a set consists of finite sums of ring multiples of its generators; in particular, elements of $(f)$ are of the form $g f$ .

[F20]

Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring: every field is an integral domain.

[F21]

A polynomial ring in finitely many indeterminates over an integral domain is an integral domain: a polynomial ring in finitely many variables over an integral domain is an integral domain.

[F22]

Locally Noetherian and Noetherian schemes: a scheme is locally Noetherian if it has an affine open cover by spectra of Noetherian rings.

[F23]

Unital left and right modules over a ring; unqualified module means left module: the left regular module has scalar action $r \cdot a = r a$ , given by ring multiplication.

Proof

technique · direct

1.1F4F5F6F7F8F10F11F22F23givenalgebra

Noetherian affine scheme. The finite-generation theorem [F4] says every ideal of $P$ is finitely generated. The left regular modules $_{P} P$ and $_{A} A$ have scalar action by multiplication [F23]; hence [F7] and [F8] identify their submodules with left ideals, which are ideals because both rings are commutative. Thus every submodule of $_{P} P$ is finitely generated; [F6] makes this module Noetherian and [F5] makes $P$ a Noetherian ring. Put $A = P / I$ and let $π : P \to A$ be the quotient map [F10, F11]. For an ideal $K \subseteq A$ , its preimage $J = π^{- 1} (K)$ is an ideal of $P$ : additivity and closure under multiplication follow by applying the homomorphism $π$ and the ideal property of $K$ . Since $P$ is Noetherian, [F7, F8] identify $J$ with a submodule of $_{P} P$ , and [F6] gives a finite generating list $p_{1}, \dots, p_{s}$ for $J$ . Every $a \in K$ has a representative $p \in P$ by [F11], and then $p \in J$ ; writing $p = \sum_{i} c_{i} p_{i}$ shows $a = \sum_{i} π (c_{i}) π (p_{i})$ . Each $π (p_{i})$ lies in $K$ , so they generate $K$ . Thus every ideal of $A$ is finitely generated. By [F7, F8, F23], every submodule of $_{A} A$ is an ideal; it too is finitely generated, so [F6] and [F5] show that $A$ is Noetherian. The affine cover $X = Spec A$ itself makes $X$ locally Noetherian by [F22]. Each argument fixes one ideal or submodule and establishes existence of a finite generating list for it; no simultaneous family of lists is chosen, and no choice principle is used.

1.2F3F9F16F17givenalgebra

The polynomial associated graded ring. Let $P_{d}$ be the homogeneous polynomials of total degree $d$ . By [F16, F17], every polynomial is a finite sum of homogeneous parts. For every $r \geq 0$ , $q^{r} = ⨁_{d \geq r} P_{d}$ : a product of $r$ elements of $q$ has no term of degree below $r$ , while every monomial of degree at least $r$ factors as $r$ variables times a monomial; the finite-sum description of ideal products [F9] gives the two inclusions. Thus the degree- $d$ map $P_{d} \to q^{d} / q^{d + 1}$ is an isomorphism. Since multiplication in the associated graded ring is induced from $P$ [F3], these maps identify ${gr}_{q} P$ with $P$ as graded rings. In degree zero this is the identification $P_{0} = k$ .

1.3F18F19F20F21givenalgebra

The principal case. Suppose $I = (f)$ for $0 \neq f \in P$ . By [F20, F21], $P$ is a domain. If $0 \neq h \in I$ , [F19] writes $h = g f$ ; since $h \neq 0$ , $g \neq 0$ . Let $d$ and $e$ be the lowest degrees of $f$ and $g$ . The lowest-degree part of $g f$ is $g_{\min} f_{\min}$ : all other products of homogeneous parts have degree greater than $d + e$ , and this product is nonzero in the domain $P$ . Thus every lowest form of a nonzero element of $I$ lies in $(f_{\min})$ , while $f_{\min}$ is itself one of those forms. By [F18], ${in}_{q} (I) = (f_{\min})$ .

2.1F1F10F11F12F13F14F15F16step 1.1givenalgebra

The rational stalk. Evaluation at the origin sends every $t_{i}$ to zero and each constant to itself. By the unique monomial expansion [F16], its kernel in $P$ is $q$ . As $I \subseteq q$ , evaluation factors through $A$ with kernel $m = q / I$ and quotient $A / m ≅ k$ ; [F12] makes $m$ maximal, and [F13] makes it prime. The stalk formula [F14] gives $O_{X, x} ≅ A_{m}$ , whose unique maximal ideal is $n = m A_{m}$ by [F15]. By [F1] and step 1.1, ${Cone}_{x} (X) = Spec ({gr}_{n} A_{m})$ .

2.2F3F9F10F11F16F17F18step 1.2givenalgebra

Quotient filtration and its kernel. The quotient operations [F10, F11] give $m^{r} = (q^{r} + I) / I$ for every $r$ , by induction on $r$ . Projection therefore defines a graded ring map $ϕ : {gr}_{q} P \to {gr}_{m} A$ , which is surjective in each degree because every class in $m^{r} / m^{r + 1}$ is represented by an element of $q^{r}$ . Use step 1.2 to identify its source with $P$ . For a homogeneous $h \in P_{d}$ , its class lies in the degree- $d$ kernel exactly when $h \in I + q^{d + 1}$ . Write $h = g + u$ with $g \in I$ and $u \in q^{d + 1}$ . If $h \neq 0$ , then $g = h - u$ has no terms below degree $d$ and its degree- $d$ part is $h$ , so $g_{\min} = h$ . Conversely, if $0 \neq g \in I$ has lowest part $g_{\min}$ of degree $d$ , then $g - g_{\min} \in q^{d + 1}$ , so $g_{\min}$ lies in the degree- $d$ kernel. The kernel is a homogeneous ideal; it contains every lowest form, and each of its homogeneous elements is itself a lowest form. It is therefore exactly ${in}_{q} (I)$ , including when $I = 0$ .

3.1F1F2F3step 2.1step 2.2

The local tangent cone. The localization maps of [F2] give an isomorphism in every degree from ${gr}_{m} A$ to ${gr}_{n} A_{m}$ . They preserve multiplication because each degree map is induced by the ring map $A \to A_{m}$ and the products in both associated graded rings are induced by ring multiplication [F3]. Thus step 2.2 yields the canonical graded $k$ -algebra isomorphism $P / {in}_{q} (I) ≅ {gr}_{m_{x}} O_{X, x}$ , sending each variable to its degree-one initial class. By [F1] and step 2.1, its spectrum is the scheme-theoretic tangent cone at $x$ .

4.1F9F16F17F18F19step 1.2step 2.2givenalgebra∎

A generating list need not suffice, and boundary cases. For $P = k [X, Y, Z]$ , set $f_{1} = X Y$ , $f_{2} = X Z + Z (Y^{2} - Z^{2})$ , and $I = (f_{1}, f_{2}) \subseteq (X, Y, Z)$ , so the origin belongs to $X = Spec (P / I)$ . The initial forms of this generating list are $X Y$ and $X Z$ , which generate an ideal contained in $(X)$ . But $h = Y Z (Y^{2} - Z^{2}) = Y f_{2} - Z f_{1}$ is a nonzero homogeneous element of $I$ , so it is its own lowest form and belongs to ${in}_{(X, Y, Z)} (I)$ . Since $(X Y, X Z) \subseteq (X)$ while $h \notin (X)$ (its image modulo $(X)$ is the same nonzero polynomial), the two displayed initial forms do not generate the initial ideal. This witness works in every characteristic because the distinct monomials $Y^{3} Z$ and $Y Z^{3}$ cannot cancel. For $n = 0$ , $q = 0$ forces $I = 0$ and $P = A = k$ , so the graded ring is $k$ in degree zero and the initial ideal is zero; the nonzero principal case is excluded. For $n = 1$ , each $q^{r} / q^{r + 1}$ is generated by $t_{1}^{r}$ , consistent with steps 1.2 and 2.2. Degree zero has $P_{0} = k$ and $I \subseteq q$ contains no nonzero constant; degree one is covered by the same kernel calculation with $q^{2}$ . If $I = 0$ , the set of nonzero elements of $I$ is empty and its generated initial ideal is zero. The condition $I \subseteq q$ implies $I \neq P$ , so the origin exists and $X$ is not empty. The lemma asserts no iff, so there are no converse directions to check.

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The scheme-theoretic linear span of the tangent cone

Statement

Let $X$ be a locally Noetherian $k$ -scheme and let $x \in X (k)$ be a $k$ -rational point. Put $C_{x} X = m_{x} / m_{x}^{2}$ and $T_{x} X = {Hom}_{k} (C_{x} X, k)$ . The module $C_{x} X$ is finite-dimensional; write $A (T_{x} X) = Spec S_{k} (C_{x} X)$ for the affine space associated to $T_{x} X$ , using the canonical evaluation isomorphism $C_{x} X ≅ (T_{x} X)^{\lor}$ . Multiplication in the local ring induces a graded surjection

S_{k} (C_{x} X) ⟶ {gr}_{m_{x}} O_{X, x}

whose degree-one map is the identity on $C_{x} X$ . It therefore defines a closed immersion of the scheme-theoretic tangent cone ${Cone}_{x} (X)$ into $A (T_{x} X)$ , and no proper linear closed subscheme of this affine space contains ${Cone}_{x} (X)$ scheme-theoretically. Here a linear closed subscheme means one defined by an ideal generated by a vector subspace of degree-one forms; the full scheme structure of the cone is retained.

Facts & Assumptions

Given: A locally Noetherian $k$ -scheme $X$ and a $k$ -rational point $x$ .

[F1]

The scheme-theoretic tangent cone at $x$ is $Spec ({gr}_{m_{x}} O_{X, x})$ (The scheme-theoretic tangent cone at a point).

[F2]

The intrinsic tangent space is $T_{x} X = {Hom}_{κ (x)} (C_{x} X, κ (x))$ (The intrinsic Zariski tangent space); at a rational point $κ (x) = k$ .

[F3]

A locally Noetherian scheme has an affine open cover by spectra of Noetherian rings (Locally Noetherian and Noetherian schemes).

[F4]

A point of an affine scheme corresponds to a prime ideal (Prime ideals and maximal ideals in a commutative ring).

[F5]

The left regular module of a left Noetherian ring is Noetherian (Left and right Noetherian rings).

[F6]

Every submodule of a Noetherian module is finitely generated (Noetherian modules: every submodule is finitely generated).

[F7]

In a commutative ring the left, right and two-sided ideal notions agree (Left, right and two-sided ideals).

[F8]

A submodule is an additive subgroup closed under scalar multiplication (Submodule of a module).

[F9]

The stalk of the affine structure sheaf at a prime $p$ is $R_{p}$ (The stalk of the affine structure sheaf at a prime is A_p).

[F10]

The localization is $R_{p} = (R ∖ p)^{- 1} R$ (Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ ).

[F11]

The unique maximal ideal of $R_{p}$ is $p R_{p}$ ( $R_{p}$ is local with unique maximal ideal $p R_{p}$ ).

[F12]

The dual-numbers scheme is $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ and its class $ϵ$ is nilpotent (The affine scheme of dual numbers).

[F13]

The symmetric algebra $S_{k} (V)$ is a commutative graded algebra generated by the degree-one image of $V$ (Symmetric algebra of a vector space).

[F14]

A linear map from $V$ to a commutative $k$ -algebra extends uniquely to an algebra map from $S_{k} (V)$ (Universal property of the symmetric algebra).

Proof

technique · direct

1.1F2F3F4F5F6F7F8F9F10F11givenalgebra

Choose an affine open $U = Spec R$ containing $x$ , available by [F3], and write $x = p$ . Then $R$ is Noetherian, and $p$ is a submodule of the left regular module $R$ by [F4, F7, F8]. By [F5, F6], it has a finite generating list. The stalk and its maximal ideal are $O_{X, x} = R_{p}$ and $m_{x} = p R_{p}$ by [F9, F10, F11], so the images of that list generate $m_{x}$ . Hence $C_{x} X = m_{x} / m_{x}^{2}$ is finite-dimensional over $k$ . By [F2], $T_{x} X$ is its dual, and the finite-dimensional evaluation map $C_{x} X \to (T_{x} X)^{\lor}$ is an isomorphism. Thus the coordinate algebra of $A (T_{x} X)$ is $S_{k} (C_{x} X)$ .

2.1F13F14step 1.1givenalgebra

By step 1.1, $S_{k} (C_{x} X)$ is the coordinate algebra of the affine tangent space. The degree-one quotient $C_{x} X = m_{x} / m_{x}^{2}$ maps to the degree-one part of ${gr}_{m_{x}} O_{X, x}$ by the identity. Its multiplication extends to a graded algebra map $ϕ : S_{k} (C_{x} X) \to {gr}_{m_{x}} O_{X, x}$ by [F13, F14]. In degree $d$ , every element of $m_{x}^{d}$ is a finite sum of products of $d$ elements of $m_{x}$ ; replacing each factor by its class modulo $m_{x}^{2}$ changes each product only by an element of $m_{x}^{d + 1}$ . Thus $ϕ$ is surjective in every degree. In degree one it is the identity, so if $J = \ker ϕ$ , then $J_{1} = 0$ .

3.1F1step 2.1algebra

By [F1], the spectrum of the target of $ϕ$ is ${Cone}_{x} (X)$ . The surjection $ϕ$ gives a closed immersion ${Cone}_{x} (X) ↪ Spec S_{k} (C_{x} X) = A (T_{x} X)$ , with scheme ideal $J$ . If a linear closed subscheme $L$ contains the cone scheme-theoretically, its ideal is generated by a subspace $W \subseteq C_{x} X$ of degree-one forms and must satisfy $W \subseteq J$ . Taking degree-one parts gives $W \subseteq J_{1} = 0$ , hence $W = 0$ and $L = A (T_{x} X)$ . This proves both the embedding and the claimed scheme-theoretic linear span without replacing the cone by its reduction.

4.1F1F12step 2.1algebra∎

If $C_{x} X = 0$ , then the target affine space is $Spec k$ and the surjection in step 2.1 forces every positive graded piece of the local associated graded ring to vanish; the cone is the whole point. For the one-dimensional nonreduced example from [F12] at $x = (ϵ)$ , the local ring is $k [ϵ] / (ϵ^{2})$ and its maximal ideal is $(ϵ)$ , so its associated graded ring is $k [ϵ] / (ϵ^{2})$ . The cone is a doubled origin in $A^{1}$ , while its reduction is only the origin; no nonzero linear equation vanishes on the scheme-theoretic cone. The point hypothesis rules out an empty $X$ at the point under discussion. The argument uses one affine neighborhood and a finite generating list for its one prime ideal, and makes no simultaneous choices, basis selection, or Axiom of Choice. The statement is not an iff.

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Multiplicity of a hypersurface equation at a rational point

Definition

Let $k$ be any field, let $n \in N$ , put $P = k [X_{1}, \dots, X_{n}]$ , and let $a = (a_{1}, \dots, a_{n}) \in k^{n}$ . Suppose $0 \neq f \in P$ and $f (a) = 0$ . Write the unique finite homogeneous decomposition

f (a_{1} + t_{1}, \dots, a_{n} + t_{n}) = \sum_{j \geq 0} f_{j} (t_{1}, \dots, t_{n}),

where $f_{j}$ is homogeneous of total degree $j$ . The multiplicity of the hypersurface equation $f$ at $a$ , denoted ${mult}_{a} (f)$ , is the least $j$ for which $f_{j} \neq 0$ .

Let $m_{a} = (X_{1} - a_{1}, \dots, X_{n} - a_{n})$ and $R = P_{m_{a}} = O_{A_{k}^{n}, a}$ , with maximal ideal $m = m_{a} R$ . For $g \in R$ of finite order, write ${ord}_{m} (g) = d$ when $g \in m^{d} ∖ m^{d + 1} .$ Then ${mult}_{a} (f) = {ord}_{m} (f)$ . In particular, replacing the local equation by $u f$ for a unit $u \in R^{\times}$ leaves its $m$ -adic order unchanged.

This is multiplicity of the equation, unchanged under a local unit. It is not an invariant of the reduced support: for every integer $r \geq 1$ , ${mult}_{a} (f^{r}) = r {mult}_{a} (f)$ .

Facts & Assumptions

Given: A field $k$ , a finite $n \in N$ , a point $a \in k^{n}$ , and a nonzero polynomial $f \in k [X_{1}, \dots, X_{n}]$ with $f (a) = 0$ .

[F1]

Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ : a polynomial has a unique finite monomial expansion; grouping its monomials by total degree gives a unique finite sum of homogeneous parts.

[F2]

homogeneous polynomial and homogeneous ideal: a polynomial is homogeneous of degree $d$ when each occurring monomial has total degree $d$ .

[F3]

Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ : $P_{m_{a}}$ consists of fractions $g / s$ with $s \notin m_{a}$ .

[F4]

$R / M$ is a field if and only if $M$ is a maximal ideal: $P / m_{a}$ is a field if and only if $m_{a}$ is maximal.

[F5]

$R_{p}$ is local with unique maximal ideal $p R_{p}$ : $P_{m_{a}}$ is a nonzero local ring with unique maximal ideal $m_{a} P_{m_{a}}$ .

[F6]

All initial forms define the tangent cone: at the rational origin of $Spec k [t_{1}, \dots, t_{n}]$ , the canonical graded map from the polynomial ring to the associated graded local ring is an isomorphism; it sends each variable to its degree-one initial class.

[F7]

The associated graded ring and associated graded module of an ideal-adic filtration: multiplication in the associated graded ring is induced by multiplication in the local ring.

[F8]

Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring: every field is an integral domain.

[F9]

A polynomial ring in finitely many indeterminates over an integral domain is an integral domain: a polynomial ring in finitely many variables over an integral domain is an integral domain.

[F10]

The stalk of the affine structure sheaf at a prime is A_p: on an affine scheme, the structure-sheaf stalk at a prime is canonically the corresponding ring localization.

[F11]

Every maximal ideal of a commutative ring is prime: every maximal ideal of a commutative ring is prime.

Proof

technique · direct

1.1F1F3F4F5F10F11givenalgebra

Translate coordinates by $t_{i} = X_{i} - a_{i}$ : the substitution $t_{i} \mapsto X_{i} - a_{i}$ is a polynomial-ring isomorphism with inverse $X_{i} \mapsto t_{i} + a_{i}$ , and evaluation at $a$ becomes evaluation at the origin. By [F1], the latter has kernel generated by the variables, since every monomial with zero constant term is divisible by some $t_{i}$ ; hence its transported kernel is $m_{a} = (X_{1} - a_{1}, \dots, X_{n} - a_{n})$ . Thus $P / m_{a} ≅ k$ , so [F4] makes $m_{a}$ maximal and [F11] makes it prime; [F3] and [F5] identify $R$ as a local ring with maximal ideal $m$ . Evaluation extends to $R$ because every denominator outside $m_{a}$ has nonzero value at $a$ , and it identifies $R / m$ with $k$ . The affine stalk identification [F10] gives $R = O_{A_{k}^{n}, a}$ ; translation identifies this filtered local ring with $k [t_{1}, \dots, t_{n}]_{(t_{1}, \dots, t_{n})}$ .

2.1F1F2F6step 1.1givenalgebra

Write $f (a + t) = \sum_{j} f_{j} (t)$ as in [F1]–[F2], and let $d$ be the least index with $f_{d} \neq 0$ ; it exists because translation is an isomorphism and $f \neq 0$ , and $d \geq 1$ because $f (a) = 0$ . Apply [F6] with $I = 0$ at the rational origin: it identifies the associated graded local ring with $k [t_{1}, \dots, t_{n}]$ , taking the degree- $j$ symbol of a polynomial to its degree- $j$ homogeneous part. Hence $f \in m^{d}$ and its class in $m^{d} / m^{d + 1}$ is the nonzero polynomial $f_{d}$ , so $f \notin m^{d + 1}$ and its $m$ -adic order is exactly the least degree $d$ .

3.1F5F6F7F8F9step 2.1algebra

Let $u \in R^{\times}$ . Its degree-zero initial class is its nonzero residue in $R / m ≅ k$ , while the degree- $d$ initial class of $f$ is $f_{d} \neq 0$ by step 2.1. By [F7], the initial class of $u f$ is the product of these classes; [F8]–[F9] make the identified graded ring $k [t_{1}, \dots, t_{n}]$ a domain, so the product is nonzero. Thus $u f \in m^{d} ∖ m^{d + 1}$ and ${ord}_{m} (u f) = d = {mult}_{a} (f)$ . For $r \geq 1$ , the initial class of $f^{r}$ is $f_{d}^{r}$ , nonzero and homogeneous of degree $r d$ in the same domain, so ${mult}_{a} (f^{r}) = r d$ .

4.1F1step 1.1step 2.1step 3.1givenalgebra∎

In one variable, $f = (X - a)^{3} + (X - a)^{4}$ has multiplicity $3$ , unchanged after multiplication by the local unit $1 + (X - a)$ by step 3.1; $X - a$ has multiplicity $1$ . Degree zero cannot occur for a nonzero polynomial vanishing at $a$ , and the zero polynomial is excluded because it has no least nonzero homogeneous part. If $n = 0$ , every polynomial is constant, so no nonzero polynomial vanishes at the unique point of $k^{0}$ . The proof uses only the fixed coordinate translation, the unique finite polynomial expansion, and a fixed local unit; it makes no family selection and uses neither AC nor DC. There is no interval or endpoint parameter and no iff assertion in this definition.

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Multiplicity one is the smooth hypersurface test

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be any field, let $n \geq 1$ be finite, let $0 \neq f \in k [X_{1}, \dots, X_{n}]$ be nonconstant, and let $a \in k^{n}$ satisfy $f (a) = 0$ . Put $X = Spec (k [X_{1}, \dots, X_{n}] / (f)),$ and let $x \in X (k)$ be the point defined by evaluation at $a$ . Then the structure morphism $X \to Spec k$ is smooth at $x$ if and only if ${mult}_{a} (f) = 1$ . Here “smooth at $x$ ” means that the structure map is standard smooth at the corresponding prime after shrinking to an open neighborhood of $x$ (Smooth morphisms via local standard smooth presentations, Standard smooth presentations and locally standard smooth maps). The equation is used with its actual scheme structure; no reducedness, perfectness, or characteristic assumption is imposed.

Facts & Assumptions

Given: A field $k$ , finite $n \geq 1$ , a nonzero nonconstant polynomial $f \in P = k [X_{1}, \dots, X_{n}]$ , a point $a \in k^{n}$ with $f (a) = 0$ , the affine $k$ -scheme $X = Spec (P / (f))$ , the corresponding point $x$ , and the Axiom of Choice. Set $m_{a} = (X_{1} - a_{1}, \dots, X_{n} - a_{n})$ , $A = P / (f)$ , $q = m_{a} / (f)$ , and $R = P_{m_{a}}$ with maximal ideal $m = m_{a} R$ .

[F1]

Multiplicity of a hypersurface equation at a rational point: the least nonzero homogeneous part of $f (a + t)$ has degree ${mult}_{a} (f)$ , and this is the $m$ -adic order of $f$ in $R$ . Thus multiplicity one means exactly that the class of $f$ in $m / m^{2}$ is nonzero.

[F2]

Standard smooth presentations and locally standard smooth maps: a finite-presentation map is standard smooth at $q$ when, after localizing away from $q$ , it has a presentation with an invertible Jacobian minor; for the one equation $f$ , the $1 \times 1$ minor is a partial derivative of $f$ .

[F3]

Smooth morphisms via local standard smooth presentations: smoothness is defined locally by standard smooth presentations at the source points. In particular, the pointwise condition at $x$ is precisely standard smoothness at $q$ after a neighborhood shrinking.

[F4]

Locally standard smooth iff flat with geometrically regular fibres: for a finite-presentation map $R_{0} \to S$ and $q_{0}$ over $p_{0}$ , standard smoothness at $q_{0}$ is equivalent to flatness of $(R_{0})_{p_{0}} \to S_{q_{0}}$ and geometric regularity of the fiber at $q_{0}$ .

[F5]

Geometrically regular algebras and geometrically regular fibres: geometric regularity of a fiber at a point requires that, after every field extension, all local rings at points above it are regular; it therefore implies regularity after the extension $k / k$ itself.

[F6]

Modules over a field are projective, flat, and injective: under AC every module over a field is flat, so the local $k$ -algebra map to any local ring of a chart is flat.

[F7]

regular local regular quotient ideal is parameter generated: under AC, if $(R_{0}, n, ℓ)$ is regular local, $J \subseteq n$ , and $R_{0} / J$ is regular, then $J$ is generated by an initial part of a regular system of parameters.

[F8]

regular system of parameters equivalent basis: under AC, the classes of a regular system of parameters form a basis of $n / n^{2}$ ; hence the classes in any initial part are linearly independent.

[F9]

localisation and polynomial extension of regular rings: under AC, finite polynomial extensions and localizations of regular Noetherian rings are regular; in particular $R = P_{m_{a}}$ is regular local.

[F10]

Localisation commutes with quotient rings: $S^{- 1} R / S^{- 1} I ≅ {\overset{ˉ}{S}}^{- 1} (R / I)$ : $(S^{- 1} P) / (S^{- 1} I) ≅ {\overset{ˉ}{S}}^{- 1} (P / I)$ for an ideal $I$ and a multiplicative set $S$ .

[F11]

The stalk of the affine structure sheaf at a prime is A_p: the local ring of $Spec A$ at $q$ is canonically $A_{q}$ .

[F12]

The Axiom of Choice: AC supplies a choice function for every family of nonempty sets; its uses here are only those explicitly inherited through [F4], [F6], [F7], [F8], and [F9].

Source qualification

Milne, Algebraic Geometry v6.10, §4b, Definition 4.9 and the following paragraph (printed pp. 83–84 / PDF pp. 83–84), defines the leading form as the least-degree nonzero homogeneous summand and uses its degree as the multiplicity of a plane-curve singularity. The book's global convention is that $k$ is algebraically closed; this is terminology and classical context, not proof of the present arbitrary-field scheme statement. Milne's Chapter 10 supplement, §f, 10.58 (PDF p. 16) and 10.64 (PDF p. 18), treats algebraic schemes over a field and states that a rational point is nonsingular exactly when its local ring is regular. These source statements motivate the criterion; the argument below proves the equation-level result for every field and retains nonreduced hypersurfaces.

Proof

technique · direct

1.1F1givenalgebra

Translate to $t_{i} = X_{i} - a_{i}$ and write $f (a + t) = L (t) +$ terms of degree at least two. By [F1], ${mult}_{a} (f) = 1$ exactly when $L \neq 0$ . For each $i$ , the coefficient of $t_{i}$ in $L$ is $\partial f / \partial X_{i} (a)$ : expanding each monomial $(a_{j} + t_{j})^{e_{j}}$ , its linear $t_{i}$ coefficient is the formal derivative coefficient, with the integer exponent interpreted in $k$ . Thus multiplicity one is equivalent, in every characteristic, to at least one partial derivative being nonzero at $a$ .

2.1F2F3step 1.1given

Suppose ${mult}_{a} (f) = 1$ , and choose the least index $i$ with $h = (\partial f / \partial X_{i}) (a) \neq 0$ . The class of the polynomial $\partial f / \partial X_{i}$ is outside $q$ in $A$ , so localizing there gives the one-equation presentation $k [X_{1}, \dots, X_{n}] / (f)$ with its $1 \times 1$ Jacobian minor invertible. Since $n \geq 1$ , this is a standard smooth chart by [F2]; hence the structure morphism is smooth at $x$ by [F3].

2.2F1F2F3F4F5F6F7F8F9F10F11step 1.1given

Conversely, suppose the structure morphism is smooth at $x$ . By [F2]–[F3], shrink once around $x$ to a finite-presentation standard smooth chart $B$ and let $q_{B}$ be its prime for $x$ . The local $k$ -module $B_{q_{B}}$ is flat by [F6]. Apply [F4] with base field $k$ and its zero prime: the fiber is $B$ , and it is geometrically regular at $q_{B}$ . Taking the extension $k / k$ in [F5] shows that $B_{q_{B}}$ is regular. The chart does not change the stalk, so [F11] gives $B_{q_{B}} ≅ A_{q}$ ; then [F10] identifies this ring with $R / (f) R$ . By [F9], $R$ is regular local, and $J = (f) R \subseteq m$ is nonzero by the finite order in [F1]. By [F7], $J$ is generated by an initial part $u_{1}, \dots, u_{r}$ of a regular system of parameters. Their classes in $m / m^{2}$ are linearly independent by [F8]. Since $J = (f) R$ and $f \in m$ , every $s f$ modulo $m^{2}$ is the residue of $s$ times the class of $f$ , so the image of $J$ in $m / m^{2}$ has dimension at most one. Hence $r \leq 1$ . If $r = 0$ , [F7] makes $J = 0$ , contrary to [F1]; therefore $r = 1$ , and the image of $J$ is nonzero. It is spanned by the class of $f$ , so $f \notin m^{2}$ . By [F1], ${mult}_{a} (f) = 1$ .

3.1F1F4F6F7F8F9F12step 2.1step 2.2givenalgebra∎

In one variable, at $a = 0$ , $f = t$ has multiplicity $1$ and derivative $1$ , giving the standard smooth chart; $f = t^{p}$ in characteristic $p > 0$ has multiplicity $p$ and derivative $0$ , and step 2.2 rules out smoothness. The zero polynomial is excluded because it has no least nonzero homogeneous part; a zero-variable polynomial cannot meet the nonconstant hypothesis. If the hypersurface has no $k$ -rational points there is no instance of this pointwise claim, and there is no interval or endpoint parameter. Steps 2.1 and 2.2 establish both iff directions. The coefficient and chart arguments make no family of choices; AC is used through [F4], [F6], [F7], [F8], and [F9], and no additional DC assumption is used.

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Products preserve smoothness

Statement

Assume the Axiom of Choice. Let $k$ be an algebraically closed field, and let $X$ and $Y$ be smooth classical varieties over $k$ . Their classical product is smooth. If $(X_{i})_{i}$ and $(Y_{j})_{j}$ are their irreducible-component decompositions, the irreducible components of $X \times_{k} Y$ are exactly the nonempty products $X_{i} \times_{k} Y_{j}$ , and $\dim (X_{i} \times_{k} Y_{j}) = \dim X_{i} + \dim Y_{j} .$ If either factor is empty, the product has no components.

More generally, for any field $k$ and finite-type $k$ -schemes $X, Y$ smooth over $k$ , the scheme-theoretic product $X \times_{k} Y$ is smooth over $k$ . In both clauses, smoothness is measured by the local-standard-smooth convention. The dimension assertion concerns only the classical-variety components; no dimension claim is made for arbitrary finite-type schemes.

Facts & Assumptions

Given: The Axiom of Choice, an algebraically closed field $k$ , smooth classical varieties $X, Y$ over $k$ , and finite-type $k$ -schemes smooth over a field in the general clause.

[F1]

A finite-type morphism of schemes is smooth when each source point has affine neighbourhoods on which the induced ring map is standard smooth at the corresponding prime; standard smoothness at a prime allows a further principal shrinking (Smooth morphisms via local standard smooth presentations).

[F2]

An affine model is a polynomial zero set in finite-dimensional affine space; its coordinate ring is a quotient of a finite-variable polynomial $k$ -algebra and is therefore finite type (The coordinate ring of a classical affine algebraic set).

[F14]

A classical variety has a finite affine-model cover (Classical algebraic prevarieties, regular maps, and varieties).

[F15]

Regular maps, including the product projections, are continuous because they are morphisms of locally ringed spaces (Classical algebraic prevarieties, regular maps, and varieties).

[F3]

Every scheme fibre product exists. For affine charts over an affine base, its open charts are spectra of the tensor-product algebras (Existence of all scheme fibre products).

[F4]

A standard-smooth algebra remains standard smooth at every point after arbitrary base change of the base ring (Base change and composition of standard smooth presentations).

[F5]

A composite of locally standard-smooth maps is locally standard smooth; the displayed standard-smooth presentation has the sum of the two relative presentation dimensions (Base change and composition of standard smooth presentations).

[F6]

Over an algebraically closed field and under AC, products of nonempty classical varieties exist; products of irreducible varieties are irreducible, and their dimensions add (Dimensions add under products).

[F7]

A classical variety has finitely many irreducible components (Classical varieties have finite irreducible decompositions).

[F8]

A constructed product has the categorical universal property; a product with a point is the other factor, and a product with an empty factor has empty underlying set (Products of classical algebraic sets and their universal property).

[F9]

AC asserts that every family of nonempty sets has a choice function (The Axiom of Choice).

[F10]

A morphism is of finite type when it is locally of finite type and quasi-compact (Locally finite type and finite type morphisms).

[F11]

The case $c = 0$ is allowed in a standard-smooth presentation; it is a localisation of a polynomial ring. In particular $n = c = 0$ presents the base ring itself (Standard smooth presentations and locally standard smooth maps).

[F13]

The affine product of classical affine varieties exists as a classical affine variety and has coordinate ring $A \otimes_{k} B$ (The product of affine varieties has coordinate ring k[X] tensor_k k[Y]).

Proof

technique · local standard-smooth presentations and product charts

1.1F1F3F4F5F10givenalgebrachoose

Let $P = X \times_{k} Y$ be the scheme-theoretic product of finite-type $k$ -schemes smooth over $k$ . For affine neighbourhoods $Spec A$ and $Spec B$ of the projections of any point, [F3] gives the product chart $Spec (A \otimes_{k} B)$ . Finite generating lists for $A$ and $B$ generate $A \otimes_{k} B$ , so $P$ is locally of finite type. Each factor has a finite affine cover because its structure morphism is quasi-compact; the resulting finite family of product charts covers $P$ , so $P$ is quasi-compact. Thus [F10] makes $P$ finite type. Fix $z \in P$ and write $x, y$ for its projections. Smoothness and [F1] give affine neighbourhoods $U = Spec A$ of $x$ and $V = Spec B$ of $y$ where $k \to A$ and $k \to B$ are standard smooth at the corresponding primes; shrink by the principal opens witnessing the presentations. By [F3], $U \times_{k} V = Spec (A \otimes_{k} B)$ is an open affine neighbourhood of $z$ . The map $A \to A \otimes_{k} B$ is the base change of $k \to B$ , so [F4] makes it standard smooth at the prime for $z$ . Its composite with $k \to A$ is standard smooth there by [F5]. Hence $P \to Spec k$ is locally standard smooth at $z$ .

1.2F6F7F8F15givenalgebrachoose

Suppose $X$ and $Y$ are nonempty. By [F7], write them as finite unions of irreducible components $X = ⋃_{i} X_{i}$ and $Y = ⋃_{j} Y_{j}$ . A point of either factor is a morphism from the one-point affine variety. Given $x \in X$ and $y \in Y$ , [F8] gives a unique point of the product whose projections are $x, y$ ; conversely, the projections of a product point determine it uniquely by the same universal property. Thus product points are pairs, and each $(x, y)$ belongs to some $X_{i} \times_{k} Y_{j}$ , so these products cover $X \times_{k} Y$ . Each product is closed as the intersection of the inverse images of the closed sets $X_{i}$ and $Y_{j}$ under the continuous projections [F15]; there are finitely many by [F7]. Each $X_{i} \times_{k} Y_{j}$ is irreducible by [F6]. Fix one point in each of the two nonempty factors; pairing that fixed point with an arbitrary point of the other factor shows that both product projections are surjective. If $X_{i} \times_{k} Y_{j} \subseteq X_{i^{'}} \times_{k} Y_{j^{'}}$ , surjectivity gives $X_{i} \subseteq X_{i^{'}}$ and $Y_{j} \subseteq Y_{j^{'}}$ , so maximality of the original components forces equality in both coordinates. Conversely, an irreducible closed subset of a finite union of closed sets lies in one member of that union, so every irreducible component of $X \times_{k} Y$ is one of these products. Applying [F6] to each irreducible pair gives $\dim (X_{i} \times_{k} Y_{j}) = \dim X_{i} + \dim Y_{j}$ . If one factor is empty, the product and the component-pair list are empty by [F8].

2.1F1F2F3F6F10F13F14algebrastep 1.1

Since $z$ was arbitrary, [F1] gives smoothness of $P$ over $k$ . If either factor is empty, then $P$ is empty and smoothness is vacuous, proving the general finite-type-scheme claim. For the classical product, [F14] gives finite affine covers; each affine chart has a finite-type coordinate algebra by [F2], so [F10] makes its associated scheme finite type. Fix a product point $z$ with projections $x, y$ , and choose affine neighbourhoods $U = Spec A$ and $V = Spec B$ . The open set $p^{- 1} (U) \cap q^{- 1} (V)$ in the classical product is itself the classical product $U \times_{k} V$ : a pair of maps into $U, V$ gives a unique map into the global product by [F6], and its image lies in this open set; uniqueness is inherited. By [F13] its coordinate ring is $A \otimes_{k} B$ , so its associated affine scheme is the same chart as the scheme product chart supplied by [F3]. The principal-open restrictions agree because both invert $f \otimes 1$ and $1 \otimes g$ on a product of $D (f)$ and $D (g)$ ; the resulting ring is $(A \otimes_{k} B)_{f \otimes 1, 1 \otimes g}$ . Therefore the local chart calculation in step 1.1 applies at every classical product point, and [F1] gives smoothness.

3.1F1F3F6F7F8F9F11step 1.1step 1.2step 2.1givenalgebra∎

The product with the zero-dimensional point $Spec k$ is the other factor by [F8], and dimensions add as $0 + d = d$ ; its map to $Spec k$ has the zero-variable, zero-equation standard-smooth presentation allowed by [F11]. The one-dimensional example $A_{k}^{1} \times_{k} A_{k}^{1} = A_{k}^{2}$ has the standard-smooth presentation with two variables and no equations by [F11], and its component dimension is $1 + 1 = 2$ by [F6]. If a factor is the empty scheme $Spec 0$ , its tensor-product chart is empty and no point requires a smoothness check [F3]. AC is propagated because the smooth-morphism convention [F1] and the classical component and dimension suppliers [F6, F7] are stated under AC [F9]; the pointwise standard-smooth argument makes no simultaneous chart choice. There is no interval endpoint, and neither smoothness preservation nor the component-dimension assertion is an iff.

Source qualification

Vakil, Foundations of Algebraic Geometry, Classes 51–52, §2.8, printed/PDF p. 5, states the smooth-product result as an exercise and points to base change and composition; it is corroboration, not a proof here. The Stacks Project, Morphisms of Schemes, Lemmas 29.35.4–5 (Section 29.35, tag 01V4, lines 51–56), states and proves composition and base-change stability for smooth morphisms. Lemma 29.35.11 (same section, lines 80–85) records the local standard-smooth chart criterion. The item proves the needed presentation steps directly from the fully written local standard-smooth result Base change and composition of standard smooth presentations; the Stacks results corroborate those operations and do not replace that argument. Milne, Algebraic Geometry v6.10, §5j, Proposition 5.35 (printed p. 115), proves dimension additivity for irreducible varieties by reducing to affine varieties and comparing transcendence bases in their tensor-product coordinate rings. Its irreducible hypotheses hold for each pair $X_{i}, Y_{j}$ ; the identification of all component products as components is proved in step 1.2.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Smooth morphisms via local standard smooth presentations

Definition

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and $f : X \to Y$ a morphism of finite-type $k$ -schemes. The morphism $f$ is smooth if every point $x \in X$ has affine neighborhoods $x \in V = Spec B$ and $f (x) \in U = Spec A$ , with $f (V) \subseteq U$ , for which the induced map $A \to B$ is standard smooth at the prime corresponding to $x$ (Standard smooth presentations and locally standard smooth maps). Thus the condition is imposed at every source point, including nonclosed points; it is local on the source and target. In this condition, standard smoothness at a prime means that after a further principal shrinking around that prime the ring map has a standard smooth presentation.

The pointwise and global equivalences in Locally standard smooth iff flat with geometrically regular fibres identify standard smoothness for finite-presentation affine charts with flatness and geometrically regular scheme-theoretic fibres. Applied after the local shrinkings above, this gives the corresponding local criterion for $f$ . The local-presentation clause itself is choice-free; AC is assumed here for that proved equivalence, through the cited theorem.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

The submersion criterion between smooth varieties

Statement

Assume the Axiom of Choice. Let $k$ be algebraically closed and let $X, Y$ be smooth classical varieties over $k$ , with their finite-type $k$ -scheme structures. Assume their structure morphisms are smooth in the sense of Smooth morphisms via local standard smooth presentations. Let $f : X \to Y$ be a finite-type morphism of these $k$ -schemes (Locally finite type and finite type morphisms) and let $x \in X$ be a classical closed point with $y = f (x)$ . All such points have residue field $k$ . Here “smooth at $x$ ” means that the induced scheme morphism is locally standard smooth at $x$ (Smooth morphisms via local standard smooth presentations); the structural morphisms $X \to Spec k$ and $Y \to Spec k$ are locally standard smooth at $x$ and $y$ , respectively. Then $f$ is smooth at $x$ if and only if its differential $d_{x} f : T_{x} X ⟶ T_{y} Y$ is surjective. If these equivalent conditions hold, the scheme-theoretic fibre $X_{y} = X \times_{Y} Spec k$ has a regular local ring at $x$ of dimension $\dim_{x} X - \dim_{y} Y$ . For every such $f$ (whether or not it is smooth at $x$ ), its fibre tangent space is canonically $T_{x} (X_{y}) = \ker (d_{x} f) .$

Facts & Assumptions

Given: AC; an algebraically closed field $k$ ; smooth classical varieties $X, Y$ over $k$ ; their locally standard-smooth structural morphisms; a finite-type morphism $f : X \to Y$ ; and a classical closed point $x \in X$ with $y = f (x)$ . Write $C_{x} X = m_{x} / m_{x}^{2}$ and $C_{y} Y = m_{y} / m_{y}^{2}$ for the cotangent spaces of the local scheme charts.

[F1]

The Axiom of Choice: every family of nonempty sets has a choice function.

[F2]

Classical algebraic prevarieties, regular maps, and varieties: classical varieties here are over an algebraically closed field; their classical points have residue field $k$ , and regular maps respect the $k$ -algebra structures.

[F3]

Global and local dimension of classical varieties: for a classical closed point $z$ , $\dim_{z} Z$ is the maximum of the dimensions of the irreducible components of $Z$ containing $z$ .

[F4]

Local dimension for a reducible classical algebraic set: for a reduced classical finite-type variety and a closed point $z$ , $\dim O_{Z, z} = \dim_{z} Z$ .

[F5]

Smooth morphisms via local standard smooth presentations: smoothness of a finite-type $k$ -scheme morphism is defined locally by standard smooth presentations; pointwise smoothness at $z$ is the standard-smooth condition at its prime after shrinking.

[F6]

Locally finite type and finite type morphisms: a morphism is of finite type when it is locally of finite type and quasi-compact.

[F7]

The intrinsic Zariski tangent space: at a rational point, $T_{z} Z = {Hom}_{k} (m_{z} / m_{z}^{2}, k)$ ; these spaces are finite-dimensional for locally finite-type schemes over $k$ .

[F8]

Differentials, open restriction, and the chain rule: for a $k$ -morphism at rational points, the differential is the dual of the induced cotangent map and agrees with post-composition on based dual-number points.

[F9]

Submersion criterion for locally standard smooth morphisms: under AC, if $X, Y$ are locally standard smooth over $k$ at rational $x, y = f (x)$ and $f$ is of finite type, then $f$ is locally standard smooth at $x$ iff $C_{y} Y \to C_{x} X$ is injective.

[F10]

Submersion criterion for locally standard smooth morphisms: in the smooth case the fibre local ring is regular of dimension $\dim O_{X, x} - \dim O_{Y, y}$ .

[F11]

Scheme-theoretic fibre: $X_{y}$ is $X \times_{Y} Spec κ (y)$ , viewed as a $κ (y)$ -scheme; here $κ (y) = k$ .

[F12]

Base change of objects, morphisms and properties: base change uses the fibre product $X \times_{Y} Spec k$ and its projection maps.

[F13]

Existence of all scheme fibre products: fibre products exist with their universal property.

Proof

technique · direct

1.1F2F5F6F9given

Put $C_{x} X = m_{x} / m_{x}^{2}$ and $C_{y} Y = m_{y} / m_{y}^{2}$ , and let $α : C_{y} Y \to C_{x} X$ be the cotangent map induced by $f$ . By [F2], $x, y$ are $k$ -rational; by [F5] their structural smoothness assumptions give standard-smooth charts, and [F6] records that $f$ is finite type. Hence [F9] applies and says $f$ is smooth at $x$ exactly when $α$ is injective.

2.1F7F8step 1.1algebra

By [F7], $C_{x} X$ and $C_{y} Y$ are finite-dimensional; the differential is $d_{x} f = α^{*}$ by [F8]. A linear map and its dual have equal rank, so $α$ is injective iff $rank (α) = \dim C_{y} Y = \dim T_{y} Y = rank (α^{*})$ , iff $d_{x} f$ is surjective. With step 1.1 this proves both directions.

3.1F3F4F10step 2.1givenalgebra

If these equivalent conditions hold (step 2.1), [F10] gives a regular local ring for the scheme-theoretic fibre at $x$ , of dimension $\dim O_{X, x} - \dim O_{Y, y}$ . By [F3] and [F4], these stalk dimensions equal $\dim_{x} X$ and $\dim_{y} Y$ . This proves the stated local fibre dimension; no regularity at other fibre points is asserted.

3.2F8F11F12F13step 2.1givenalgebra

Let $D = Spec (k [ϵ] / (ϵ^{2}))$ . By [F8], $v \in T_{x} X$ is represented by a based map $γ : D \to X$ , and $d_{x} f (v)$ by $f \circ γ$ . By [F11]–[F13] and the fibre-product universal property, based maps $D \to X_{y}$ at $x$ correspond to based $γ : D \to X$ whose composite is the constant map $D \to Spec k \overset{y}{\to} Y$ . That constant map represents zero in $T_{y} Y$ , so these are exactly the $v$ with $d_{x} f (v) = 0$ . The correspondence is linear and canonical, proving $T_{x} (X_{y}) = \ker (d_{x} f)$ even when $f$ is not smooth at $x$ .

4.1F1F4F5F7F8F9F10step 2.1step 3.1step 3.2givenalgebra∎

If $T_{y} Y = 0$ , every differential to it is surjective and $C_{y} Y = 0$ makes $α$ injective, so [F9] gives smoothness; if $T_{x} X = 0$ but $T_{y} Y \neq 0$ , neither condition holds, and when both vanish the fibre has local dimension zero. The identity $A_{k}^{1} \to A_{k}^{1}$ at $0$ has differential $1$ and point fibre $Spec k$ , regular of dimension zero. For $t \mapsto t^{2}$ at $0$ , the derivative $2 t d t$ vanishes in every characteristic, so [F8, F9] show the differential is zero and the map is not smooth. Its fibre is $Spec (k [t] / (t^{2}))$ ; since $t$ is nilpotent the only prime is $(t)$ , so the local dimension is zero, while its maximal ideal $m = (t)$ has $m^{2} = 0$ and $m / m^{2} ≅ k$ . Thus the local ring is not regular and its one-dimensional tangent space is the full kernel. This checks the degenerate case and shows regularity is asserted only under smoothness. If $X$ has no classical points there is no $x$ to check; the dimension difference is nonnegative when $f$ is smooth by steps 2.1 and 3.1. AC is inherited only through [F1], [F4], [F5], and [F9]; linear algebra and the fibre-product argument add no choice principle.

Source qualification

Vakil, Foundations of Algebraic Geometry, Classes 51–52, §2.2, printed and PDF p. 5, calls the related result a “Trickier Exercise”: it assumes pure-dimensional smooth varieties and surjectivity at every closed point, then asks for smoothness of relative dimension $\dim X - \dim Y$ ; it gives the local flatness criterion as a hint, not a proof. The present pointwise proof uses the complete local argument in [F9], so it does not infer the result from that exercise or require global pure dimension. Stacks Project Algebra Lemma 10.128.2 (tag 07DY), statement and proof, independently gives flatness when parameters of a regular local base map to a regular sequence. That is corroboration for the parameter-flatness step inside [F9], not a premise used directly here; the local-flatness and regular-sequence inputs are proved in the cited published supplier. The separate fibre-tangent identity above follows from the fibre-product universal property and the dual-number description.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

A transverse hyperplane slice is smooth at the chosen point

Statement

Assume the Axiom of Choice. Let $k$ be an algebraically closed field, let $n \geq 1$ , and let $X \subseteq A_{k}^{n}$ be a classical variety over $k$ , embedded as a closed subvariety and carrying its reduced finite-type $k$ -scheme structure. Suppose that $X$ is smooth at the classical closed point $x \in X$ (Smooth morphisms via local standard smooth presentations) and put $d = \dim_{x} X$ , assumed to satisfy $d \geq 1$ . Let $h = ℓ - c \in k [x_{1}, \dots, x_{n}]$ be an affine-linear polynomial whose linear part $ℓ$ is nonzero and which satisfies $h (x) = 0$ , and let $H = V (h)$ be the closed subscheme of $A_{k}^{n}$ cut out by the principal ideal $(h)$ --- the fibre of $h : A_{k}^{n} \to A_{k}^{1}$ over the origin $0$ , that is, the affine hyperplane through $x$ . Assume that $ℓ$ is nonzero on $T_{x} X$ : under the identification $T_{x} A_{k}^{n} = k^{n}$ obtained from Tangent vectors at rational points are dual-number points and Universal property of a polynomial ring on an arbitrary family of indeterminates, the composite $T_{x} X \overset{d_{x} ι}{\to} T_{x} A_{k}^{n} = k^{n} \overset{ℓ}{\to} k$ is not the zero map, where $ι : X ↪ A_{k}^{n}$ is the inclusion. Then the restricted morphism $h ∣_{X} : X \to A_{k}^{1}$ is smooth at $x$ ; the scheme-theoretic intersection $Z = X \times_{A_{k}^{n}} H$ is canonically the scheme-theoretic fibre of $h ∣_{X}$ over $0$ , its structure morphism $Z \to Spec k$ is smooth at $x$ , and $O_{Z, x}$ is a regular local ring of dimension $d - 1$ ; and $T_{x} Z = \ker (d_{x} (h ∣_{X})),$ the kernel of the composite displayed above (so that, under that identification, $T_{x} Z$ is the subspace ${v \in T_{x} X : d_{x} (h ∣_{X}) (v) = 0}$ of $T_{x} X$ ).

Facts & Assumptions

Given: AC; an algebraically closed field $k$ ; $n \geq 1$ ; a classical variety $X \subseteq A_{k}^{n}$ with closed point $x$ at which $X$ is smooth; $d = \dim_{x} X \geq 1$ ; the affine-linear polynomial $h = ℓ - c$ with nonzero linear part $ℓ$ and $h (x) = 0$ ; the closed subscheme $H = V (h)$ ; and the transversality assumption that the composite $T_{x} X \to T_{x} A_{k}^{n} = k^{n} \to k$ induced by $ℓ$ is not the zero map.

[F1]

The Axiom of Choice: every family of nonempty sets has a choice function.

[F2]

Classical algebraic prevarieties, regular maps, and varieties: a classical algebraic variety over an algebraically closed field $k$ is a separated classical prevariety; varieties may be reducible or empty, their affine models are polynomial zero sets whose points have residue field canonically $k$ , and these definitions use no Axiom of Choice.

[F3]

The coordinate ring of a classical affine algebraic set: for an affine algebraic set $X \subseteq k^{n}$ the coordinate ring is $k [X] = k [x_{1}, \dots, x_{n}] / I (X)$ ; it is reduced, and the finite coordinate classes generate it as a $k$ -algebra.

[F4]

Global and local dimension of classical varieties: for a classical variety $X$ with irreducible components $X_{1}, \dots, X_{m}$ and a closed point $x$ one has $\dim_{x} X = \max_{x \in X_{i}} \dim X_{i}$ , where $\dim X$ is the chain dimension.

[F5]

Local dimension for a reducible classical algebraic set: under AC, for a reduced classical finite-type space $X$ over an algebraically closed field and a closed point $x$ one has $\dim O_{X, x} = \max_{x \in X_{i}} \dim X_{i}$ .

[F6]

Smooth morphisms via local standard smooth presentations: a morphism of finite-type $k$ -schemes is smooth when every source point has affine neighbourhoods on which the induced ring map is standard smooth at the prime for that point; the condition is local on source and target, and the definition assumes AC.

[F7]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation of an $R$ -algebra $S$ is an isomorphism $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with an invertible $c \times c$ Jacobian minor; the case $c = 0$ is exactly a localisation of a polynomial ring, and standard smoothness at a prime holds after a principal shrinking.

[F8]

Fibres of standard smooth algebras are regular of relative dimension: under AC, if $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ is standard smooth over the commutative ring $R$ , then for every prime of $R$ and every field extension of its residue field, every local ring of the corresponding base-changed fibre is a regular local ring.

[F9]

The stalk of the affine structure sheaf at a prime is A_p: for $p \in Spec A$ , the affine structure-sheaf stalk is canonically $O_{Spec A, p} ≅ A_{p}$ .

[F10]

Regular points of locally Noetherian schemes: for a point $x$ of a locally Noetherian scheme, $x$ is regular exactly when $O_{X, x}$ is a regular local ring, and then the intrinsic tangent space $T_{x} X$ is finite-dimensional over $κ (x)$ with $\dim_{κ (x)} T_{x} X = \dim O_{X, x}$ .

[F11]

The affine scheme of dual numbers and Tangent vectors at rational points are dual-number points: $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ is the dual-numbers scheme, and for a $k$ -scheme $X$ with $x \in X (k)$ the intrinsic tangent space $T_{x} X$ is naturally isomorphic, as a $k$ -vector space, to the fibre over $x$ of ${Hom}_{k} (D_{k}, X) \to X (k)$ ; equivalently $T_{x} X ≅ {Der}_{k} (O_{X, x}, k)$ .

[F12]

Universal property of a polynomial ring on an arbitrary family of indeterminates: a $k$ -algebra map from $k [X_{1}, \dots, X_{n}]$ to a commutative $k$ -algebra is uniquely determined by arbitrary images of its $n$ variables.

[F13]

Differentials, open restriction, and the chain rule: the differential $d_{x} f$ is the dual of the induced cotangent map, it agrees with post-composition by $f$ on based dual-number points, it satisfies the chain rule, and it is an isomorphism for isomorphisms of $k$ -schemes; no choice is used.

[F14]

Scheme-theoretic fibre: for a morphism $f : X \to S$ and a point $s \in S$ , the scheme-theoretic fibre is $X_{s} = X \times_{S} Spec κ (s)$ .

[F15]

Intersections of subschemes: the scheme-theoretic intersection of finitely many closed subschemes of a scheme is their iterated fibre product over it, cut out by the sum of their ideal sheaves.

[F16]

Fibre product of schemes and Existence of all scheme fibre products: a fibre product of $X \to S \leftarrow Y$ is a scheme $P$ with projections $p : P \to X$ , $q : P \to Y$ such that $f p = g q$ and, for every test scheme $T$ and morphisms $a : T \to X$ , $b : T \to Y$ with $f a = g b$ , there is exactly one $h : T \to P$ with $p h = a$ and $q h = b$ ; every such diagram of schemes has a fibre product.

[F17]

Affine fibre products are spectra of tensor products: the fibre product of affine schemes over an affine base is $Spec B \times_{Spec A} Spec C ≅ Spec (B \otimes_{A} C)$ , with projections corresponding to $b \mapsto b \otimes 1$ and $c \mapsto 1 \otimes c$ .

[F18]

$M \otimes_{R} R / I ≅ M / I M$ naturally: for a commutative ring $R$ , an ideal $I \subseteq R$ and an $R$ -module $M$ there is a natural isomorphism $M \otimes_{R} (R / I) ≅ M / I M$ , $m \otimes (r + I) \mapsto r m + I M$ ; it is $R / I$ -linear, for $I = 0$ it is the tensor-unit isomorphism, and for $I = R$ both sides are zero.

[F19]

Base change and composition of standard smooth presentations: base change of a standard smooth presentation along an arbitrary ring map is standard smooth with the same parameters, so locally standard smooth maps are stable under base change of the base ring.

[F20]

Submersion criterion for locally standard smooth morphisms: under AC, let $X, Y$ be $k$ -schemes locally standard smooth over $k$ at $k$ -rational points $x \in X$ and $y = f (x)$ , and let $f$ be of finite type; then $f$ is locally standard smooth at $x$ if and only if the induced map $m_{y} / m_{y}^{2} \to m_{x} / m_{x}^{2}$ is injective; if so, and $m = \dim O_{X, x}$ , $n = \dim O_{Y, y}$ , then the local ring $S / m_{y} S$ of the scheme-theoretic fibre $X_{y}$ at $x$ is a regular local ring of dimension $m - n$ .

[F21]

embedding dimension and regular local ring: for a nonzero commutative Noetherian local ring $(R, m, k)$ one has $edim R = \dim_{k} (m / m^{2})$ , and $R$ is regular local exactly when $edim R = \dim R$ .

[F22]

Locally finite type and finite type morphisms and Subalgebra generated by a subset, algebras of finite type, and module-finite algebras: a morphism is of finite type when it is locally of finite type and quasi-compact, and an $R$ -algebra is of finite type over $R$ when it is generated as an $R$ -algebra by finitely many elements, equivalently a quotient of a polynomial ring in finitely many variables.

[F23]

Classical varieties have finite irreducible decompositions: every classical variety is Noetherian with finitely many irreducible components, and every open or closed subvariety has a finite affine cover.

[F24]

The affine line $A_{k}^{1}$ has coordinate ring $k [t]$ by The coordinate ring of a classical affine algebraic set, and its local ring at a closed point $a$ is $k [t]_{(t - a)}$ with residue field $k$ by The classical affine local ring is localization at the point's maximal ideal.

[F25]

[F26]

Finite type is affine-local on source and target: being locally of finite type is affine-local on source and target, and a quasi-compact morphism locally of finite type is of finite type; equivalently, over each affine target open this may be tested on a finite affine source cover.

[F27]

Affine and projective n-space have dimension n: for every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ .

[F28]

Every algebra of finite type over a Noetherian ring is a Noetherian ring: a commutative algebra of finite type over a Noetherian commutative ring is a Noetherian ring.

[F29]

Locally Noetherian and Noetherian schemes: a scheme is locally Noetherian if it has an affine open cover by spectra of Noetherian rings.

Proof

technique · direct

1.1F2F3F4F5F6F7F9F23given

Setup. The closed subvariety $X \subseteq A_{k}^{n}$ is an affine model with its reduced finite-type structure and function sheaf [F2], and its coordinate ring $k [X] = k [x_{1}, \dots, x_{n}] / I (X)$ is reduced and generated as a $k$ -algebra by the finitely many classes ${\overset{ˉ}{x}}_{1}, \dots, {\overset{ˉ}{x}}_{n}$ [F3]. Its points have residue field $k$ [F2], so $x$ is a $k$ -rational point. The variety $X$ is Noetherian with finitely many irreducible components [F23], and [F5] with [F4] gives $\dim O_{X, x} = \max_{x \in X_{i}} \dim X_{i} = \dim_{x} X = d,$ the maximum being over the components containing $x$ , a nonempty finite family. Smoothness of $X$ at $x$ means that the structure morphism $X \to Spec k$ is locally standard smooth at $x$ [F6]; fix an affine chart $Spec A$ containing $x$ on which $k \to A$ is standard smooth [F7], and write $m$ for the prime of $x$ in $A$ , so that $O_{X, x} ≅ A_{m}$ [F9]. The subscheme $H = V (h)$ is cut out by the principal ideal $(h) \subseteq k [x_{1}, \dots, x_{n}]$ , and its defining equation vanishes at $x$ : $h (x) = 0$ .

1.2F10F11F12F13F25givenalgebra

The differential of $h ∣_{X}$ . Write $f = h ∣_{X}$ . Every $v \in T_{x} X$ is represented by a based dual-number point $γ : D_{k} \to X$ at $x$ [F11]. Its composite $γ^{'} = ι \circ γ : D_{k} \to A_{k}^{n}$ corresponds to a $k$ -algebra map $k [X_{1}, \dots, X_{n}] \to k [ϵ] / (ϵ^{2})$ [F25]. By [F12] this map is uniquely determined by the images of the $X_{i}$ . Since reduction modulo $ϵ$ gives the point $x = (x_{1}, \dots, x_{n})$ , these images have the unique form $X_{i} \mapsto x_{i} + ϵ w_{i}$ for $w = (w_{1}, \dots, w_{n}) \in k^{n}$ . Conversely every $w \in k^{n}$ gives such a based point by [F12], and the $k$ -linear dual-number correspondence [F11] identifies $T_{x} A_{k}^{n}$ with $k^{n}$ in these coordinates. Call $w$ the image of $v$ under $d_{x} ι$ . By [F13] the differential $d_{x} f (v)$ is represented by the composite $f \circ γ$ , and substituting $γ^{'}$ in the affine-linear form $h = ℓ - c$ gives $h (x + ϵ w) = h (x) + ϵ ℓ (w) = ϵ ℓ (w),$ because $h (x) = 0$ and $ℓ$ is $k$ -linear. Hence $d_{x} f (v) = ℓ (w)$ , that is, $d_{x} f = ℓ \circ d_{x} ι$ and $d h_{x} = ℓ$ ; the transversality hypothesis is therefore exactly the condition $d_{x} f \neq 0$ . Taking $n = 1$ in the same calculation gives $T_{0} A_{k}^{1} = k$ , which is one-dimensional, and $T_{x} X$ is finite-dimensional [F10]. Thus a nonzero $d_{x} f$ is surjective, and by [F13] the induced cotangent map $m_{0} / m_{0}^{2} \to m_{x} / m_{x}^{2}$ is its dual, hence injective.

1.3F3F22F24F25F26givenalgebra

The morphism $f$ is of finite type. The affine model $X$ has coordinate ring $k [X] = k [x_{1}, \dots, x_{n}] / I (X)$ , generated as a $k$ -algebra by the finitely many classes ${\overset{ˉ}{x}}_{1}, \dots, {\overset{ˉ}{x}}_{n}$ [F3], and the affine line $A_{k}^{1}$ has coordinate ring $k [t]$ [F24]; by [F25] the $k$ -morphism $f = h ∣_{X}$ from the affine chart $X$ to $A_{k}^{1}$ corresponds to the $k$ -algebra map $k [t] \to k [X]$ sending $t$ to the class $\overset{ˉ}{h}$ of $h$ , the pullback of the coordinate function. This ring map is of finite type: the same finite family generates $k [X]$ over $k$ [F3], hence over $k [t]$ [F22]. By [F26] finiteness of type may be tested over each affine target open on a finite affine source cover; the target $Spec k [t] = A_{k}^{1}$ is affine and the single chart $X$ is such a cover, so $f$ is of finite type.

2.1F11F12F13F14F15F16F17F18step 1.2givenalgebra

Regularity of $X$ at $x$ and of the affine line at the origin. The coordinate ring $A$ of the chart of step 1.1 is a finitely generated $k$ -algebra [F3], hence a Noetherian ring by [F28] because $k$ is a field and therefore Noetherian; so $Spec A$ is locally Noetherian [F29]. Applying clause 1 of [F8] to the standard smooth presentation of step 1.1 with $R = k$ , $p = (0)$ and $K = k$ shows that every local ring of that chart, in particular $O_{X, x} = A_{m}$ , is a regular local ring; by [F10] therefore $\dim_{k} T_{x} X = \dim O_{X, x} = d$ , so $T_{x} X \neq 0$ because $d \geq 1$ . The affine line has coordinate ring $k [t]$ and local ring $k [t]_{(t)}$ at the origin with residue field $k$ [F24], and $k \to k [t]$ is standard smooth with one variable and no equation [F7]; hence $A_{k}^{1} \to Spec k$ is locally standard smooth at $0$ [F6] and $O_{A_{k}^{1}, 0}$ is a regular local ring [F8]. The affine line is irreducible with $\dim A_{k}^{1} = 1$ [F27], so [F5] with [F4] gives $\dim O_{A_{k}^{1}, 0} = \dim_{0} A_{k}^{1} = 1$ . [F3, F4, F5, F6, F7, F8, F10, F24, F27, F28, F29, step 1.1, given, algebra] 2.2 The slice is the fibre. First, $H = V (h)$ is the fibre of $h$ over the origin: the fibre product of $h : A_{k}^{n} \to A_{k}^{1}$ and the point $0 : Spec k \to A_{k}^{1}$ is $Spec (k [x_{1}, \dots, x_{n}] \otimes_{k [t]} k)$ with the projections of [F17], and [F18] identifies $k [x_{1}, \dots, x_{n}] \otimes_{k [t]} k ≅ k [x_{1}, \dots, x_{n}] / (h)$ , where $k [x_{1}, \dots, x_{n}]$ is a $k [t]$ -algebra through $t \mapsto h$ , the ideal $I M$ is generated by $h$ for $I = (t)$ and $M = k [x_{1}, \dots, x_{n}]$ , and the isomorphism is one of $k$ -algebras; hence $H = Spec (k [x_{1}, \dots, x_{n}] / (h))$ is this fibre, with projections $π : H \to A_{k}^{n}$ and $ρ : H \to Spec k$ satisfying $h \circ π = (0) \circ ρ$ [F16]. Second, $Z = X \times_{A_{k}^{n}} H$ is the scheme-theoretic intersection of the closed subschemes $X$ and $H$ of affine space, with projections $π_{X} : Z \to X$ , $π_{H} : Z \to H$ satisfying $ι \circ π_{X} = π \circ π_{H}$ [F15]. Third, the fibre $X_{0} = X \times_{A_{k}^{1}} Spec k$ of $f$ over $0$ has projections $p : X_{0} \to X$ , $q : X_{0} \to Spec k$ satisfying $f \circ p = (0) \circ q$ [F14]. All three fibre products exist, and a morphism into any of them is determined by its projections [F16]. The morphisms $ι \circ p$ and $q$ have equal composites to $A_{k}^{1}$ , namely $h \circ ι \circ p = f \circ p = (0) \circ q$ , so the universal property of $H$ gives a unique $θ_{H} : X_{0} \to H$ with $π \circ θ_{H} = ι \circ p$ and $ρ \circ θ_{H} = q$ ; since $ι \circ p = π \circ θ_{H}$ , the pair $(p, θ_{H})$ induces a unique $θ : X_{0} \to Z$ with $π_{X} \circ θ = p$ and $π_{H} \circ θ = θ_{H}$ . Conversely the morphisms $π_{X}$ and $ρ \circ π_{H}$ have equal composites to $A_{k}^{1}$ , namely $f \circ π_{X} = h \circ ι \circ π_{X} = h \circ π \circ π_{H} = (0) \circ ρ \circ π_{H}$ , so the universal property of $X_{0}$ gives a unique $ψ : Z \to X_{0}$ with $p \circ ψ = π_{X}$ and $q \circ ψ = ρ \circ π_{H}$ . By the uniqueness clauses $ψ \circ θ = {id}_{X_{0}}$ and $θ \circ ψ = {id}_{Z}$ : both composites induce the same projections, and a morphism into $H$ is determined by its composites with $π$ and $ρ$ . Hence $θ$ is a canonical isomorphism $X_{0} \to Z$ over $X$ and over $Spec k$ . The $k$ -point $x : Spec k \to X$ satisfies $f \circ x = (0) \circ (structure map)$ because $h (x) = 0$ , so it induces a $k$ -point of $X_{0}$ , carried by $θ$ to a $k$ -point of $Z$ mapping to $x$ ; this is the point at which all local statements are taken. Finally the tangent space. A $k$ -morphism $D_{k} \to X_{0}$ is by the universal property a pair $(γ, δ)$ with $γ : D_{k} \to X$ and $δ : D_{k} \to Spec k$ such that $f \circ γ = (0) \circ δ$ [F16]; the morphism $δ$ is unique, and being based at $x$ means that $γ$ is based at $x$ and $δ$ is the structure morphism. Hence based dual-number points of $X_{0}$ at $x$ correspond bijectively to based dual-number points $γ : D_{k} \to X$ of $X$ at $x$ whose composite $f \circ γ$ is the constant point at $0$ . Under the identifications of [F11] this correspondence is $k$ -linear and identifies $T_{x} X_{0}$ with $\ker (d_{x} f)$ : by [F13], $d_{x} f (v)$ is represented by $f \circ γ$ , and the constant point at $0$ represents the zero vector of $T_{0} A_{k}^{1}$ [F12]. Since $θ$ is an isomorphism, its differential at $x$ is an isomorphism [F13], so $T_{x} Z = \ker (d_{x} f) .$

3.1F1F5F6F8F15F16F19F20F23F24step 2.1step 3.1step 2.2step 4.1givenalgebra∎

The submersion criterion. Take $Y = A_{k}^{1}$ and $y = 0 = f (x)$ : the point $y$ has residue field $k$ [F24], and both $x$ and $y$ are $k$ -rational [step 1.1]. The schemes $X$ and $Y$ are locally standard smooth over $k$ at $x$ and $y$ [step 1.1, step 2.1], the morphism $f$ is of finite type [step 1.3], and the cotangent map $m_{y} / m_{y}^{2} \to m_{x} / m_{x}^{2}$ of [F20] is injective by [step 1.2]. Clause 1 of [F20] therefore makes $f$ locally standard smooth at $x$ , that is, smooth at $x$ in the sense of [F6]. [F6, F20, F24, step 1.1, step 2.1, step 1.2, step 1.3, given, algebra] 4.1 Dimension of the slice. By step 3.1, clause 2 of [F20] applies at $x$ with $S = O_{X, x}$ , $A = O_{A_{k}^{1}, 0}$ , $m = \dim S = \dim O_{X, x} = d$ [step 1.1] and $n = \dim A = \dim O_{A_{k}^{1}, 0} = 1$ [step 2.1]; it makes the local ring $S / m_{y} S = O_{X_{0}, x}$ of the fibre at $x$ a regular local ring of dimension $m - n = d - 1$ . By step 2.2, $O_{Z, x} ≅ O_{X_{0}, x}$ , so $O_{Z, x}$ is a regular local ring of dimension $d - 1$ in the sense of [F21]. Consistently, rank-nullity for the surjective differential [step 1.2] gives $\dim_{k} \ker (d_{x} f) = \dim_{k} T_{x} X - 1 = d - 1$ [step 2.1], and $\dim_{k} T_{x} Z = \dim_{k} \ker (d_{x} f)$ by step 2.2, so the tangent dimension of the slice agrees with the local dimension. [F20, F21, step 2.1, step 1.2, step 3.1, step 2.2, given, algebra] 5.1 Smoothness of the slice and conclusion. Since $f$ is smooth at $x$ [step 3.1] and $Z ≅ X_{0}$ is the base change of $f$ along the point $0 : Spec k \to A_{k}^{1}$ [step 2.2], clause 1 of [F19] makes $Z \to Spec k$ locally standard smooth at $x$ , so the slice is smooth at $x$ [F6]. Together with steps 3.1, 2.2 and 4.1 this proves the assertions of the statement, including $T_{x} Z = \ker (d_{x} (h ∣_{X}))$ and its description as the set of $v \in T_{x} X$ with $d_{x} (h ∣_{X}) (v) = 0$ . Boundaries. The hypothesis $d \geq 1$ guards against vacuity: if $d = 0$ then $\dim_{k} T_{x} X = d = 0$ [step 2.1], so $T_{x} X$ carries no nonzero linear functional and the transversality hypothesis fails. For $d = 1$ the conclusion gives $\dim O_{Z, x} = 0$ [step 4.1], so the slice is isolated at $x$ in the local sense. The ambient endpoint $n = 1$ forces $d \leq 1$ , hence $d = 1$ and the same zero-dimensional conclusion. For $X = A_{k}^{n}$ the inclusion is the identity, the slice $Z ≅ H$ is the hyperplane itself (the projection $Z \to H$ is an isomorphism by the universal property [F16] applied to ${id}_{A_{k}^{n}}$ ), the transversality condition is exactly $ℓ \neq 0$ , and step 2.2 gives $T_{x} Z = \ker (ℓ)$ . No characteristic hypothesis is used: the argument never divides by an integer, so all characteristics are covered. The variety $X$ may be reducible, and nothing is asserted in the nontransverse case where $ℓ$ vanishes on $T_{x} X$ . AC enters the statement through [F1] and is used only through the suppliers that assume it, namely [F5], [F6], [F8], [F20], [F23] and [F24], each cited at the step that uses it; the affine chart, its presentation, the polynomial $h$ and the point $x$ are single given objects, so no further selection is made and [F13], [F15], [F16] and [F18] are choice-free.

Source qualification

Milne, Algebraic Geometry v6.10, Exercise 4-2 (printed pp. 98-99; PDF pages 98-99) assumes $V$ irreducible and $P$ nonsingular on $V$ , and asks only that $P$ be nonsingular on each irreducible component of $V \cap H$ on which it lies, adding "you may assume" that each component has codimension one in $V$ ; the official solution (printed p. 222) argues from $T_{a} (V \cap H) \subset T_{a} (V) \cap T_{a} (H)$ and the dimension inequality. The item above instead treats the scheme-theoretic intersection of an arbitrary closed subvariety with the hyperplane cut out by an affine-linear equation, allows a reducible $X$ , and proves the tangent identity by the fibre-product universal property and dual-number points. Regularity and the local dimension $d - 1$ are taken from the locally standard smooth submersion criterion Submersion criterion for locally standard smooth morphisms, whose pointwise hypotheses suffice; the earlier scaffold planned to route them through The submersion criterion between smooth varieties, which assumes globally smooth varieties. The converse questions of the exercise -- an example with $H \supset T_{P} (V)$ and $P$ singular on $V \cap H$ , and whether $P$ must be singular in that case -- are not asserted here; the affine-linear form, the characteristic and the ambient dimension are unrestricted.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

A tangent direction is realized by a local smooth curve

Statement

Assume the Axiom of Choice. Let $k$ be an algebraically closed field, let $n \geq 1$ , and let $X \subseteq A_{k}^{n}$ be a classical variety over $k$ , embedded as a closed subvariety and carrying its reduced finite-type $k$ -scheme structure. Suppose that $X$ is smooth at the classical closed point $x \in X$ (Smooth morphisms via local standard smooth presentations) and put $d = \dim_{x} X$ , assumed to satisfy $d \geq 1$ . Then for every vector $v \in T_{x} X$ there is a reduced closed subvariety $C \subseteq X$ (The reduction of a scheme) with $x \in C$ which is smooth at $x$ and satisfies $\dim_{x} C = 1$ , and the differential $d_{x} ι_{C}$ of the closed immersion $ι_{C} : C ↪ X$ maps $T_{x} C$ isomorphically onto the line $k v \subseteq T_{x} X$ when $v \neq 0$ ; in particular $v \in d_{x} ι_{C} (T_{x} C)$ for every $v$ . When $v = 0$ the construction produces a curve whose tangent space $T_{x} C$ is a line through the origin, so that $0 \in d_{x} ι_{C} (T_{x} C)$ . The curve $C$ is closed, hence locally closed, in $X$ ; no characteristic, perfectness (beyond algebraic closedness), irreducibility or separatedness hypothesis on $X$ beyond the standing conventions is used. The dimension-zero case admits no such curve: if $d = 0$ , then no reduced closed subvariety $C \subseteq X$ with $x \in C$ and $\dim_{x} C = 1$ exists, so the hypothesis $d \geq 1$ is necessary.

Facts & Assumptions

Given: AC; an algebraically closed field $k$ ; an integer $n \geq 1$ ; a classical variety $X \subseteq A_{k}^{n}$ with closed point $x \in X$ at which $X$ is smooth; $d = \dim_{x} X \geq 1$ ; and a tangent vector $v \in T_{x} X$ . Write $a = (a_{1}, \dots, a_{n}) \in k^{n}$ for the coordinates of the point $x$ and $P = k [x_{1}, \dots, x_{n}]$ for the polynomial ring. Throughout this proof, coordinate vectors and standard basis vectors are indexed by $1, \dots, n$ : the coordinate $u_{i}$ means the value $u (i - 1)$ in the function-on- $n$ convention, and $e_{i}$ is the unit vector at $i - 1$ .

[F1]

The Axiom of Choice: every family of nonempty sets has a choice function.

[F2]

[F3]

[F4]

[F5]

[F6]

[F7]

[F8]

[F9]

[F10]

The intrinsic cotangent space and The intrinsic Zariski tangent space: $C_{x} X = m_{x} / m_{x}^{2}$ is the cotangent space and $T_{x} X = {Hom}_{κ (x)} (m_{x} / m_{x}^{2}, κ (x))$ the intrinsic tangent space; at a $k$ -rational point these are $k$ -vector spaces.

[F11]

The affine scheme of dual numbers and Tangent vectors at rational points are dual-number points: $D_{k} = Spec (k [ϵ] / (ϵ^{2}))$ , and for a $k$ -scheme $X$ and $x \in X (k)$ the intrinsic tangent space $T_{x} X$ is naturally isomorphic, as a $k$ -vector space, to the fibre over $x$ of ${Hom}_{k} (D_{k}, X) \to X (k)$ ; equivalently $T_{x} X ≅ {Der}_{k} (O_{X, x}, k)$ .

[F12]

Universal property of a polynomial ring on an arbitrary family of indeterminates: a $k$ -algebra map from $P = k [x_{1}, \dots, x_{n}]$ to a commutative $k$ -algebra is uniquely determined by arbitrary images of its $n$ variables.

[F13]

[F14]

The reduction of a scheme: the reduction $X_{r e d}$ is the closed subscheme with structure sheaf $O_{X} / N_{X}$ , where the germs of $N_{X}$ are the nilpotent elements of the local rings; on $Spec A$ it is $Spec (A / \sqrt{(0)})$ , so the stalk at $x$ is $O_{X, x} / nil (O_{X, x})$ .

[F15]

Affine schemes are contravariantly equivalent to commutative rings: for commutative unital rings $A, B$ the assignment $φ \mapsto Spec (φ)$ gives a natural bijection ${Hom}_{C R i n g} (A, B) ≅ {Hom}_{L R S} (Spec B, Spec A)$ , a contravariant equivalence on affine schemes; hence a closed subscheme of $A_{k}^{n}$ is $Spec (P / J)$ for its ideal $J \subseteq P$ and morphisms into it are the ring maps out of $P / J$ .

[F16]

[F17]

A transverse hyperplane slice is smooth at the chosen point: under AC, for $X \subseteq A_{k}^{n}$ a classical variety over an algebraically closed $k$ smooth at a closed point $x$ with $d = \dim_{x} X \geq 1$ , and an affine-linear $h = ℓ - c$ with nonzero linear part $ℓ$ , $h (x) = 0$ , such that $ℓ$ is nonzero on $T_{x} X$ , the scheme-theoretic intersection $Z = X \times_{A_{k}^{n}} V (h)$ is canonically the scheme-theoretic fibre of $h ∣_{X}$ over $0$ , its structure morphism is smooth at $x$ , $O_{Z, x}$ is a regular local ring of dimension $d - 1$ , and $T_{x} Z = \ker (d_{x} (h ∣_{X})) = {u \in T_{x} X : d_{x} (h ∣_{X}) (u) = 0}$ .

[F18]

Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: every algebraically closed field is perfect.

[F19]

Jacobian criterion and openness of the regular locus over a perfect field: under AC, for a perfect field $k$ , $P = k [x_{1}, \dots, x_{n}]$ , an ideal $I \subseteq P$ , $A = P / I$ and $q \in Spec A$ with $A_{q}$ regular, there is $t \in A ∖ q$ such that $A_{t}$ is a standard smooth $k$ -algebra; in particular such an $A$ is locally standard smooth over $k$ at every prime at which it is regular.

[F20]

The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ : the standard unit vectors $e_{i}$ form an ordered basis of $F^{n}$ with $\dim_{F} F^{n} = n$ ; a vector $u \in k^{n}$ has coordinates $u_{i} = u (i)$ and $(\sum_{i < n} λ_{i} e_{i}) (j) = λ_{j}$ , so each coordinate projection $u \mapsto u_{i}$ is a linear functional and $k^{1} = k$ has dimension $1$ .

[F21]

Linear subspace of a vector space and Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis: a linear subspace is a subset closed under the vector-space operations, and $\dim_{F} V$ is the cardinality of a basis of $V$ when $V$ is finite-dimensional.

[F22]

Rank-nullity: $\dim_{F} V = nullity T + rank T$ : for a linear map $T : V \to W$ with $V$ finite-dimensional, $\dim_{F} V = \dim_{F} (\ker T) + \dim_{F} (im T)$ ; the theorem is choice-free.

[F23]

Existence and basic properties of irreducible components: under AC, every irreducible subset of a topological space is contained in an irreducible component, and every irreducible component is closed.

[F24]

The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution and Universal property of a polynomial ring on an arbitrary family of indeterminates: $P = k [x_{1}, \dots, x_{n}]$ is the polynomial ring in the $n$ variables over $k$ , and for every commutative $k$ -algebra $S$ and every family $(s_{1}, \dots, s_{n})$ in $S$ there is a unique $k$ -algebra map $P \to S$ with $x_{i} \mapsto s_{i}$ ; in particular evaluation at $a = (a_{1}, \dots, a_{n})$ is the $k$ -algebra map $x_{i} \mapsto a_{i}$ , and expressions such as $x_{j} - λ x_{s} - μ$ are elements of $P$ .

[F25]

regular local rings are domains and cohen macaulay: under AC, every regular local ring is a domain, hence reduced.

Proof

technique · direct

1.1F2F3F4F5F6F7F8F9F11F12F21givenalgebra

Setup and dimensions. By [F2] and [F3] the closed subvariety $X \subseteq A_{k}^{n}$ is the affine model with its reduced finite-type structure, $k [X] = P / I (X)$ is reduced and generated by the finitely many classes ${\overset{ˉ}{x}}_{1}, \dots, {\overset{ˉ}{x}}_{n}$ , and the closed points of $X$ have residue field $k$ ; thus $x$ is a $k$ -rational point with coordinates $a = (a_{1}, \dots, a_{n})$ . Smoothness at $x$ means that the structure morphism is locally standard smooth at $x$ [F6], so some principal shrinking around $x$ has a standard smooth presentation [F7]; applying [F8] with $R = k$ , $p = (0)$ and $K = k$ shows that $O_{X, x}$ is a regular local ring, so $x$ is a regular point and $\dim_{k} T_{x} X = \dim O_{X, x}$ [F9], while $\dim O_{X, x} = \max_{x \in X_{i}} \dim X_{i} = \dim_{x} X = d$ by [F5] with [F4]. A based dual-number point of $A_{k}^{n}$ at $a$ is, by the affine anti-equivalence [F15] and the polynomial universal property [F12], a map $P \to k [ϵ] / (ϵ^{2})$ whose reduction modulo $ϵ$ sends $x_{i}$ to $a_{i}$ . Its variable images are therefore uniquely $x_{i} \mapsto a_{i} + ϵ w_{i}$ for a vector $w \in k^{n}$ , and conversely every such vector determines a based point. Under the $k$ -linear dual-number bijection [F11], the tangent space $T_{a} A_{k}^{n}$ is thus $k^{n}$ in these coordinates, and the composite with $X ↪ A_{k}^{n}$ sends a tangent vector of $X$ to the unique $w \in k^{n}$ whose map factors through $X$ ; write $T = {w \in k^{n} : the based dual-number point X_{i} \mapsto a_{i} + ϵ w_{i} lies in X} \subseteq k^{n} .$ Since the correspondence is $k$ -linear and bijective, $T$ is a linear subspace [F21] with $\dim_{k} T = d$ , so $T \neq 0$ because $d \geq 1$ ; let $v \in T$ denote the image of the given tangent vector $v \in T_{x} X$ , and note that all tangent-space identifications below are made with these coordinates, so that $T = T_{x} X$ as subspaces of $k^{n}$ .

1.2F6F10F11F12F13F14F15F18F19givenalgebra

Reduction does not change the tangent subspace. Let $Y \subseteq A_{k}^{n}$ be a closed subscheme of finite type over $k$ with $x \in Y (k)$ , say $Y = Spec (P / J)$ for its ideal $J$ [F15], and suppose that the local ring $O_{Y, x}$ is reduced. Then the closed immersion $Y_{r e d} ↪ Y$ has $O_{Y_{r e d}, x} = O_{Y, x} / nil (O_{Y, x}) = O_{Y, x}$ [F14], so it induces an isomorphism of local rings at $x$ , hence an isomorphism of cotangent spaces $m_{x} / m_{x}^{2}$ [F10] and, dualizing, an isomorphism $T_{x} Y_{r e d} \to T_{x} Y$ [F13]; since the composite $Y_{r e d} \to Y \to A_{k}^{n}$ is the closed immersion $Y_{r e d} ↪ A_{k}^{n}$ , the chain rule of [F13] shows that the identification of $T_{x} Y_{r e d}$ with a subspace of $T_{x} A_{k}^{n} = k^{n}$ [F11, F12] agrees with the composite of the identifications for $Y_{r e d} \to Y$ and $Y \to A_{k}^{n}$ , and since the first of these is an isomorphism the two subspaces of $k^{n}$ coincide: $T_{x} Y_{r e d} = T_{x} Y$ . This equality is the form in which the invariance under reduction is used below, and it also shows that a reduced closed subscheme with regular local ring at $x$ is smooth at $x$ : if $O_{Y, x}$ is regular, then $x$ is a regular point of $Y$ , and $Y = Spec (P / J)$ is locally standard smooth at $x$ by [F19] because $k$ is perfect [F18]; by [F6] that is smoothness of $Y$ at $x$ .

1.3F20F21F24givenalgebra

The linear forms. Assume first that $v \neq 0$ ; since $v \neq 0$ , there is a least index $s \in {1, \dots, n}$ with $v_{s} \neq 0$ in the relabelled coordinates. For every index $j \neq s$ define $ℓ_{j} (u) = u_{j} - \frac{v_{j}}{v_{s}} u_{s} (u \in k^{n}),$ a $k$ -linear functional on $k^{n}$ [F20, F21] satisfying $ℓ_{j} (v) = v_{j} - \frac{v_{j}}{v_{s}} v_{s} = 0$ ; consequently $k v \subseteq \ker ℓ_{j}$ for every $j \neq s$ , and conversely if $u \in T$ satisfies $ℓ_{j} (u) = 0$ for all $j \neq s$ , then $u_{j} = \frac{v_{j}}{v_{s}} u_{s}$ for all $j \neq s$ and therefore $u = \frac{u_{s}}{v_{s}} v$ , so $T \cap ⋂_{j \neq s} \ker ℓ_{j} = k v .$ For each $j \neq s$ put $h_{j} = x_{j} - \frac{v_{j}}{v_{s}} x_{s} - (a_{j} - \frac{v_{j}}{v_{s}} a_{s}) \in P$ [F24]; then $h_{j} (a) = 0$ and $h_{j}$ is affine-linear with linear part $ℓ_{j}$ , because $h_{j} (a + ϵ w) = h_{j} (a) + ϵ ℓ_{j} (w)$ for every $w \in k^{n}$ , and $ℓ_{j} \neq 0$ because $ℓ_{j} (v) = 0$ and $ℓ_{j} (e_{j}) = 1$ for $j \neq s$ .

2.1F20F22step 1.3givenalgebra

The active indices. Recursively for $j = 1, \dots, n$ , put $T^{(0)} : = T$ and $T^{(j)} : = T^{(j - 1)} \cap \ker ℓ_{j} if j \neq s and ℓ_{j} \neq 0 on T^{(j - 1)}, T^{(j)} : = T^{(j - 1)} otherwise,$ the first case being called active at $j$ ; let $J$ be the finite set of active indices, listed in increasing order as $J = {j_{1} < \dots < j_{m}}$ . At an active index the restriction $ℓ_{j} ∣_{T^{(j - 1)}}$ is a nonzero linear map to $k = k^{1}$ , so its image has dimension $1$ [F20] and rank-nullity [F22] gives $\dim_{k} T^{(j)} = \dim_{k} T^{(j - 1)} - 1$ , while at an inactive index $T^{(j)} = T^{(j - 1)}$ ; hence $\dim_{k} T^{(j)} = d - # {i : j_{i} \leq j}$ for every $j$ , and $\dim_{k} T^{(n)} = d - m$ . Moreover $T^{(n)} = k v$ : on the one hand every $T^{(j)}$ contains $v$ because $v \in T$ and $ℓ_{j^{'}} (v) = 0$ for all $j^{'} \neq s$ [step 1.3], so $k v \subseteq T^{(n)}$ ; on the other hand if $u \in T^{(n)}$ and $j \neq s$ , then either $j$ is active, in which case $T^{(n)} \subseteq T^{(j)} \subseteq \ker ℓ_{j}$ , or $j$ is inactive, in which case $ℓ_{j}$ vanishes on $T^{(j - 1)} \supseteq T^{(n)}$ ; so $u \in T \cap ⋂_{j \neq s} \ker ℓ_{j} = k v$ by step 1.3, giving $T^{(n)} \subseteq k v$ . Therefore $\dim_{k} T^{(n)} = 1$ and $m = d - 1$ .

3.1F4F5F14F15F16F17F25step 1.2step 1.3step 2.1givenalgebra

The induction on the active slices. Put $Y_{0} : = X$ , and for $i = 1, \dots, m$ define $Z_{i} : = Y_{i - 1} \times_{A_{k}^{n}} H_{i}$ , where $H_{i} = V (h_{j_{i}})$ is the hyperplane cut out by the affine-linear polynomial of step 1.3 for the index $j_{i}$ , and put $Y_{i} : = (Z_{i})_{r e d}$ [F14, F15, F16]. Each $Z_{i}$ is a closed subscheme of $Y_{i - 1}$ (base change of the closed immersion $H_{i} ↪ A_{k}^{n}$ ) and each $Y_{i}$ is a reduced closed subvariety of $X$ containing $x$ , because $x \in X = Y_{0}$ and each $h_{j_{i}}$ vanishes at $x$ [step 1.3] so the $k$ -point $x$ lifts to $Z_{i}$ and to $Y_{i}$ . The induction claim is: $T_{x} Y_{i} = T^{(j_{i})}$ as subspaces of $k^{n}$ , $\dim_{k} T_{x} Y_{i} = d - i$ , $O_{Y_{i}, x}$ is a regular local ring of dimension $d - i$ , $\dim_{x} Y_{i} = d - i$ , and $Y_{i}$ is smooth at $x$ . For $i = 0$ this is step 1.1 together with the given smoothness. Assume the claim for $i - 1$ with $1 \leq i \leq m = d - 1$ ; then $d - i + 1 \geq 2 \geq 1$ , so the slice lemma [F17] applies to the variety $Y_{i - 1}$ , smooth at $x$ by the induction claim, with the affine-linear form $h_{j_{i}}$ of linear part $ℓ_{j_{i}}$ : the index $j_{i}$ is active, which means $ℓ_{j_{i}}$ is nonzero on $T^{(j_{i} - 1)}$ , and $T^{(j_{i} - 1)} = T^{(j_{i - 1})} = T_{x} Y_{i - 1}$ (for $i = 1$ , $T^{(j_{1} - 1)} = T^{(0)} = T_{x} X$ because all indices below $j_{1}$ are inactive), so the transversality hypothesis holds. The slice lemma gives that $Z_{i}$ is smooth at $x$ over $k$ with $O_{Z_{i}, x}$ a regular local ring of dimension $(d - i + 1) - 1 = d - i$ and $T_{x} Z_{i} = \ker (d_{x} (h_{j_{i}} ∣_{Y_{i - 1}})) = {u \in T_{x} Y_{i - 1} : ℓ_{j_{i}} (u) = 0} = T^{(j_{i} - 1)} \cap \ker ℓ_{j_{i}} = T^{(j_{i})} .$ Since $O_{Z_{i}, x}$ is regular, it is reduced by [F25], so step 1.2 applied to $Y = Z_{i}$ gives $T_{x} Y_{i} = T_{x} (Z_{i})_{r e d} = T_{x} Z_{i} = T^{(j_{i})}$ and $O_{Y_{i}, x} = O_{Z_{i}, x}$ , a regular local ring of dimension $d - i$ ; moreover $Y_{i}$ is reduced, so [F5] with [F4] gives $\dim_{x} Y_{i} = \dim O_{Y_{i}, x} = d - i$ , and $Y_{i}$ is smooth at $x$ by the second part of step 1.2. This proves the claim for $i$ and completes the induction.

4.1step 1.1step 3.1algebra

The case $v = 0$ . If $v = 0$ , then $T \neq 0$ by step 1.1, so choose any nonzero $u \in T$ and apply the construction of steps 1.3, 2.1 and 3.1 to $u$ in place of $v$ ; it yields a curve $C$ with $T_{x} C$ mapped isomorphically onto the line $k u$ , and $v = 0 \in k u = d_{x} ι_{C} (T_{x} C)$ .

4.2F13step 1.1step 1.2step 2.1step 3.1givenalgebra

Conclusion for nonzero $v$ . Let $v \neq 0$ and put $C : = Y_{m} = Y_{d - 1}$ with the notation of step 3.1; then $C \subseteq X$ is a reduced closed subvariety with $x \in C$ , smooth at $x$ , and step 3.1 at $i = m = d - 1$ gives $T_{x} C = T^{(n)} = k v$ (if $m > 0$ , the last active slice has $T_{x} C = T^{(j_{m})} = T^{(n)}$ ; if $m = 0$ , no slice occurs and $T_{x} C = T = T^{(n)}$ ) [step 2.1] and $\dim_{x} C = 1$ . Under the closed immersion $C ↪ X$ the differential $d_{x} ι_{C}$ is injective and the diagram with the two ambient identifications commutes [F13, step 1.1, step 1.2], so $d_{x} ι_{C}$ carries $T_{x} C$ isomorphically onto the subspace of $T_{x} X$ whose ambient image is $k v$ , namely onto $k v$ itself; in particular $v \in d_{x} ι_{C} (T_{x} C)$ .

5.1F1F4F5F6F8F17F19F23step 1.1step 1.3step 2.1givenalgebra∎

Boundaries. If $d = 1$ then $m = 0$ [step 2.1] and $C : = Y_{0} = X$ works: $X$ is reduced with $x \in X$ , smooth at $x$ by hypothesis with $\dim_{x} X = 1$ , and $T_{x} X = T$ is one-dimensional, so $T = k v$ for the nonzero $v$ [step 1.1]. If $d = 0$ no such curve exists: a reduced closed subvariety $C \subseteq X$ with $x \in C$ and $\dim_{x} C = 1$ has, by [F5] with [F4], an irreducible component of $C$ containing $x$ of dimension $1$ , which is an irreducible closed subset of $X$ passing through $x$ and hence is contained in an irreducible component of $X$ containing $x$ [F23], so that $\dim_{x} X \geq 1$ by [F4], a contradiction; this is why the hypothesis $d \geq 1$ is stated. The construction divides only by $v_{s} \neq 0$ [step 1.3], so no characteristic hypothesis is needed and the affine-linear forms $h_{j}$ are available in every characteristic; the ambient dimension $n \geq 1$ may equal $1$ , in which case $d \leq 1$ and the case $d = 1$ above applies; $X$ may be reducible, and the curve $C$ produced is closed in $X$ , hence locally closed. The Axiom of Choice enters the statement through [F1] and is used only through the suppliers that assume it, namely [F4], [F5], [F6], [F8], [F17], [F18 as used through F19] and [F23], [F25], each cited at the step that uses it; the explicit linear forms $ℓ_{j}$ , the finite recursion defining the active set, the polynomials $h_{j}$ , the enumerations and all tangent identifications involve no selection, and [F20], [F22] and the reductions of [F14] are choice-free.

Source qualification

Milne, Algebraic Geometry v6.10, Exercise 4-3 (printed p. 98) asks: "Given a smooth point on a variety and a tangent vector at the point, show that there is a smooth curve passing through the point with the given vector as its tangent vector (see mo111467)." The solution printed at p. 222 argues by choosing suitable hypersurfaces through the point with linearly independent differentials and citing the predecessor Exercise 4-2; the item above makes that argument scheme-precise: it constructs the hyperplanes $H_{i}$ from explicit coordinate forms $ℓ_{j} (u) = u_{j} - \frac{v_{j}}{v_{s}} u_{s}$ , replaces the intermediate intersections by their reductions so that each step can invoke A transverse hyperplane slice is smooth at the chosen point verbatim, and obtains the curve as a reduced closed subvariety (the exercise asks only for a locally closed curve). Milne works over an algebraically closed field with classical varieties and radical vanishing ideals; the item allows reducible $X$ and records the dimension-zero obstruction. No smoothness of the curve away from $x$ is claimed, and no characteristic hypothesis is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

A dominant map has a surjective differential on a dense source open

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field of characteristic $0$ , let $X$ and $Y$ be irreducible classical varieties over $k$ , and let $f ⁣ : X \to Y$ be a dominant morphism (Dominant classical morphisms and rational maps). Regard $X$ and $Y$ as integral finite-type $k$ -schemes under Irreducible classical varieties and integral separated finite-type schemes, and let $X_{r e g}$ , $Y_{r e g}$ be their regular loci (Regular and singular loci). Then there exist nonempty open subsets $U \subseteq X_{r e g}$ and $V \subseteq Y_{r e g}$ with $f (U) \subseteq V$ such that for every closed point $x \in U$ , with $y = f (x)$ , the differential $d_{x} f ⁣ : T_{x} X ⟶ T_{y} Y$ of Differentials, open restriction, and the chain rule is surjective. In the construction $V$ is taken to be $Y_{r e g}$ , and $U$ is of the form $U = X_{r e g} \cap f^{- 1} (Y_{r e g}) \cap D (H)$ for a nonempty affine chart $Spec S \subseteq X$ lying inside an affine chart $Spec R \subseteq Y$ and a nonzero element $H \in S$ ; thus $U$ is a nonempty open subset of $X_{r e g}$ , dense in $X$ . No smoothness of $X$ or of $Y$ is assumed on the complements of the two regular loci.

Facts & Assumptions

Given: An algebraically closed field $k$ of characteristic $0$ , irreducible classical varieties $X, Y$ over $k$ , a dominant morphism $f ⁣ : X \to Y$ , and the Axiom of Choice.

[F1]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function.

[F2]

Irreducible classical varieties and integral separated finite-type schemes: over the algebraically closed field $k$ , under AC the closed-point construction and its inverse give an equivalence between irreducible classical $k$ -varieties and integral finite-type $k$ -schemes satisfying the affine-overlap separation condition, each original point being identified with its singleton, so classical points correspond to closed points.

[F3]

Dominant classical morphisms and rational maps: a morphism $ϕ ⁣ : U \to Y$ from a nonempty open subset of an affine variety to an affine variety is dominant when the closure of $ϕ (U)$ is $Y$ ; for morphisms of varieties this is density of the image.

[F4]

Classical varieties have finite irreducible decompositions: every classical variety is Noetherian and has finitely many irreducible components; every open or closed subvariety has a finite affine cover.

[F5]

The coordinate ring of a classical affine algebraic set: for an affine algebraic set $W \subseteq k^{n}$ the coordinate ring is $k [W] = k [x_{1}, \dots, x_{n}] / I (W)$ , it is reduced, and the finitely many coordinate classes generate it as a $k$ -algebra.

[F6]

A classical affine variety has a domain coordinate ring, and conversely: under AC an affine algebraic set $W$ is a classical affine variety if and only if $k [W]$ is a nonzero integral domain.

[F7]

Classical affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms: under AC, pullback gives a natural bijection ${Mor}_{k} (W, W^{'}) ≅ {Hom}_{k -alg} (k [W^{'}], k [W])$ for affine algebraic sets, reversing composition and preserving identities.

[F8]

Irreducibility via nonempty open subsets, connectedness and open subspaces: a space is irreducible if and only if it is nonempty and every two nonempty open subsets meet, if and only if it is nonempty and every nonempty open subset is dense; a nonempty open subspace of an irreducible space is irreducible.

[F9]

Function fields and dominant pullbacks on general varieties: under AC, for irreducible classical $X$ the fraction fields of all nonempty affine charts identify canonically with $k (X)$ , and a dominant morphism $f ⁣ : X \to Y$ between irreducible classical varieties induces an injection $f^{*} ⁣ : k (Y) ↪ k (X)$ .

[F10]

Finitely generated field extensions $F (a_{1}, \dots, a_{r})$ : $L / K$ is finitely generated when $L = K (a_{1}, \dots, a_{r})$ for a finite list.

[F11]

Dimension equals transcendence degree: under AC, if $X$ is an irreducible classical variety then $\dim X = {trdeg}_{k} k (X) < \infty$ .

[F12]

Transcendence degree is additive in finite towers: for a tower $k \subseteq K \subseteq L$ with finite transcendence degrees, ${trdeg}_{k} L = {trdeg}_{k} K + {trdeg}_{K} L$ .

[F13]

Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: every field of characteristic zero is perfect.

[F14]

Finitely generated extensions of a perfect field are separably generated: a finitely generated field extension of a perfect field has a separating transcendence basis.

[F15]

Differentials of a separably generated field extension: if $K \subseteq L$ is a finitely generated field extension separably generated over $K$ by $t_{1}, \dots, t_{r}$ , then $d t_{1}, \dots, d t_{r}$ are an $L$ -basis of $Ω_{L / K}$ .

[F16]

Localization, base change and functoriality of differentials: for a ring map $A \to B$ : base change gives $B^{'} \otimes_{B} Ω_{B / A} ≅ Ω_{B^{'} / A^{'}}$ ; localization gives $U^{- 1} Ω_{B / A} ≅ Ω_{U^{- 1} B / V^{- 1} A}$ ; and a ring map $B \to C$ carries a canonical $C$ -linear functoriality map $C \otimes_{B} Ω_{B / A} \to Ω_{C / A}$ , $c \otimes d b \mapsto c d (image of b)$ .

[F17]

Localisation of modules is extension of scalars: for a multiplicative subset $S \subseteq R$ the map $S^{- 1} R \otimes_{R} M \to S^{- 1} M$ , $(a / s) \otimes m \mapsto a m / s$ , is an isomorphism.

[F18]

Localisation of a module at a multiplicative subset: elements of $S^{- 1} M$ are fractions $m / s$ , and $m / 1 = 0$ exactly when $u m = 0$ for some $u \in S$ .

[F19]

The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain: for an integral domain $D$ , its field of fractions is $Frac (D) = (D ∖ {0})^{- 1} D$ , with elements $a / b$ , $b \neq 0$ .

[F20]

Differentials of a polynomial quotient and the Jacobian cokernel: for $P = A [x_{1}, \dots, x_{n}]$ and $B = P / I$ , the module $Ω_{P / A}$ is free with basis $d x_{1}, \dots, d x_{n}$ , and the sequence $I / I^{2} \to B \otimes_{P} Ω_{P / A} \to Ω_{B / A} \to 0$ is exact.

[F21]

Universal property of a polynomial ring on an arbitrary family of indeterminates: a ring map $R \to S$ and elements $s_{i} \in S$ extend uniquely to a ring map $R [x_{i}] \to S$ with $x_{i} \mapsto s_{i}$ .

[F22]

Tensoring is right exact: tensoring a right-exact sequence $A \to B \to C \to 0$ by any module preserves right exactness.

[F23]

Every spanning subset of a vector space contains a basis: under AC, every subset of a vector space that spans it contains a basis.

[F24]

Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis: for a finite-dimensional vector space $V$ over a field, $\dim_{F} V$ is the number of elements of a basis.

[F25]

Change of rings: $N \otimes_{R} M ≅ N \otimes_{S} (S \otimes_{R} M)$ : for $R \to S$ , a right $S$ -module $N$ and a left $R$ -module $M$ , there is a natural isomorphism $N \otimes_{R} M ≅ N \otimes_{S} (S \otimes_{R} M)$ .

[F26]

Cotangent space at a rational point: at a $k$ -rational point $x$ of a $k$ -scheme, the map $m / m^{2} \to Ω_{X / k} \otimes_{O_{X, x}} κ (x)$ , $[a] \mapsto d a \otimes 1$ , is an isomorphism.

[F27]

Differentials, open restriction, and the chain rule: at $k$ -rational points the local map induces ${\overset{ˉ}{f}}_{x}^{♯} ⁣ : C_{y} Y \to C_{x} X$ , whose dual is $d_{x} f = ({\overset{ˉ}{f}}_{x}^{♯})^{*} ⁣ : T_{x} X \to T_{y} Y$ .

[F28]

Transitivity sequence for differentials: for ring maps $A \to B \to C$ the sequence $C \otimes_{B} Ω_{B / A} \to Ω_{C / A} \to Ω_{C / B} \to 0$ is exact.

[F29]

The map of affine spectra induced by a ring homomorphism: a ring map $φ ⁣ : A \to B$ gives the contraction map $Spec B \to Spec A$ , $q \mapsto φ^{- 1} q$ , whose sheaf map on $D (f)$ is the localization map $A_{f} \to B_{φ (f)}$ .

[F30]

The stalk maps induced by a ring map are local: the induced stalk homomorphism $A_{p} \to B_{q}$ at $p = φ^{- 1} q$ is local.

[F31]

The stalk of the affine structure sheaf at a prime is A_p: for $p \in Spec A$ there is a canonical isomorphism $O_{Spec A, p} ≅ A_{p}$ .

[F32]

Affine charts recover the algebraic module of differentials: for $Spec B \to Spec A$ one has $Γ (Spec B, Ω) = Ω_{B / A}$ and $Ω (D (g)) ≅ Ω_{B_{g} / A}$ , compatibly with the universal derivations and localization.

[F33]

Assuming choice, $\ker T^{*} = (im T)^{\circ}$ and $im T^{*} = (\ker T)^{\circ}$ ; in finite dimensions $rank T^{*} = rank T$ : under AC, for a linear map $T ⁣ : V \to W$ of finite-dimensional spaces, $rank T^{*} = rank T$ .

[F34]

Rank-nullity: $\dim_{F} V = nullity T + rank T$ : for a linear map $T$ with $V$ finite-dimensional, $\dim_{F} V = nullity T + rank T$ .

[F35]

Regular and singular loci: the regular locus is the set of points with regular local ring; under AC, for a reduced classical finite-type space over an algebraically closed field and a closed point $x$ , $x \in X_{r e g}$ if and only if $\dim_{κ (x)} T_{x} X = \dim_{x} X$ .

[F36]

Dense regular loci on every component: under AC, for a perfect field $k$ and a reduced finite-type $k$ -scheme $X$ , the regular locus is open, meets every irreducible component in a dense open subset of it, and is nonempty when $X \neq \emptyset$ .

[F37]

Global and local dimension of classical varieties: for a classical variety with irreducible components $X_{i}$ and a closed point $x$ , $\dim_{x} X = \max_{x \in X_{i}} \dim X_{i}$ , and $\dim X$ is the chain dimension.

[F38]

The spectrum of a principal localisation is the distinguished open D(f): the localization map $R \to R_{f}$ induces a homeomorphism from $Spec R_{f}$ onto the distinguished open subset $D (f) = {p : f \notin p}$ .

[F39]

The intrinsic Zariski tangent space: for a locally finite-type $k$ -scheme the intrinsic tangent space at any point is finite-dimensional over the residue field, and at a $k$ -rational point the intrinsic and relative tangent spaces agree.

Proof

technique · direct

1.1F2F3F4F5F6F7F8F9givenalgebra

The variety $Y$ is nonempty and irreducible, so it has a nonempty affine chart $Y_{0} = Spec R$ [F4], and by dominance [F3] the open subset $f^{- 1} (Y_{0}) \subseteq X$ is nonempty; it therefore contains a nonempty affine chart $X_{0} = Spec S \subseteq X$ [F4]. The rings $R = k [Y_{0}]$ and $S = k [X_{0}]$ are finitely generated $k$ -algebras given by finitely many coordinate classes [F5], and because $Y_{0}$ and $X_{0}$ are nonempty open subsets of irreducible spaces they are themselves irreducible [F8], so $R$ and $S$ are nonzero integral domains [F6]. The restriction $f ∣_{X_{0}} ⁣ : X_{0} \to Y_{0}$ is a morphism of affine varieties, so its pullback is a $k$ -algebra homomorphism $φ ⁣ : R \to S$ [F7]. The fraction fields of the charts identify canonically with $k (Y)$ and $k (X)$ , and dominance makes $f^{*} ⁣ : k (Y) ↪ k (X)$ injective [F9]; carrying the affine pullback through these identifications exhibits the map $Frac R \to Frac S$ induced by $φ$ as $f^{*}$ , so $φ$ is injective. Write $K : = Frac R = k (Y)$ and $L : = Frac S = k (X)$ .

2.1F5F10F11F12step 1.1givenalgebra

Write the coordinate generators of $S$ as $s_{1}, \dots, s_{m}$ , so that $S = k [s_{1}, \dots, s_{m}]$ and $L = k (s_{1}, \dots, s_{m})$ is finitely generated over $k$ [F5, F10]; in the same way $K = k (r_{1}, \dots, r_{n})$ for the coordinate generators of $R$ [F5, F10]. Since $φ$ is injective, $K \subseteq L$ , so $L / K$ is finitely generated [F10]. By [F11], $\dim Y = {trdeg}_{k} K$ and $\dim X = {trdeg}_{k} L$ are finite, so the tower $k \subseteq K \subseteq L$ has, by [F12], $r : = {trdeg}_{K} L = {trdeg}_{k} L - {trdeg}_{k} K = \dim X - \dim Y \geq 0.$ Moreover, since $k \subseteq R$ and the $s_{i}$ generate $S$ as a $k$ -algebra, they generate $S$ as an $R$ -algebra: the $R$ -subalgebra they generate is a $k$ -subalgebra containing every $s_{i}$ , hence equals $S$ [F5].

3.1F13F14F15step 2.1givenalgebra

The field $K$ contains $k$ and so has characteristic $0$ , hence is perfect [F13]. By [F14] the finitely generated extension $L / K$ has a separating transcendence basis $t_{1}, \dots, t_{r}$ ; a separating transcendence basis is in particular a transcendence basis, so its length is ${trdeg}_{K} L = r$ as computed in step 2.1. By [F15] the differentials $d t_{1}, \dots, d t_{r}$ form an $L$ -basis of $Ω_{L / K}$ . In particular $\dim_{L} Ω_{L / K} = r$ ; for $r = 0$ the empty list is the basis and $Ω_{L / K} = 0$ .

3.2F20F21step 2.1givenalgebra

By step 2.1 the elements $s_{1}, \dots, s_{m}$ generate $S$ as an $R$ -algebra, so by the universal property [F21] there is a surjective $R$ -algebra map $R [x_{1}, \dots, x_{m}] \to S$ with $x_{i} \mapsto s_{i}$ , whose kernel we call $I$ . By [F20], $Ω_{R [x_{1}, \dots, x_{m}] / R}$ is free with basis $d x_{1}, \dots, d x_{m}$ , and the sequence $I / I^{2} \to S \otimes_{R [x_{1}, \dots, x_{m}]} Ω_{R [x_{1}, \dots, x_{m}] / R} \to Ω_{S / R} \to 0$ is exact. Therefore $Ω_{S / R}$ is a quotient of the free $S$ -module with basis $d x_{1}, \dots, d x_{m}$ , and in particular it is generated as an $S$ -module by the $m$ elements $g_{1}, \dots, g_{m}$ , where $g_{i}$ is the image of $d x_{i}$ .

4.1F16F17F19step 3.1givenalgebra

Apply the localization clause of [F16] to the injective ring map $φ ⁣ : R \to S$ with $U = S ∖ {0}$ and $V = R ∖ {0}$ : the image of $V$ in $S$ is contained in $U$ because $φ$ is injective, and $V^{- 1} R = K$ , $U^{- 1} S = L$ are the fraction fields [F19]. This gives an isomorphism $U^{- 1} Ω_{S / R} ≅ Ω_{L / K}$ , and [F17] identifies $U^{- 1} Ω_{S / R}$ with $L \otimes_{S} Ω_{S / R}$ . Hence $Ω_{S / R} \otimes_{S} L ≅ Ω_{L / K}$ and, by step 3.1, the $L$ -vector space $Ω_{S / R} \otimes_{S} L$ has dimension $r$ .

5.1F18step 3.1step 4.1givenalgebra

For each $i$ , the element $d t_{i}$ of $Ω_{S / R} \otimes_{S} L = U^{- 1} Ω_{S / R}$ from step 4.1 has the form $ω_{i} / h_{i}$ with $ω_{i} \in Ω_{S / R}$ and $0 \neq h_{i} \in S$ [F18]. If $λ ⁣ : Ω_{S / R} \to Ω_{S / R} \otimes_{S} L$ is the localization map, then $λ (ω_{i}) = h_{i} d t_{i}$ , which is nonzero because $h_{i} \neq 0$ in the field $L$ and $d t_{i}$ is part of an $L$ -basis (step 3.1). The two $L$ -spans agree, ${span}_{L} (λ (ω_{1}), \dots, λ (ω_{r})) = {span}_{L} (d t_{1}, \dots, d t_{r}) = Ω_{S / R} \otimes_{S} L,$ since each $λ (ω_{i}) = h_{i} d t_{i}$ lies in the span of the $d t_{i}$ and each $d t_{i} = h_{i}^{- 1} λ (ω_{i})$ lies in the span of the $λ (ω_{i})$ .

6.1F18F19step 3.2step 5.1givenalgebra

Each generator $g_{j}$ of step 3.2 satisfies $λ (g_{j}) \in {span}_{L} (λ (ω_{1}), \dots, λ (ω_{r}))$ by step 5.1, say $λ (g_{j}) = \sum_{i = 1}^{r} c_{j i} λ (ω_{i})$ with $c_{j i} \in L$ . Since $L = Frac S = (S ∖ {0})^{- 1} S$ [F19], the finitely many coefficients have a common denominator: $c_{j i} = a_{j i} / h$ with $a_{j i} \in S$ and $0 \neq h \in S$ . Then $λ (h g_{j} - \sum_{i} a_{j i} ω_{i}) = 0$ , so by the kernel criterion for localizations [F18] there is $0 \neq u_{j} \in S$ with $u_{j} (h g_{j} - \sum_{i} a_{j i} ω_{i}) = 0$ in $Ω_{S / R}$ . Put $H : = h \prod_{j = 1}^{m} u_{j} \neq 0$ . In the localization $Ω_{S / R} [1 / H]$ the element $u_{j} h$ is invertible, and the relations rewrite as $g_{j} = \sum_{i} (u_{j} a_{j i} / (u_{j} h)) ω_{i}$ with coefficients in $S_{H}$ . Hence $Ω_{S / R} [1 / H]$ is generated over $S_{H}$ by $ω_{1}, \dots, ω_{r}$ . (For $r = 0$ , step 5.1 gives $λ (g_{j}) \in {span}_{L} \emptyset = 0$ , so $u_{j} g_{j} = 0$ for suitable $0 \neq u_{j} \in S$ and $Ω_{S / R} [1 / H] = 0$ with $H : = \prod_{j} u_{j}$ , generated by the empty family.)

7.1F22F23F24F25step 6.1givenalgebra

Let $p \in Spec S$ satisfy $H \notin p$ . Applying [F25] with $R \to S$ the localization $S \to S_{H}$ , $N = κ (p)$ and $M = Ω_{S / R}$ , and identifying $S_{H} \otimes_{S} Ω_{S / R}$ with $Ω_{S / R} [1 / H]$ via [F17], shows $Ω_{S / R} \otimes_{S} κ (p) ≅ κ (p) \otimes_{S_{H}} Ω_{S / R} [1 / H] .$ The right-hand side is generated as a $κ (p)$ -vector space by the images of $ω_{1}, \dots, ω_{r}$ , because $Ω_{S / R} [1 / H]$ is generated over $S_{H}$ by these $r$ elements (step 6.1) and $κ (p) \otimes_{S_{H}} -$ is right exact [F22]. A vector space spanned by $r$ elements contains a basis inside that spanning set [F23] and therefore has dimension at most $r$ [F24]. Thus $\dim_{κ (p)} (Ω_{S / R} \otimes_{S} κ (p)) \leq r .$

7.2F2F3F8F13F36F38step 6.1givenalgebra

Let $X_{r e g}$ and $Y_{r e g}$ be the regular loci of the schemes $X$ and $Y$ [F35]. The field $k$ is perfect [F13], and $X$ , $Y$ are reduced finite-type $k$ -schemes via [F2]; hence by [F36] the two regular loci are open, and each meets every irreducible component of its scheme in a dense open subset. Since $X$ and $Y$ are irreducible, $X_{r e g}$ and $Y_{r e g}$ are nonempty dense open subsets of $X$ and $Y$ . The preimage $f^{- 1} (Y_{r e g}) \subseteq X$ is nonempty, because the dense image $f (X)$ meets the nonempty open set $Y_{r e g}$ [F3, F8], and it is open. The principal open $D (H) = {p \in Spec S : H \notin p}$ is an open subset of the chart $X_{0}$ [F38] and it is nonempty because $H \neq 0$ (step 6.1), so $D (H)$ is a nonempty open subset of $X$ . Define $U : = D (H) \cap X_{r e g} \cap f^{- 1} (Y_{r e g}) \subseteq X, V : = Y_{r e g} \subseteq Y .$ The three sets displayed are nonempty open subsets of the irreducible space $X$ [F8], so $U$ is a nonempty open subset of $X$ with $U \subseteq X_{r e g}$ and $f (U) \subseteq V$ .

8.1F2F16F22F25F26F27F28F29F30F31F32step 7.2givenalgebra

Fix a closed point $x \in U$ and put $y = f (x) \in V = Y_{r e g}$ ; both are $k$ -rational points of the respective schemes [F2]. Let $m_{x} \subseteq S$ and $m_{y} \subseteq R$ be the corresponding maximal ideals. The local rings are $O_{Y, y} = R_{m_{y}}$ and $O_{X, x} = S_{m_{x}}$ [F31], and the induced map of local rings is the localization of $φ$ at these primes, which sends $a$ to $φ (a)$ [F29] and is local [F30]; hence the cotangent map ${\overset{ˉ}{f}}_{x}^{♯} ⁣ : C_{y} Y \to C_{x} X$ of [F27] sends the class of $a \in m_{y}$ to the class of $φ (a)$ . The cotangent isomorphism [F26] at the $k$ -rational points, applied on the charts $Spec R$ and $Spec S$ and combined with [F32], gives identifications $θ_{y} ⁣ : C_{y} Y ≅ Ω_{R / k} \otimes_{R} κ (y), θ_{x} ⁣ : C_{x} X ≅ Ω_{S / k} \otimes_{S} κ (x),$ both sending $[a] \mapsto d a \otimes 1$ . Let $α ⁣ : S \otimes_{R} Ω_{R / k} \to Ω_{S / k}$ be the functoriality map of F16, $b \otimes d a \mapsto b d φ (a)$ ; it is the first map of the exact sequence $S \otimes_{R} Ω_{R / k} \overset{α}{\to} Ω_{S / k} \to Ω_{S / R} \to 0$ of [F28]. Base changing this sequence along $S \to κ (x)$ and using the identification $(S \otimes_{R} Ω_{R / k}) \otimes_{S} κ (x) ≅ Ω_{R / k} \otimes_{R} κ (x)$ of [F25], together with the fact that $R \to κ (x)$ is evaluation at $y$ , yields the exact sequence $Ω_{R / k} \otimes_{R} κ (y) \overset{α_{x}}{\to} Ω_{S / k} \otimes_{S} κ (x) \to Ω_{S / R} \otimes_{S} κ (x) \to 0$ [F22]. For $a \in m_{y}$ one computes $α_{x} (θ_{y} ([a])) = α_{x} (d a \otimes 1) = d φ (a) \otimes 1 = θ_{x} ([φ (a)]) = θ_{x} ({\overset{ˉ}{f}}_{x}^{♯} ([a]))$ , so under the identifications $θ_{y}$ , $θ_{x}$ the map $α_{x}$ is precisely the cotangent map ${\overset{ˉ}{f}}_{x}^{♯}$ ; in particular $rank α_{x} = rank {\overset{ˉ}{f}}_{x}^{♯}$ .

9.1F26F33F34F39step 8.1givenalgebra

By [F27], $d_{x} f$ is the transpose of ${\overset{ˉ}{f}}_{x}^{♯}$ , so by [F33] its rank equals $rank {\overset{ˉ}{f}}_{x}^{♯}$ ; all tangent and cotangent spaces here are finite-dimensional [F39, F26]. Hence $rank d_{x} f = rank α_{x}$ . Since $Ω_{R / k} \otimes_{R} κ (y) \to Ω_{S / k} \otimes_{S} κ (x) \to Ω_{S / R} \otimes_{S} κ (x) \to 0$ is exact (step 8.1), the rank-nullity theorem [F34] applied to $α_{x}$ gives $rank d_{x} f = \dim_{k} (Ω_{S / k} \otimes_{S} κ (x)) - \dim_{κ (x)} (Ω_{S / R} \otimes_{S} κ (x)) = \dim_{k} T_{x} X - \dim_{κ (x)} (Ω_{S / R} \otimes_{S} κ (x)),$ the last equality because $θ_{x}$ identifies $Ω_{S / k} \otimes_{S} κ (x)$ with $C_{x} X$ [F26] and $C_{x} X$ is the dual of the finite-dimensional space $T_{x} X$ [F39].

10.1F35F37step 2.1step 7.1step 9.1givenalgebra

Since $x \in X_{r e g}$ , the classical dimension test [F35] gives $\dim_{k} T_{x} X = \dim_{x} X$ , and since $X$ is irreducible its only irreducible component is $X$ itself, so $\dim_{x} X = \dim X$ by [F37]. Likewise $y \in Y_{r e g}$ gives $\dim_{k} T_{y} Y = \dim_{y} Y = \dim Y$ by [F35, F37]. The point $x$ lies in $D (H)$ , so $H \notin m_{x}$ and step 7.1 bounds $\dim_{κ (x)} (Ω_{S / R} \otimes_{S} κ (x)) \leq r$ . Therefore step 9.1 and $r = \dim X - \dim Y$ (step 2.1) give $rank (d_{x} f) = \dim X - \dim_{κ (x)} (Ω_{S / R} \otimes_{S} κ (x)) \geq \dim X - r = \dim Y = \dim_{k} T_{y} Y .$ A linear map has rank at most the dimension of its target, so $rank (d_{x} f) = \dim_{k} T_{y} Y$ and $d_{x} f ⁣ : T_{x} X \to T_{y} Y$ is surjective. This holds at every closed point $x$ of the nonempty open set $U$ , and $f (U) \subseteq V \subseteq Y_{r e g}$ , which is the assertion.

11.1F1F2F6F7F8F9F11F23F33F36step 3.1step 6.1step 7.1step 7.2step 10.1givenalgebra∎

Boundary and scope dispositions. Empty: $X$ and $Y$ are nonempty because irreducible means nonempty [F8], so the charts and the open $U$ of step 7.2 are nonempty, and no empty-case convention is needed; the sets $X_{r e g}$ , $Y_{r e g}$ are nonempty by [F36], and if $Y$ is a point then $r = \dim X$ , and the separating basis in step 3.1 is empty exactly when $\dim X = 0$ . Zero: the case $r = \dim X - \dim Y = 0$ is covered by the empty-list convention of steps 3.1 and 6.1: then $Ω_{L / K} = 0$ , $Ω_{S / R} [1 / H] = 0$ , so step 7.1 gives $\dim_{κ (x)} (Ω_{S / R} \otimes_{S} κ (x)) = 0$ , and step 10.1 concludes $rank (d_{x} f) = \dim X = \dim Y$ with no modification; likewise $\dim Y = 0$ forces $Y$ to be a single point in the present irreducible setting, $T_{y} Y = 0$ , and surjectivity is the equality $rank = 0$ already obtained. One: nothing in the argument divides by a natural number or assumes a generator count $\geq 1$ ; the lists $s_{1}, \dots, s_{m}$ , $r_{1}, \dots, r_{n}$ , $g_{1}, \dots, g_{m}$ and $t_{1}, \dots, t_{r}$ may have length one or zero, and the length-one case $r = 1$ has $\dim X - \dim Y = 1$ ; generically finite maps instead have $r = 0$ , and both cases are covered by the same argument. Degenerate: the proof does not require $φ$ to be surjective or the charts to be smooth, and it does not require $f$ to be finite or flat; the degenerate dominant case $X \to Y = Spec k$ (so $K = k$ , $r = \dim X$ , $H$ and the nonempty open $U$ are obtained as in steps 6.1 and 7.2) is covered by steps 6.1-10.1, and characteristic $0$ is essential, the Frobenius example $k [u] \to k [t]$ , $u \mapsto t^{p}$ showing failure in characteristic $p$ and lying outside the hypothesis of characteristic $0$ . Endpoints: the two inequalities used in step 10.1 are the endpoint bounds $\dim_{κ (x)} (Ω_{S / R} \otimes_{S} κ (x)) \leq r$ of step 7.1 and $rank (d_{x} f) \leq \dim_{k} T_{y} Y$ ; at $r = 0$ the first is tight and at $r = \dim X$ the second is tight, in both cases producing the stated equality rather than a strict inequality. Nonempty-choice: AC is declared in [F1] and is used in this proof only through the AC-assuming suppliers [F36] (regular loci), [F33] (transpose rank), [F23] (bases inside spanning sets), [F2] (the classical-scheme dictionary), [F7] (affine antiequivalence), [F6] (domain criterion), [F9] (function fields of charts) and [F11] (dimension equals transcendence degree), each cited at the step that uses it; the field-theoretic steps 3.1, 3.2-6.1 and the linear algebra of steps 8.1-10.1 make no further choice. Biconditional directions: no biconditional is asserted by this lemma; the only implications are the chain of equalities and the single inequality of step 10.1, whose forward reading gives surjectivity, and no converse is claimed.

Source qualification

The classical statement proved here is the source-open form of generic smoothness in characteristic $0$ . Vakil proves at §3.1, Proposition 3.1, for a dominant morphism of integral finite-type $k$ -schemes that there is a nonempty open set $U \subseteq X$ on which the morphism is smooth; his proof defines the relative dimension $n = \dim X - \dim Y$ , notes that the relative differential module has rank $n$ at the generic point and rank at least $n$ everywhere, and uses upper semicontinuity of fibre rank and constant rank to conclude local freeness and flatness on a dense open set. The present item records only the source-side differentiability conclusion and is proved without local freeness, flatness or the smoothness of the structure morphisms: the spreading-out step 6.1 produces a nonempty principal open on which the fibre of the relative differential module is generated by the $r$ lifted elements, and the final comparison step 10.1 uses the tangent-space criterion through Differentials, open restriction, and the chain rule. The smoothness conclusion that Vakil draws from that criterion is taken up by the consumer Generic smoothness on the source through The submersion criterion between smooth varieties, not asserted here. The characteristic- $0$ hypothesis enters only through perfectness of $K$ and the separating transcendence basis of [F14]; positive characteristic is genuinely different, as recorded on the counterexample page. The source works with schemes; the translation to irreducible classical varieties is the equivalence of [F2], and the affine charts, their coordinate rings and the canonical function fields are those of [F5] and [F9].

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Generic smoothness on the source

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field of characteristic $0$ , let $X$ and $Y$ be irreducible classical varieties over $k$ , and let $f ⁣ : X \to Y$ be a dominant morphism. Regard $X$ and $Y$ as integral finite-type $k$ -schemes under Irreducible classical varieties and integral separated finite-type schemes, let $X_{r e g}, Y_{r e g}$ be their regular loci (Regular and singular loci), and let $U = X_{r e g} \cap f^{- 1} (Y_{r e g}) \cap D (H)$ be the nonempty open subset of $X$ produced by A dominant map has a surjective differential on a dense source open, so that $f (U) \subseteq Y_{r e g}$ and $d_{x} f$ is surjective at every closed point $x \in U$ .

Then there are a nonempty affine open subvariety $V \subseteq U$ of $X$ — explicitly a nonempty principal open $D_{X_{0}} (h^{'})$ of an affine chart $X_{0}$ of $X$ — and a nonempty affine open subvariety $W \subseteq Y_{r e g}$ — a nonempty principal open of an affine chart of $Y$ — such that:

$f (V) \subseteq W$ , and $V \to Spec k$ and $W \to Spec k$ are smooth, so that $V$ and $W$ are smooth affine classical varieties over $k$ ;
the restriction $f ∣_{V} ⁣ : V \to W$ is a morphism of finite type and is smooth in the sense of Smooth morphisms via local standard smooth presentations: it is locally standard smooth at every point of $V$ ; consequently the restriction $f ∣_{V} ⁣ : V \to Y$ (equivalently $V \to Y_{r e g}$ ) is smooth as well.

In particular $f$ is smooth at every point of the nonempty open subset $V$ of its source. Neither $X$ nor $Y$ is assumed smooth outside its regular locus, and the target-side statement — a dense open subset of $Y$ over which the source is smooth — is not asserted here: it requires a smooth source and fails without that hypothesis.

Facts & Assumptions

Given: The Axiom of Choice; an algebraically closed field $k$ of characteristic $0$ ; irreducible classical varieties $X$ and $Y$ over $k$ ; a dominant morphism $f ⁣ : X \to Y$ ; and the open subset $U = X_{r e g} \cap f^{- 1} (Y_{r e g}) \cap D (H)$ supplied by [F3].

[F1]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function.

[F2]

Irreducible classical varieties and integral separated finite-type schemes: under AC the closed-point construction and its inverse give an equivalence between irreducible classical $k$ -varieties and integral finite-type $k$ -schemes satisfying the affine-overlap separation condition, each original point being identified with its singleton; classical points correspond to closed points, and classical regular maps to scheme $k$ -morphisms.

[F3]

A dominant map has a surjective differential on a dense source open: under AC, for $k$ algebraically closed of characteristic $0$ and $f ⁣ : X \to Y$ dominant between irreducible classical varieties, the set $U = X_{r e g} \cap f^{- 1} (Y_{r e g}) \cap D (H)$ for a nonempty affine chart $Spec S \subseteq X$ over an affine chart $Spec R \subseteq Y$ and $0 \neq H \in S$ is a nonempty open subset of $X$ with $U \subseteq X_{r e g}$ , $f (U) \subseteq Y_{r e g}$ , and $d_{x} f ⁣ : T_{x} X \to T_{f (x)} Y$ surjective at every closed point $x \in U$ .

[F4]

Regular and singular loci: for a locally Noetherian scheme $X$ , $X_{r e g} = {x \in ∣ X ∣ : O_{X, x} is a regular local ring}$ .

[F5]

Affine open subschemes: for a scheme $X$ and open $U \subseteq X$ , the open subscheme is $(U, O_{X} ∣_{U})$ , with the restricted structure sheaf.

[F6]

The stalk of a presheaf at a point: the stalk at $x$ is the filtered colimit of the sections over open neighbourhoods of $x$ ; the neighbourhoods of $x$ contained in an open $U ∋ x$ are cofinal, so for the restricted sheaf $O_{U, x} ≅ O_{X, x}$ canonically.

[F7]

Regular points of locally Noetherian schemes: a point $x$ of a locally Noetherian scheme is regular exactly when $O_{X, x}$ is a regular local ring; this is absolute regularity of the local ring.

[F8]

Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: every field of characteristic zero is perfect, and every algebraically closed field is perfect.

[F9]

Regular equals smooth over a perfect field: under AC, for a perfect field $k$ and a finite-type $k$ -scheme $X$ , $X$ is regular (every local ring $O_{X, x}$ is regular) if and only if $X \to Spec k$ is smooth in the local-standard-smooth sense.

[F10]

Classical algebraic prevarieties, regular maps, and varieties: a classical algebraic prevariety over an algebraically closed $k$ is a quasi-compact locally ringed space with a sheaf of $k$ -algebras covered by open subspaces isomorphic to affine models (polynomial zero sets, including empty and reducible ones), whose points have residue field canonically $k$ ; a classical algebraic variety is a separated prevariety; polynomial principal opens form a basis of the topology; zero loci of regular functions are closed. These definitions use no Axiom of Choice.

[F11]

[F12]

Irreducibility via nonempty open subsets, connectedness and open subspaces: a nonempty open subspace of an irreducible space is irreducible; an irreducible space is nonempty.

[F13]

Every nonempty principal open is a classical affine variety: under AC, for an affine variety $X$ and $0 \neq h \in A = k [X]$ , the principal open $D_{X} (h)$ , with its regular functions, is isomorphic to the closed graph $Z = {(x, t) \in X \times k : t h (x) = 1}$ ; its coordinate ring is canonically $A [T] / (T h - 1) ≅ A_{h}$ , a nonzero domain, and $D_{X} (h)$ is affine.

[F14]

The coordinate ring of a classical affine algebraic set: for an affine algebraic set $X \subseteq k^{n}$ , $k [X] = k [x_{1}, \dots, x_{n}] / I (X)$ is reduced and generated as a $k$ -algebra by the finitely many coordinate classes.

[F15]

Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals: under AC, $J \mapsto V (J)$ and $X \mapsto I (X)$ are inverse inclusion-reversing bijections between radical ideals and algebraic sets; nonempty irreducible algebraic sets correspond precisely to proper prime ideals, and points to maximal ideals.

[F16]

The closed points of the prime spectrum are exactly the maximal ideals: under AC, for a commutative ring $R$ and $p \in Spec R$ , the singleton ${p}$ is closed if and only if $p$ is a maximal ideal.

[F17]

In a finite-type algebra over a field, closed points are dense in every closed subset of the spectrum: under AC, for a finite-type $k$ -algebra $A$ and a closed subset $Z \subseteq Spec A$ , every nonempty open subset of $Z$ contains a closed point of $Spec A$ .

[F18]

The submersion criterion between smooth varieties: under AC, for smooth classical varieties $X, Y$ over algebraically closed $k$ whose structure morphisms are smooth, and a finite-type morphism $f ⁣ : X \to Y$ , at a classical closed point $x$ with $y = f (x)$ the morphism $f$ is smooth at $x$ if and only if $d_{x} f ⁣ : T_{x} X \to T_{y} Y$ is surjective.

[F19]

Differentials, open restriction, and the chain rule: at $k$ -rational points the differential is the dual of the induced cotangent map and is functorial under composition; every $k$ -open immersion induces an isomorphism on tangent spaces at each rational point.

[F20]

Smooth morphisms via local standard smooth presentations: a finite-type $k$ -scheme morphism is smooth if at every source point there are affine neighbourhoods for which the induced ring map has a standard smooth presentation after principal shrinking; the condition is local on the source and on the target.

[F21]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation of an $R$ -algebra $S$ is an isomorphism $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ whose Jacobian matrix has a $c \times c$ minor that is a unit in $S$ ; standard smoothness at a prime holds after localizing at an element outside that prime, and a further principal localization may be absorbed into the presentation.

[F22]

The intrinsic Zariski tangent space: $T_{x} X$ is the dual of $m_{x} / m_{x}^{2}$ ; for a locally finite-type $k$ -scheme it is finite-dimensional over $κ (x)$ .

[F23]

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras: $R [a_{1}, \dots, a_{n}]$ is the smallest $R$ -subalgebra containing the $a_{i}$ , and an $R$ -algebra is of finite type exactly when it is generated by finitely many elements.

[F24]

Locally finite type and finite type morphisms: a morphism is locally of finite type when locally on affine charts the ring maps are of finite type, and of finite type when it is locally of finite type and quasi-compact.

Proof

technique · direct

1.1F2F3F4F8given

The field $k$ is perfect by [F8]. By [F2] the varieties $X$ and $Y$ are integral finite-type $k$ -schemes with $f$ a $k$ -morphism; the regular loci $X_{r e g}, Y_{r e g}$ are defined by [F4], and by [F3] the set $U = X_{r e g} \cap f^{- 1} (Y_{r e g}) \cap D (H)$ is a nonempty open subset of $X$ contained in $X_{r e g}$ with $f (U) \subseteq Y_{r e g}$ , and $d_{x} f ⁣ : T_{x} X \to T_{f (x)} Y$ is surjective at every closed point $x \in U$ . We keep these notations throughout.

1.2F3F10F11F12F13

Affine pieces. By [F11] the variety $X$ has a finite affine cover; choose a chart $X_{0}$ with $U \cap X_{0} \neq \emptyset$ and a point $x_{0} \in U \cap X_{0}$ . Since principal opens form a basis of the topology [F10], there is $0 \neq h \in k [X_{0}]$ with $x_{0} \in D_{X_{0}} (h) \subseteq U \cap X_{0}$ . Similarly $y_{0} = f (x_{0}) \in f (U) \subseteq Y_{r e g}$ ; choose an affine chart $Y_{0}$ of $Y$ with $y_{0} \in Y_{0}$ [F11] and $0 \neq g \in k [Y_{0}]$ with $y_{0} \in D_{Y_{0}} (g) \subseteq Y_{r e g} \cap Y_{0}$ [F10]. The set $D_{X_{0}} (h) \cap f^{- 1} (D_{Y_{0}} (g))$ is a nonempty open subset of the affine variety $X_{0}$ containing $x_{0}$ , so by [F10] there is $0 \neq h^{'} \in k [X_{0}]$ with $x_{0} \in D_{X_{0}} (h^{'}) \subseteq D_{X_{0}} (h) \cap f^{- 1} (D_{Y_{0}} (g))$ . Put $V : = D_{X_{0}} (h^{'})$ and $W : = D_{Y_{0}} (g)$ , so that $V$ is a nonempty open subvariety of $X$ with $V \subseteq D_{X_{0}} (h) \subseteq U$ , and $f (V) \subseteq W \subseteq Y_{r e g}$ . By [F13] the principal opens $V$ and $W$ are affine varieties with coordinate rings $k [V] = k [X_{0}]_{h^{'}}$ and $k [W] = k [Y_{0}]_{g}$ . Since $X_{0}$ and $Y_{0}$ are nonempty open subsets of the irreducible varieties $X$ and $Y$ , all four are irreducible [F12].

2.1F2F13F14F15F16F23step 1.2

Points of $V$ and finite generation. By [F14] the coordinate ring $k [X_{0}]$ is reduced and generated over $k$ by finitely many coordinate classes; hence so is its localization $k [V] = k [X_{0}]_{h^{'}}$ , generated by those classes together with the inverse of $h^{'}$ [F23]. So $V = Spec k [V]$ is a finite-type $k$ -scheme, and likewise $W = Spec k [W]$ . By [F13] and [F15] the points of the affine variety $V$ are the maximal ideals of $k [V]$ , and by [F16] these are exactly the closed points of the scheme $V$ ; under the equivalence [F2] they are the classical points of $V$ , hence closed points of the scheme $X$ lying in $V \subseteq U$ . In particular every point $x$ of the classical variety $V$ is a closed point of $X$ and satisfies the conclusion of [F3], and its image $f (x)$ lies in $W$ .

3.1F4F5F6F7F8F9F10F20step 2.1given

The varieties $V$ and $W$ are smooth over $k$ . Let $z \in V$ . Since $V \subseteq U \subseteq X_{r e g}$ , the open subscheme description [F5] and the cofinality of the neighbourhoods inside $V$ [F6] give $O_{V, z} ≅ O_{X, z}$ , which is regular because $z \in X_{r e g}$ [F4, F7]. Hence the finite-type $k$ -scheme $V$ is regular, and $V \to Spec k$ is smooth by [F9]. The same argument with $W \subseteq Y_{r e g}$ gives $O_{W, z} ≅ O_{Y, z}$ regular for $z \in W$ , and $W \to Spec k$ smooth by [F9]. Each of $V$ and $W$ is a classical algebraic variety in the sense of [F10]: as an affine model it is a quasi-compact locally ringed space covered by itself, and it is separated because for regular maps $φ, ψ ⁣ : Z \to V$ from any classical prevariety $Z$ the coordinate components $φ_{i}, ψ_{i}$ are regular functions on $Z$ (pullback of the coordinate functions of the affine model), so the equalizer is the finite intersection of the closed zero loci ${φ_{i} - ψ_{i} = 0}$ [F10]. In particular $V$ and $W$ are smooth classical varieties over $k$ in the sense of [F10] and [F20].

3.2F23F24step 2.1

The restriction is finite type. Write $g = f ∣_{V} : V \to W$ . Let $ι ⁣ : V ↪ X$ and $κ ⁣ : W ↪ Y$ be the open immersions, so that $κ \circ (f ∣_{V}) = f \circ ι$ . Write $B = k [V]$ and $C = k [W]$ , affine coordinate rings as in step 2.1, and let $φ ⁣ : C \to B$ be the $k$ -algebra map induced by $f ∣_{V} ⁣ : V \to W$ . Choose finitely many $k$ -algebra generators $b_{1}, \dots, b_{m}$ of $B$ [F23]. Since $k \subseteq C$ and $C [b_{1}, \dots, b_{m}]$ is a $C$ -subalgebra of $B$ containing $k$ and all $b_{i}$ , it contains the $k$ -subalgebra generated by the $b_{i}$ , which is $B$ ; hence $B = C [b_{1}, \dots, b_{m}]$ is generated by finitely many elements over $C$ [F23]. Thus $φ$ is of finite type, the morphism $f ∣_{V}$ is locally of finite type on the affine charts, and it is quasi-compact because its source is affine; by [F24] the restriction $f ∣_{V} ⁣ : V \to W$ is of finite type.

4.1F3F19F22step 2.1step 3.2

Differential comparison. Let $x$ be a point of the classical variety $V$ and put $y = f (x) \in W$ . By step 2.1, $x$ is a closed point of $X$ lying in $U$ , so $d_{x} f$ is surjective [F3]; in particular the case $T_{f (x)} Y = 0$ is allowed and the conclusion is unaffected. The identity $κ \circ (f ∣_{V}) = f \circ ι$ of step 3.2, the functoriality of the differential, and the fact that the $k$ -open immersions $ι, κ$ induce isomorphisms on tangent spaces [F19] give $d_{x} (f ∣_{V}) = (d_{g (x)} κ)^{- 1} \circ d_{x} f \circ d_{x} ι$ , where $d_{g (x)} κ ⁣ : T_{g (x)} W \to T_{f (x)} Y$ is an isomorphism; the tangent spaces are finite-dimensional over $k$ [F22]. Therefore $rank d_{x} (f ∣_{V}) = rank d_{x} f = \dim_{k} T_{f (x)} Y = \dim_{k} T_{g (x)} W$ , and $d_{x} (f ∣_{V}) ⁣ : T_{x} V \to T_{g (x)} W$ is surjective at every point $x$ of the classical variety $V$ .

5.1F18step 2.1step 3.1step 3.2step 4.1

The criterion at every point of $V$ . Let $x$ be a point of the classical variety $V$ ; by step 2.1 it is a classical closed point of the affine variety $V$ . The structure morphisms $V \to Spec k$ and $W \to Spec k$ are smooth [3.1], so $V$ and $W$ are smooth classical varieties over $k$ in the sense of [F18]; the morphism $f ∣_{V} ⁣ : V \to W$ is of finite type [3.2] and its differential at $x$ is surjective [4.1]. By the submersion criterion [F18], the restriction $f ∣_{V}$ is smooth at $x$ . As $x$ was an arbitrary point of the classical variety $V$ , the restriction is smooth at every point of $V$ in the classical sense.

6.1F13F15F16F17F20F21step 2.1step 5.1

Upgrade to scheme points. Let $Sm \subseteq V$ be the set of points at which $f ∣_{V}$ is locally standard smooth, so that $Sm$ contains every point of the classical variety $V$ by step 5.1. If $z \in Sm$ , then by [F20] there are affine neighbourhoods of $z$ and of $f ∣_{V} (z)$ and a principal shrinking on which the induced ring map has a standard smooth presentation [F21]; the Jacobian minor of that presentation is a unit on the whole shrinking, hence remains a unit in every further localization, so the same presentation witnesses standard smoothness at every point of that shrinking. Therefore $Sm$ is open in $V$ . Suppose $V ∖ Sm$ were nonempty. It is a nonempty closed subset of the affine finite-type $k$ -scheme $V = Spec B$ of step 2.1 and is a nonempty open subset of itself; by [F17] it contains a closed point $z$ of $Spec B$ . By [F16] the point $z$ is a maximal ideal of $B$ , and by [F15] applied to the affine variety $V$ with coordinate ring $B$ [F13] it is a point of the classical variety $V$ ; this contradicts step 5.1. Hence $V ∖ Sm = \emptyset$ , and $f ∣_{V} ⁣ : V \to W$ is smooth in the sense of [F20].

7.1F20step 1.2step 3.1step 6.1given

Conclusion. The restriction $f ∣_{V} ⁣ : V \to W$ is smooth [6.1], and $W$ is an open subscheme of $Y$ with $f (V) \subseteq W$ . Since smoothness is local on the target [F20], the same standard smooth presentations witness smoothness of the restriction $f ∣_{V} ⁣ : V \to Y$ at every point of $V$ ; the same applies to $V \to Y_{r e g}$ because $W \subseteq Y_{r e g}$ . Thus $f$ is smooth at every point of the nonempty open subset $V$ of its source, with $V \subseteq U$ a principal open of an affine chart of $X$ . Neither $X$ nor $Y$ is assumed smooth outside $X_{r e g}$ , $Y_{r e g}$ , and no target-side generic smoothness is claimed here.

8.1F1F2F3F9F10F12F13F15F17F18step 1.2step 2.1step 5.1∎

Boundary and scope dispositions. Empty: $X$ and $Y$ are nonempty because irreducible means nonempty [F12], so the charts $X_{0}, Y_{0}$ and the sets $U$ , $V$ , $W$ of steps 1.2 and 2.1 are nonempty; there is no empty-case convention to invoke, and the empty scheme is excluded by the hypothesis. Zero: relative dimension $0$ is allowed — if $\dim X = \dim Y$ the differential is an isomorphism at the points of $V$ and the conclusion is unaffected; if $Y$ is a point then $Y_{r e g} = Y$ , the chart $Y_{0}$ is the whole point, $W = Y_{0}$ , and step 3.1 shows directly that $V \to W = Spec k$ is smooth, so the criterion's conclusion in step 5.1 is consistent. One: nothing in the argument divides by a natural number or requires a generator count or relative dimension at least one; the lists $b_{1}, \dots, b_{m}$ of step 3.2 may have any finite length, and the case $\dim_{k} T_{f (x)} Y = 1$ is the first instance in which surjectivity of $d_{x} f$ is a genuine condition. Degenerate: neither $X$ nor $Y$ is assumed smooth, and $f$ is neither assumed finite nor flat; the set $V$ must be allowed to be a proper subset of $X$ , as the example $t \mapsto t^{2}$ on $A_{k}^{1}$ shows, where the differential vanishes at the origin and $V$ lies in $U = A_{k}^{1} ∖ {0}$ ; a differential of rank zero is compatible with $V \subseteq U$ exactly when the target tangent space is zero; in particular the structure map to $Spec k$ has this property, and the case where $U$ is not affine is handled by passing to the principal open $V$ of a chart. Endpoints: the argument uses no closed-range or dimension endpoint claim; at one extreme $f$ may already be smooth on all of $X$ , in which case the construction still returns some nonempty principal open $V$ , and every such $V$ is dense in $X$ because $X$ is irreducible and $V$ is nonempty and open [F12]. Nonempty-choice: AC is declared in [F1] and is used exactly through the AC-assuming suppliers [F3] (generic differential surjectivity), [F18] (submersion criterion), [F9] (regularity versus smoothness), [F2] (classical-scheme dictionary), [F13] and [F15] (principal opens and the Nullstellensatz correspondence), [F17] (density of closed points), and [F12]/[F10] as used in steps 1.2 and 2.1; the finite choices of charts and principal open generators in step 1.2 and the localization argument of step 2.1 add no further choice principle. Biconditional directions: the corollary asserts only existence of $V$ and smoothness, with no converse; the only biconditional used as a supplier is the submersion criterion [F18], and step 5.1 applies its forward direction (surjective differential implies smooth at the point), never its reverse.

Source qualification

Vakil, Classes 51–52, §3.1, Proposition 3.1 proves generic smoothness on the source: for a dominant morphism of integral finite-type $k$ -schemes over a field of characteristic $0$ there is a nonempty dense open $U \subseteq X$ with $π ∣_{U}$ smooth. The source works throughout with schemes and takes the smoothness conclusion directly from the same local analysis of the relative differential module; the present corollary instead records the conclusion that follows from the authored differential-surjectivity lemma on this page's pair by restriction to an affine principal open and the submersion criterion, and therefore also covers the classical-variety formulation with the standard-smooth convention of Smooth morphisms via local standard smooth presentations. The source asserts only that the smooth locus is a nonempty open subset of the source; it claims nothing about the size of $U$ , about smoothness of $X$ or $Y$ , or about a target-side open set, and neither does this item. The characteristic- $0$ hypothesis enters through perfectness of $k$ and through the separating-transcendence-basis input of the differential lemma; the positive-characteristic failure of the source-side statement is recorded on the counterexample page of the pair. The dictionary between classical varieties and integral finite-type schemes used for the translation is Irreducible classical varieties and integral separated finite-type schemes, and the affine chart, coordinate-ring and principal-open interfaces are those of The coordinate ring of a classical affine algebraic set and Every nonempty principal open is a classical affine variety.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Critical loci have small images in characteristic zero

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field of characteristic $0$ , let $X$ and $Y$ be smooth classical varieties over $k$ , so that their structure morphisms $X \to Spec k$ and $Y \to Spec k$ are smooth in the sense of Smooth morphisms via local standard smooth presentations, and let $f ⁣ : X \to Y$ be a morphism of classical varieties. For a classical point $x$ of $X$ the residue field is $κ (x) = k$ , and the differential $d_{x} f ⁣ : T_{x} X ⟶ T_{f (x)} Y$ of Differentials, open restriction, and the chain rule is a $k$ -linear map. For $r \geq 0$ define $C_{r} = {x \in X : rank d_{x} f \leq r},$ the set of classical points $x$ of $X$ at which that differential has rank at most $r$ . Then:

$C_{r}$ is closed in $X$ ; explicitly, $C_{r}$ is the set of classical points of a closed subvariety of the smooth classical variety $X$ ;
$\dim \overline{f (C_{r})} \leq r$ , where the closure is taken in the classical variety $Y$ and $\dim$ is the dimension of Global and local dimension of classical varieties.

Neither irreducibility, connectedness, equidimensionality nor nonemptiness of $X$ or of $Y$ is assumed, and the empty case is permitted. The characteristic- $0$ hypothesis is used only for claim 2: claim 1 holds over any algebraically closed field.

Facts & Assumptions

Given: The Axiom of Choice; an algebraically closed field $k$ of characteristic $0$ ; smooth classical varieties $X$ and $Y$ over $k$ ; a morphism $f ⁣ : X \to Y$ ; an integer $r \geq 0$ .

[F1]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function.

[F2]

Smooth morphisms via local standard smooth presentations: a morphism of finite-type $k$ -schemes is smooth when every source point has affine neighbourhoods on which the induced ring map has a standard smooth presentation at the prime of that point; the condition is local on the source and on the target, and the definition assumes AC.

[F3]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation of an $R$ -algebra $S$ is an isomorphism $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with an invertible $c \times c$ Jacobian minor; the invertible minor may be assumed to occupy the first $c$ columns, a further principal localisation may be absorbed into the presentation, and the relative dimension is $n - c$ .

[F4]

Irreducible classical varieties and integral separated finite-type schemes: under AC the closed-point construction and its inverse give an equivalence between irreducible classical $k$ -varieties and integral finite-type $k$ -schemes, each original point being identified with its singleton, so classical points correspond to closed points.

[F5]

Classical algebraic prevarieties, regular maps, and varieties: a classical algebraic prevariety over $k$ is covered by affine models whose points have residue field canonically $k$ ; a classical algebraic variety is a separated prevariety; polynomial principal opens form a basis of the topology; these definitions use no Axiom of Choice.

[F6]

[F7]

Every nonempty principal open is a classical affine variety: under AC, for an affine variety $X$ and $0 \neq h \in k [X]$ , the principal open $D_{X} (h)$ is an affine variety with coordinate ring canonically $k [X]_{h}$ .

[F8]

Classical affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms and Affine schemes are contravariantly equivalent to commutative rings: pullback gives a natural bijection between morphisms of affine algebraic sets and $k$ -algebra maps of their coordinate rings, and a ring map $A \to B$ corresponds contravariantly to a morphism $Spec B \to Spec A$ ; an affine classical variety is thus described by its coordinate ring and its spectrum.

[F9]

The coordinate ring of a classical affine algebraic set: for an affine algebraic set $W \subseteq k^{N}$ the coordinate ring is $k [W] = k [x_{1}, \dots, x_{N}] / I (W)$ ; it is reduced and generated as a $k$ -algebra by the finitely many coordinate classes.

[F10]

Differentials, open restriction, and the chain rule: at $k$ -rational points the differential is the dual of the induced cotangent map, it is functorial under composition, and every $k$ -open immersion induces an isomorphism on tangent spaces at each rational point.

[F11]

The intrinsic Zariski tangent space: $T_{x} X$ is the dual of $m_{x} / m_{x}^{2}$ ; for a $k$ -scheme locally of finite type it is finite-dimensional, and at a $k$ -rational point the intrinsic and relative tangent spaces agree.

[F12]

Cotangent space at a rational point: at a $k$ -rational point $x$ of a $k$ -scheme, the map $m / m^{2} \to Ω_{X / k} \otimes_{O_{X, x}} κ (x)$ , $[a] \mapsto d a \otimes 1$ , is an isomorphism of $k$ -vector spaces, natural in the pair $(X, x)$ .

[F13]

Jacobian presentation of Ω: for $P = A [x_{1}, \dots, x_{n}]$ and $B = P / I$ with $I = (f_{1}, \dots, f_{r})$ , the module $Ω_{B / A}$ is the cokernel of the $B$ -linear map $B^{r} \to B^{n}$ whose $j$ -th column is the vector of partial derivatives $(\partial f_{j} / \partial x_{i})$ ; in particular $Ω_{B / A}$ is generated by $d x_{1}, \dots, d x_{n}$ .

[F14]

Relative differential-rank condition: if $B$ is a standard smooth $k$ -algebra with presentation $B ≅ (k [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ whose leading $c \times c$ Jacobian minor $h = \det (\partial f_{j} / \partial x_{i})_{1 \leq i, j \leq c}$ maps to a unit of $B$ , then $Ω_{B / k}$ is free of rank $n - c$ , and in the computation the localising isomorphism $B^{n} \to B^{n - c}$ , $(u^{'}, v^{'}) \mapsto v^{'} - D C^{- 1} u^{'}$ , restricted to the complementary coordinates is the identity; accordingly the images $d u_{c + 1}, \dots, d u_{n}$ of the differentials of the free coordinates form a $B$ -basis of $Ω_{B / k}$ .

[F15]

Localization, base change and functoriality of differentials: an $A$ -algebra homomorphism $B \to C$ induces a canonical $C$ -linear functoriality map $C \otimes_{B} Ω_{B / A} \to Ω_{C / A}$ , $c \otimes d b \mapsto c d (image of b)$ .

[F16]

Change of rings: $N \otimes_{R} M ≅ N \otimes_{S} (S \otimes_{R} M)$ : for a ring homomorphism $R \to S$ , a right $S$ -module $N$ and a left $R$ -module $M$ there is a natural isomorphism $N \otimes_{R} M ≅ N \otimes_{S} (S \otimes_{R} M)$ .

[F17]

Assuming choice, $\ker T^{*} = (im T)^{\circ}$ and $im T^{*} = (\ker T)^{\circ}$ ; in finite dimensions $rank T^{*} = rank T$ : for a linear map $T ⁣ : V \to W$ of finite-dimensional vector spaces, $rank T^{*} = rank T$ .

[F18]

[algebra] Field linear algebra. For a matrix over a field, its rank is at most $r$ if and only if every $(r + 1) \times (r + 1)$ minor vanishes; for composable linear maps $rank (B \circ A) \leq rank B$ , and if $B$ is injective then $rank (B \circ A) = rank A$ ; the dual of a surjective linear map is injective; and the rank of a linear map equals the rank of any matrix of it whose selected source vectors generate the source and whose selected target vectors form a basis.

[F19]

[algebra] Quotients of local rings. If $(R, m)$ is a local ring and $I \subseteq m$ an ideal, then $R / I$ is local with maximal ideal $m / I$ and $(m / I) / (m / I)^{2} ≅ m / (m^{2} + I)$ ; hence the natural map $m / m^{2} \to (m / I) / (m / I)^{2}$ is surjective.

[F20]

The local ring at a point of an affine variety is the localization at its maximal ideal: under AC, for a classical affine variety $X$ over an algebraically closed field and $x \in X$ , there is a canonical isomorphism of local rings $O_{X, x} ≅ k [X]_{m_{x}}$ .

[F21]

Localisation commutes with kernels images and cokernels: for an $R$ -module homomorphism $f ⁣ : M \to N$ , localisation identifies $S^{- 1} (coker f) ≅ coker (S^{- 1} f)$ .

[F22]

Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals: under AC, for $A = R / I (X)$ the radical ideals of $A$ correspond bijectively to the closed subsets of $X$ ; points correspond to maximal ideals, and a closed subvariety $V (H)$ has coordinate ring $A / \sqrt{H}$ .

[F23]

Global and local dimension of classical varieties: for a classical variety $X$ with irreducible components $X_{i}$ and a closed point $x$ one has $\dim_{x} X = \max_{x \in X_{i}} \dim X_{i}$ , while $\dim X$ is its chain dimension.

[F24]

Chain dimension and the empty-space convention: for a Noetherian topological space, $\dim T$ is the supremum of the lengths of strict chains of nonempty irreducible closed subsets of $T$ , and $\dim \emptyset = - \infty$ .

[F25]

Dimension of a finite closed union: if a Noetherian space $T$ is a finite union of closed subsets $T_{1}, \dots, T_{m}$ , then $\dim T = \max_{i} \dim T_{i}$ , with both sides $- \infty$ for $m = 0$ .

[F26]

Interior, closure, boundary, exterior, derived set and isolated point in a topological space: the closure $\overline{A}$ is the smallest closed superset of $A$ , and $A$ is closed if and only if $A = \overline{A}$ .

[F27]

Regular and singular loci: for a reduced classical finite-type space over an algebraically closed field $k$ and a closed point $x$ , one has $x \in X_{r e g}$ if and only if $\dim_{κ (x)} T_{x} X = \dim_{x} X$ ; this classical component-dimension test assumes AC.

[F28]

A dominant map has a surjective differential on a dense source open: under AC, for $k$ algebraically closed of characteristic $0$ and a dominant morphism of irreducible classical varieties, the set $U = X_{r e g} \cap f^{- 1} (Y_{r e g}) \cap D (H)$ is a nonempty open subset of $X$ with $f (U) \subseteq Y_{r e g}$ and $d_{x} f$ surjective at every closed point $x \in U$ .

[F29]

Dominant classical morphisms and rational maps: a morphism is dominant when the closure of its image is the target; for morphisms of varieties this is density of the image.

[F30]

Irreducibility via nonempty open subsets, connectedness and open subspaces: an irreducible space is nonempty and every nonempty open subspace of it is irreducible.

[F31]

Existence and basic properties of irreducible components: under AC, the closure of an irreducible subset is irreducible.

[F32]

[topology] The continuous image of an irreducible space is irreducible: the inverse image of a finite closed cover of the image is a finite closed cover of the source.

Proof

technique · direct

1.1F1F2F3F4F5F10F11given

Setting. By [F2] the structure morphisms of $X$ and $Y$ are smooth, so every point of either variety has an affine neighbourhood carrying a standard smooth presentation [F3]; by [F4] the classical points of $X$ are its closed points and have residue field $k$ , and by [F5] the classical varieties have the affine-model topology with residue field $k$ , principal opens forming a basis. For a classical point $x$ of $X$ the differential $d_{x} f$ is the dual of the induced cotangent map [F10] and $T_{x} X$ is the finite-dimensional dual of $m_{x} / m_{x}^{2}$ [F11]. Define $C_{r} = {x : rank d_{x} f \leq r}$ . We prove (1) that $C_{r}$ is closed in $X$ , and (2) that $\dim \overline{f (C_{r})} \leq r$ .

1.2F2F3F5F6F7F9

Charts at a prescribed point. Let $x_{0} \in X$ . The structure morphism of $X$ is smooth, so [F2] provides an affine neighbourhood of $x_{0}$ whose coordinate ring is standard smooth after a principal shrinking; [F6] supplies finite affine covers of $X$ and of $Y$ , [F5] lets us shrink to a principal open contained in any prescribed open neighbourhood, [F7] makes principal opens affine, and [F3] absorbs the principal shrinking into the presentation. Choose in this way an affine chart $V = Spec C \subseteq Y$ containing $f (x_{0})$ , where $C = k [V]$ is standard smooth over $k$ , and an affine chart $U = Spec B \subseteq X$ containing $x_{0}$ with $f (U) \subseteq V$ and $B = k [U]$ standard smooth over $k$ ; write $B ≅ (k [x_{1}, \dots, x_{n}] / (F_{1}, \dots, F_{c}))_{g}$ with leading $c \times c$ Jacobian minor mapping to a unit of $B$ , and $d = n - c$ . Let $y_{1}, \dots, y_{m} \in C$ be coordinate classes generating $C$ as a $k$ -algebra [F9].

1.3F10F11F12F13F14

Free differentials on the chart. By [F13] the module $Ω_{B / k}$ is the cokernel of the transposed Jacobian map $B^{c} \to B^{n}$ of the presentation of $B$ , and since the leading $c \times c$ minor is invertible, [F14] shows that this cokernel is free with $B$ -basis the images $d u_{c + 1}, \dots, d u_{n}$ of the differentials of the free coordinates $u_{c + 1}, \dots, u_{n} \in B$ . For a closed point $x \in U$ , [F12] identifies $C_{x} U = m_{x} / m_{x}^{2}$ with $Ω_{B / k} \otimes_{B} κ (x)$ naturally, so that $T_{x} U ≅ (Ω_{B / k} \otimes_{B} κ (x))^{\lor}$ by [F11], and the open immersion $U ↪ X$ identifies $T_{x} U$ with $T_{x} X$ and $d_{x} (f ∣_{U})$ with $d_{x} f$ [F10].

1.4F8F13F14F15

The pullback matrix. Let $ψ ⁣ : C \to B$ be the $k$ -algebra map induced by $f ∣_{U} ⁣ : U \to V$ , under the correspondence of [F8]. By [F15] there is a canonical $B$ -linear functoriality map $γ ⁣ : B \otimes_{C} Ω_{C / k} \to Ω_{B / k}$ , $b \otimes d c \mapsto b d (ψ c)$ ; by [F13] the differentials $d y_{1}, \dots, d y_{m}$ generate the $C$ -module $Ω_{C / k}$ , so their images generate $B \otimes_{C} Ω_{C / k}$ . Since $d u_{c + 1}, \dots, d u_{n}$ is a $B$ -basis of $Ω_{B / k}$ [F14], there are unique regular functions $a_{i j} \in B$ with $d (ψ y_{j}) = \sum_{i = c + 1}^{n} a_{i j} d u_{i} (c < i \leq n, 1 \leq j \leq m);$ denote by $A = (a_{i j})$ the resulting $(n - c) \times m$ matrix over $B$ .

1.5F10F11F12F13F14F16F17F18

Rank identity on the chart. For every closed point $x \in U$ with $y = f (x)$ one has $rank d_{x} f = rank A (x)$ , where $A (x)$ is the matrix over $κ (x) = k$ obtained by reducing the entries of $A$ modulo $m_{x}$ ; more precisely the two ranks equal the rank of the fibre $γ_{x} = γ \otimes_{B} κ (x)$ . Indeed, the cotangent map ${\overset{ˉ}{ψ}}_{x}^{♯} ⁣ : C_{y} V \to C_{x} U$ corresponds under the natural isomorphisms of [F12] to $γ_{x} ⁣ : Ω_{C / k} \otimes_{C} κ (y) \to Ω_{B / k} \otimes_{B} κ (x)$ — the source is identified with $(B \otimes_{C} Ω_{C / k}) \otimes_{B} κ (x)$ by [F16] — and $d_{x} (f ∣_{U})$ is the dual of ${\overset{ˉ}{ψ}}_{x}^{♯}$ by [F10], hence has the same rank as ${\overset{ˉ}{ψ}}_{x}^{♯}$ by [F17] because these spaces are finite-dimensional [F11]; moreover $γ_{x} (d y_{j} \otimes 1) = d (ψ y_{j}) \otimes 1 = \sum_{i} a_{i j} (x) (d u_{i} \otimes 1)$ , the elements $d y_{j} \otimes 1$ generate the source over $k$ [F13], and $(d u_{i} \otimes 1)_{i > c}$ is a $k$ -basis of the target [F14], so by [F18] the rank of $γ_{x}$ is the rank of the matrix $A (x)$ ; finally $rank d_{x} (f ∣_{U}) = rank d_{x} f$ because $U ↪ X$ and $V ↪ Y$ are open immersions inducing tangent isomorphisms [F10].

1.6F6F29F31F32

Decomposition into components. As a closed subvariety of $X$ , the set $C_{r}$ is Noetherian with finitely many irreducible components $Z_{1}, \dots, Z_{s}$ [F6]; each $Z_{i}$ is an irreducible closed subvariety of $X$ , hence an irreducible classical variety. The image $f (Z_{i})$ is irreducible as the continuous image of an irreducible space [F32], so its closure $W_{i} : = \overline{f (Z_{i})}$ in $Y$ is an irreducible closed subvariety of $Y$ , i.e. an irreducible classical variety [F31]; the induced morphism $f ∣_{Z_{i}} ⁣ : Z_{i} \to W_{i}$ is dominant because $f (Z_{i})$ is dense in $W_{i}$ [F29].

1.7F28F30

Generic surjectivity on a component. Fix $i$ . By [F28] applied to the dominant morphism $f ∣_{Z_{i}}$ of irreducible classical varieties there is a nonempty open subset $U_{i} \subseteq Z_{i}$ with $U_{i} \subseteq (Z_{i})_{r e g} \cap f^{- 1} ((W_{i})_{r e g})$ and with $d_{x} (f ∣_{Z_{i}}) ⁣ : T_{x} Z_{i} \to T_{f (x)} W_{i}$ surjective at every closed point $x \in U_{i}$ ; in particular $f (x) \in (W_{i})_{r e g}$ . Choose a point $x \in U_{i}$ , which is possible because $U_{i}$ is nonempty [F30], and put $y = f (x) \in (W_{i})_{r e g}$ .

1.8F9F10F18F19F20F21F22

Closed subvarieties have injective differentials. Let $j ⁣ : Z^{'} ↪ U^{'}$ be the inclusion of a closed subvariety of an affine chart $U^{'}$ of $X$ , with vanishing ideal $I \subseteq B^{'} = k [U^{'}]$ , so that $k [Z^{'}] = B^{'} / I$ [F9, F22], and let $z \in Z^{'}$ be a point. Then $O_{Z^{'}, z} ≅ (B^{'} / I)_{m_{z}} ≅ B_{m_{z}}^{'} / I B_{m_{z}}^{'} = O_{U^{'}, z} / I O_{U^{'}, z}$ by [F20] and [F21], the maximal ideal of this quotient is $m_{z} / I O_{U^{'}, z}$ with $I \subseteq m_{z}$ , and $(m_{z} / I O) / (m_{z} / I O)^{2} ≅ m_{z} / (m_{z}^{2} + I O_{U^{'}, z})$ by [F19]; hence the natural map $m_{z} / m_{z}^{2} \to m_{Z^{'}} / m_{Z^{'}}^{2}$ is surjective [F19], and its dual is injective [F18]. By [F10] that dual is exactly the differential $d_{z} j ⁣ : T_{z} Z^{'} \to T_{z} U^{'}$ of the inclusion, so $d_{z} j$ is injective.

2.1F18F22F9step 1.5

Determinantal description and local closedness. Let $x \in U$ be a closed point. Since the entries $a_{i j} (x) \in k$ are the images of the regular functions $a_{i j}$ under $B \to B / m_{x} = k$ , the $(r + 1) \times (r + 1)$ minors of $A (x)$ are the images of the corresponding minors of $A$ ; by [F18] the inequality $rank A (x) \leq r$ holds if and only if every one of those minors vanishes. By step 1.5 this says $rank d_{x} f \leq r$ if and only if every $(r + 1) \times (r + 1)$ minor of $A$ lies in $m_{x}$ . Let $J_{U} \subseteq B$ be the ideal generated by all these minors; by [F22] the closed subvariety $V (J_{U}) \subseteq U$ has as its points exactly the maximal ideals of $B$ containing $J_{U}$ , which are precisely the closed points $x \in U$ with $rank d_{x} f \leq r$ . Hence $C_{r} \cap U = V (J_{U})$ is closed in $U$ .

2.2F5F10F18step 1.2step 1.8

Rank comparison. Let $U$ and $V$ be the affine charts around $x$ and $f (x)$ with $f (U) \subseteq V$ constructed in step 1.2; note $y = f (x) \in V$ . The restrictions $Z_{i} \cap U \subseteq U$ and $W_{i} \cap V \subseteq V$ are closed subvarieties of these affine charts [F5] with inclusion differentials that, under the open-immersion tangent isomorphisms $Z_{i} \cap U \subseteq Z_{i}$ , $U \subseteq X$ , $W_{i} \cap V \subseteq W_{i}$ and $V \subseteq Y$ [F10], are the differentials $d_{x} ι$ and $d_{y} κ$ of the closed inclusions $ι ⁣ : Z_{i} ↪ X$ and $κ ⁣ : W_{i} ↪ Y$ ; by step 1.8 both $d_{x} ι$ and $d_{y} κ$ are injective. The chain rule of [F10] applied to the identity $κ \circ (f ∣_{Z_{i}}) = f \circ ι$ gives $d_{y} κ \circ d_{x} (f ∣_{Z_{i}}) = d_{x} f \circ d_{x} ι$ ; since $d_{y} κ$ is injective, [F18] yields $rank d_{x} (f ∣_{Z_{i}}) = rank (d_{x} f \circ d_{x} ι) \leq rank d_{x} f$ . As $x \in Z_{i} \subseteq C_{r}$ we have $rank d_{x} f \leq r$ , so $rank d_{x} (f ∣_{Z_{i}}) \leq r$ .

3.1F5F26step 1.2step 2.1

Global closedness. The charts $U$ produced by step 1.2, as $x_{0}$ ranges over the classical points of $X$ , cover $X$ ; by step 2.1 each $C_{r} \cap U$ is closed in $U$ , hence $X ∖ C_{r}$ is open: a point $x \notin C_{r}$ lies in one of these charts $U$ , and $U ∖ (C_{r} \cap U)$ is an open neighbourhood of $x$ contained in $X ∖ C_{r}$ . By [F26] the set $C_{r}$ therefore equals its closure in $X$ and is a closed subset of the classical variety $X$ ; with the reduced structure induced from $X$ it is a closed subvariety of $X$ [F5], which is claim 1.

3.2F18F23F27step 1.7step 2.2

Dimension of the image of each component. By step 1.7 the differential $d_{x} (f ∣_{Z_{i}})$ is surjective, so by [F18] $rank d_{x} (f ∣_{Z_{i}}) = \dim_{k} T_{y} W_{i} .$ Since $y \in (W_{i})_{r e g}$ and $W_{i}$ is irreducible, [F27] gives $\dim_{k} T_{y} W_{i} = \dim_{y} W_{i}$ , and [F23] gives $\dim_{y} W_{i} = \dim W_{i}$ . Therefore $\dim W_{i} = rank d_{x} (f ∣_{Z_{i}}) \leq r$ by step 2.2.

4.1F6F23F24F25F26step 3.2

Assembling the closure. Since $C_{r} = Z_{1} \cup \dots \cup Z_{s}$ , one has $f (C_{r}) = f (Z_{1}) \cup \dots \cup f (Z_{s})$ . The finite union $W_{1} \cup \dots \cup W_{s}$ of the closed sets $W_{i} = \overline{f (Z_{i})}$ is closed and contains $f (C_{r})$ , so $\overline{f (C_{r})} \subseteq W_{1} \cup \dots \cup W_{s}$ by [F26]; conversely each $W_{i} = \overline{f (Z_{i})}$ is contained in $\overline{f (C_{r})}$ because $f (Z_{i}) \subseteq f (C_{r})$ and $\overline{f (C_{r})}$ is closed [F26]. Hence $\overline{f (C_{r})} = W_{1} \cup \dots \cup W_{s}$ , a finite union of closed subsets of the Noetherian space $Y$ [F6]; by [F25] its dimension is $\max_{i} \dim W_{i}$ , which is at most $r$ by step 3.2, and for $C_{r} = \emptyset$ , when the list $W_{1}, \dots, W_{s}$ is empty, both the maximum and $\dim \emptyset$ are $- \infty$ by [F24], which is at most $r$ . This is claim 2.

5.1F1F4F6F7F14F18F19F20F22F24F27F28F31step 1.2step 4.1∎

Boundary and scope dispositions. Empty: $X$ and $Y$ may be empty or reducible, and no irreducibility is assumed; for $X = \emptyset$ the set $C_{r}$ is empty and closed and $\dim \overline{f (C_{r})} = \dim \emptyset = - \infty \leq r$ by [F24], while for $Y = \emptyset$ also $X = \emptyset$ because a morphism into the empty scheme has empty source. Zero: $r = 0$ is allowed, and then $C_{0}$ consists of the points where the differential vanishes; the relative dimension $n - c = 0$ of step 1.2 is allowed, in which case $Ω_{B / k} = 0$ by [F14], the matrix $A$ has no rows, its rank is $0$ , and $C_{r} \cap U = U$ for every $r \geq 0$ with $\dim \overline{f (U)} \leq \dim V = 0$ when $V$ is a point. One: nothing in the argument divides by a natural number or requires a positive relative dimension or a positive number of coordinate functions; for $m = 0$ , when $V$ is a single point, the matrix $A$ has no columns, $γ = 0$ and $C_{r} \cap U = U$ , in agreement with claim 2. Degenerate: the smoothness of $X$ in claim 1 is essential and cannot be dropped — for the singular closed subvariety $X = V (x y) \subseteq A_{k}^{2}$ , the morphism $f ⁣ : X \to A_{k}^{1}$ , $(x, y) \mapsto x$ , has $rank d_{(0, b)} f = 0$ for every $b \neq 0$ while $rank d_{(0, 0)} f = 1$ , so the corresponding $C_{0}$ is the punctured $y$ -axis, not closed; reducible and disconnected smooth $X$ are nevertheless allowed, and claim 2 does not use the smoothness of $Y$ , the argument for it needing only the local presentation of $Y$ with coordinate functions generating its coordinate ring. Endpoints: the integer $r$ ranges over $r \geq 0$ with no upper bound, and claim 2 is trivial for $r \geq \dim Y$ since $\dim \overline{f (C_{r})} \leq \dim Y$ ; the marginal case $r = 0$ with $C_{0} = \emptyset$ is covered by the empty-list convention $- \infty \leq 0$ of step 4.1, while the case in which $U_{i}$ is a single point is covered because the surjectivity conclusion of [F28] persists at every closed point of $U_{i}$ . Nonempty-choice: AC is declared as [F1] and is used exactly through the AC-assuming suppliers [F4] (classical-scheme dictionary), [F6] (finite component decompositions), [F7] (principal opens), [F20] (local rings), [F22] (Nullstellensatz correspondence), [F27] (regular-point tangent criterion), [F28] (generic differential surjectivity) and [F31] (closures of irreducible sets), cited at steps 1.1, 1.2, 1.6, 1.7, 1.8 and 4.1; the chart, matrix, determinantal and rank-comparison computations of steps 1.3, 1.4, 1.5, 2.1, 2.2 and the local-ring duality of step 1.8 are choice-free, and no family of nonempty sets is selected anywhere. Biconditional directions: the statement asserts no equivalence, so the forward and reverse directions of a biconditional are not applicable; the only equivalence used inside the proof is the determinantal criterion of [F18], applied in step 2.1 in the direction "all $(r + 1)$ -minors vanish implies rank at most $r$ " and conversely, and the cotangent isomorphisms of [F12] are used only through their naturality.

Source qualification

Vakil, Classes 51-52, §3.4 proves the corresponding statement for morphisms of finite-type $k$ -schemes over an algebraically closed (or at least perfect) field of characteristic $0$ : the locus $X_{r}$ where the rank of the tangent map is at most $r$ is a closed subset, and the dimension of the image of $X_{r}$ is at most $r$ ; the proof replaces the source by an irreducible component of $X_{r}$ and the target by the closure of the image of that component, and then applies generic smoothness on the source together with the linear-algebra observation that restricting a linear map to subspaces cannot increase its rank. The present lemma keeps the smoothness of $X$ and $Y$ from the section's standing hypotheses: the source's assertion that the critical locus is cut out by determinantal equations is not available for an arbitrary singular source — the degenerate case recorded in step 5.1 shows this — so claim 1 is proved here from the standard smooth charts of $X$ , on which the differential is described by regular functions, and the rank comparison of step 1.8 supplies the subspace step of the source's reduction directly. The argument uses the smoothness of $Y$ only to choose a local presentation with coordinate functions generating its coordinate ring, so the same proof covers an arbitrary classical $Y$ . Characteristic $0$ enters only through A dominant map has a surjective differential on a dense source open in step 1.7; claim 1 is characteristic-free. The determinantal closedness of step 2.1 is the classical Jacobian-minor computation (Milne, Algebraic Geometry, §4d, Definition 4.22, in the equation-row convention) applied on the charts, and the local-ring duality of step 1.8 replaces the source's implicit identification of the tangent space of a closed subvariety with a subspace of the tangent space of the ambient variety.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

Generic smoothness over a dense target open

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field of characteristic $0$ , let $X$ and $Y$ be irreducible classical varieties over $k$ (Classical algebraic prevarieties, regular maps, and varieties), and let $f ⁣ : X \to Y$ be a morphism of classical varieties. Assume $X$ is smooth over $k$ , that is, the structure morphism $X \to Spec k$ is smooth in the locally-standard-smooth sense of Smooth morphisms via local standard smooth presentations (equivalently, since $k$ is perfect, $X$ is regular). Then:

there is a dense open subvariety $U \subseteq Y$ such that the restriction $f^{- 1} (U) ⟶ U$ is a smooth morphism of finite-type $k$ -schemes; when $f$ is not dominant one may take $U$ with $f^{- 1} (U) = \emptyset$ , the empty morphism being smooth;
if in addition $f$ is dominant, there is a nonempty open (hence dense) $V \subseteq U$ such that for every closed point $y \in V$ the scheme-theoretic fibre $X_{y} = X \times_{Y} Spec k (y)$ (Scheme-theoretic fibre) is nonempty, smooth over $k$ , and of pure dimension $r = \dim X - \dim Y$ .

Neither $Y$ nor $f$ is required to be smooth or flat, the fibres are not required to be irreducible or connected, and no statement is made about the size of $U$ or $V$ . The characteristic- $0$ hypothesis enters through the critical-locus dimension bound of Critical loci have small images in characteristic zero in claim 1 and through the perfectness of $k$ ; the failure of the target-open statement for a non-smooth source and the positive-characteristic failure of the corresponding source-side statement are recorded on the counterexample page of this pair.

Facts & Assumptions

Given: The Axiom of Choice; an algebraically closed field $k$ of characteristic $0$ ; irreducible classical varieties $X$ and $Y$ over $k$ ; the hypothesis that $X \to Spec k$ is smooth in the sense of Smooth morphisms via local standard smooth presentations; and a morphism $f ⁣ : X \to Y$ of classical varieties.

[F1]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function.

[F2]

Classical algebraic prevarieties, regular maps, and varieties: a classical algebraic prevariety over $k$ is a quasi-compact locally ringed space with a structure sheaf of $k$ -algebras, covered by open subspaces isomorphic to affine models (polynomial zero sets, including empty and reducible ones), whose points have residue field canonically $k$ and whose sections are functions; principal opens form a basis of the topology, zero loci of regular functions are closed, a classical algebraic variety is a separated prevariety, and these definitions use no Axiom of Choice.

[F3]

Smooth morphisms via local standard smooth presentations: for a finite-type morphism $f ⁣ : X \to Y$ of $k$ -schemes, smoothness means that every source point has affine neighbourhoods on which the induced ring map is standard smooth at the prime of that point, where standard smoothness at a prime allows a further principal shrinking; the condition is imposed at every source point and is local on the source and on the target.

[F4]

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras: a $k$ -algebra is of finite type when it is generated by finitely many elements, so a finite-type $k$ -algebra $B$ contained in a field or ring with $k \subseteq A \subseteq B$ is generated as an $A$ -algebra by the same finite list.

[F5]

Locally finite type and finite type morphisms: a morphism is locally of finite type when it is described on affine charts by finite-type ring maps, and of finite type when it is locally of finite type and quasi-compact.

[F6]

[F7]

Global and local dimension of classical varieties: for a classical variety $X$ and a closed point $x$ , $\dim_{x} X$ is the maximum of the dimensions of the irreducible components containing $x$ , while $\dim X$ is its chain dimension; for irreducible $X$ one has $\dim_{x} X = \dim X$ at every point.

[F8]

Nonempty opens preserve irreducible dimension: if $U$ is a nonempty open of an irreducible classical variety $X$ , then $\dim U = \dim X$ , and every proper closed subvariety $Z ⊊ X$ has $\dim Z < \dim X$ .

[F9]

Irreducibility via nonempty open subsets, connectedness and open subspaces: an irreducible space is nonempty and every nonempty open subset of it is dense and irreducible.

[F10]

[F11]

Dense regular loci on every component: for a reduced $k$ -scheme $X$ of finite type over a perfect field, the regular locus $X_{r e g}$ is open, its intersection with every irreducible component is a dense open subset of that component, and $X_{r e g} \neq \emptyset$ whenever $X \neq \emptyset$ .

[F12]

Regular and singular loci: for a locally Noetherian scheme, $X_{r e g} = {x : O_{X, x} is a regular local ring}$ ; for a reduced classical finite-type space over an algebraically closed field and a closed point $x$ , one has $x \in X_{r e g}$ if and only if $\dim_{κ (x)} T_{x} X = \dim_{x} X$ .

[F13]

Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: every field of characteristic zero is perfect, and every algebraically closed field is perfect.

[F14]

Regular equals smooth over a perfect field: under AC, for a perfect field $k$ and a finite-type $k$ -scheme $X$ , $X$ is regular (every local ring $O_{X, x}$ is regular local) if and only if $X \to Spec k$ is smooth in the local-standard-smooth sense.

[F15]

Affine open subschemes: for a scheme $X$ and open $U \subseteq X$ , the open subscheme is $(U, O_{X} ∣_{U})$ , with the restricted structure sheaf.

[F16]

The stalk of a presheaf at a point: the stalk at a point is the filtered colimit of the sections over open neighbourhoods of that point; the neighbourhoods contained in an open $U$ are cofinal, so $O_{U, x} ≅ O_{X, x}$ canonically for $x \in U$ .

[F17]

Every nonempty principal open is a classical affine variety: under AC, for an affine variety $Z$ and $0 \neq h \in k [Z]$ , the principal open $D_{Z} (h)$ is an affine variety with coordinate ring canonically $k [Z]_{h}$ .

[F18]

Critical loci have small images in characteristic zero: under AC, for $k$ algebraically closed of characteristic $0$ , smooth classical varieties $X$ and $Y$ over $k$ , a morphism $f ⁣ : X \to Y$ and $r \geq 0$ , the set $C_{r} = {x \in X : rank d_{x} f \leq r}$ is closed in $X$ and $\dim \overline{f (C_{r})} \leq r$ , the closure being taken in $Y$ .

[F19]

[F20]

The submersion criterion between smooth varieties: under AC, for smooth classical varieties $X, Y$ over algebraically closed $k$ whose structure morphisms are smooth, a finite-type morphism $f ⁣ : X \to Y$ and a classical point $x \in X$ , the morphism $f$ is smooth at $x$ if and only if $d_{x} f ⁣ : T_{x} X \to T_{f (x)} Y$ is surjective.

[F21]

Scheme-theoretic fibre: for a morphism $f ⁣ : X \to S$ and a point $s \in S$ , the scheme-theoretic fibre is $X_{s} = X \times_{S} Spec κ (s)$ , viewed as a $κ (s)$ -scheme; empty fibres are allowed.

[F22]

Points and topology of a fibre: for $f ⁣ : X \to S$ and $s \in S$ , the projection $X_{s} \to X$ is a homeomorphism onto $f^{- 1} (s)$ with the subspace topology and preserves residue fields.

[F23]

Restricting fibre products to open subschemes: for $f ⁣ : X \to S$ and an open $U \subseteq S$ , the open subscheme $f^{- 1} (U)$ represents the fibre product $X \times_{S} U$ .

[F24]

Existence of all scheme fibre products: fibre products of schemes exist with their universal property, so iterated fibre products over compatible bases are canonically isomorphic.

[F25]

Base change and composition of standard smooth presentations: a standard smooth algebra remains standard smooth after arbitrary base change of the base ring, and locally standard smooth morphisms are stable under arbitrary base change of the base ring; this uses no Axiom of Choice.

[F26]

Fibres have pure expected dimension over a dense open: for a dominant morphism $f ⁣ : X \to Y$ between irreducible classical varieties there is a nonempty open $U \subseteq Y$ , contained in $f (X)$ , such that every fibre $X_{y}$ with $y \in U$ is nonempty and has pure dimension $r = \dim X - \dim Y$ .

[F27]

Irreducible classical varieties and integral separated finite-type schemes: the closed-point construction and its inverse give an equivalence between irreducible classical $k$ -varieties and integral finite-type $k$ -schemes satisfying the affine-overlap separation condition, each original point being identified with its singleton; classical points correspond to closed points, and classical regular maps to scheme $k$ -morphisms.

[F28]

In a finite-type algebra over a field, closed points are dense in every closed subset of the spectrum: under AC, every nonempty closed subset of the spectrum of a finite-type algebra over a field contains a closed point; closed points are dense in each closed subset.

Proof

technique · direct

1.1F2F3F4F5F6F7F13F14F27given

Setup and conventions. By [F2] the classical varieties $X$ and $Y$ are quasi-compact locally ringed spaces over $k$ covered by affine models, every point of either is a closed point with residue field $k$ , polynomial principal opens form a basis of the topology, and their structure sheaves are sheaves of $k$ -valued functions; by [F27] the irreducible classical varieties $X$ and $Y$ correspond to integral, hence reduced, finite-type $k$ -schemes and $f$ to a $k$ -morphism of those schemes; they are Noetherian by [F6]. Write $m = \dim X$ and $n = \dim Y$ [F7]. The field $k$ is perfect by [F13], and by hypothesis the structure morphism $X \to Spec k$ is smooth [F3], so [F14] makes $X$ regular. Every $k$ -morphism of finite-type $k$ -schemes is of finite type: on affine charts $Spec A \subseteq Y$ and $Spec B \subseteq X$ with $f (Spec B) \subseteq Spec A$ , the algebra $B$ is generated as an $A$ -algebra by finitely many $k$ -algebra generators [F4], so $f$ is locally of finite type [F5], and it is quasi-compact because $X$ is Noetherian, so that every open subset of $X$ is quasi-compact [F5, F6]; the same argument applies to the restriction of $f$ to any open subvariety of $X$ .

1.2F2F3F9F10given

The non-dominant case. Suppose $f$ is not dominant, so the closure $Z : = \overline{f (X)}$ is a closed subset of $Y$ with $Z \neq Y$ [F10]. Its complement $U : = Y ∖ Z$ is open and nonempty, and it is dense in $Y$ because a nonempty open subset of the irreducible space $Y$ is dense [F9]; moreover $f (X) \subseteq Z$ , so $f^{- 1} (U) = \emptyset$ . The empty morphism $\emptyset \to U$ is smooth by [F3], the standard-smooth condition being imposed at every source point and the empty source having none; the empty scheme is a classical variety and the morphism is of finite type because its source is quasi-compact. Thus claim 1 holds in this case with this $U$ .

1.3F2F3F4F5F6F8F9F10F11F12F13F14F15F16F17F27given

The dominant case: reduction to the regular locus of the target. Suppose now that $f$ is dominant. The regular locus $Y_{r e g}$ [F12] of $Y$ , a reduced finite-type $k$ -scheme over the perfect field $k$ [F27], is a nonempty open subset of $Y$ whose intersection with every irreducible component of $Y$ is dense open in that component [F11]; since $Y$ is irreducible, $Y_{r e g}$ is nonempty, open and dense, hence irreducible [F9], and $\dim Y_{r e g} = \dim Y = n$ [F8]. At every $y \in Y_{r e g}$ the local ring $O_{Y_{r e g}, y} ≅ O_{Y, y}$ is regular [F12, F15, F16], and $Y_{r e g}$ is of finite type over the perfect field $k$ [F2, F13], so $Y_{r e g} \to Spec k$ is smooth by [F14]. The open subvariety $Y_{r e g}$ is itself a classical variety over $k$ : it is quasi-compact because $Y$ is Noetherian [F6], and its intersections with the affine models of $Y$ are covered by principal opens, which are affine models by [F17]. Similarly $X_{0} : = f^{- 1} (Y_{r e g})$ is an open subvariety of $X$ , hence a classical variety over $k$ , it is nonempty because the dense subset $f (X)$ meets the nonempty open set $Y_{r e g}$ [F9, F10], it is irreducible with $\dim X_{0} = \dim X = m$ [F8, F9], and its structure morphism $X_{0} \to Spec k$ is smooth by locality on the source [F3]. The restriction $f_{0} : = f ∣_{X_{0}} ⁣ : X_{0} \to Y_{r e g}$ is a finite-type morphism of classical varieties: on affine charts $Spec A \subseteq Y_{r e g}$ and $Spec B \subseteq X_{0}$ with $f_{0} (Spec B) \subseteq Spec A$ the algebra $B$ is generated as an $A$ -algebra by finitely many $k$ -algebra generators [F4, F5], and $f_{0}$ is quasi-compact because $X_{0}$ is Noetherian, so that every open subset of $X_{0}$ is quasi-compact [F5, F6].

2.1F9F10F18F19step 1.3given

The rank- $(n - 1)$ locus and the open set. Let $C = {x \in X_{0} : rank d_{x} f_{0} \leq n - 1}$ . If $n \geq 1$ , then [F18] applied to the morphism $f_{0}$ of smooth classical varieties with $r = n - 1 \geq 0$ shows that $C$ is closed in $X_{0}$ and that the closed subvariety $Z : = \overline{f_{0} (C)} \subseteq Y_{r e g}$ satisfies $\dim Z \leq n - 1$ ; if $n = 0$ , then $C = \emptyset$ because ranks are nonnegative, so $Z = \emptyset$ and $\dim Z = - \infty \leq - 1$ [F19]. In either case $Z \neq Y_{r e g}$ : when $n \geq 1$ because $\dim Y_{r e g} = n > n - 1 \geq \dim Z$ by step 1.3, and when $n = 0$ because $Z = \emptyset$ while $Y_{r e g} \neq \emptyset$ by step 1.3. Put $U : = Y_{r e g} ∖ Z$ ; then $U$ is open in $Y_{r e g}$ and in $Y$ , it is nonempty because $Z \neq Y_{r e g}$ , and it is dense in $Y$ because a nonempty open subset of the irreducible space $Y$ is dense [F9]. Also $X^{'} : = f^{- 1} (U) = f_{0}^{- 1} (U)$ is a nonempty open subvariety of $X_{0}$ because $f$ is dominant and $U$ is nonempty open [F9, F10].

3.1F7F8F12step 2.1given

The rank equals $n$ on $f^{- 1} (U)$ . Let $x$ be a classical closed point of $X^{'} = f^{- 1} (U)$ , so $x \in X_{0}$ and $f (x) \in U \subseteq Y_{r e g}$ by step 2.1. Then $x \notin C$ , so $rank d_{x} f_{0} \geq n$ by the definition of $C$ ; on the other hand $rank d_{x} f_{0} \leq \dim_{k} T_{f (x)} Y_{r e g}$ because $d_{x} f_{0}$ is a $k$ -linear map into that finite-dimensional space. Since $f (x)$ is a regular point of the classical variety $Y_{r e g}$ we have $\dim_{k} T_{f (x)} Y_{r e g} = \dim_{f (x)} Y_{r e g}$ [F12], and since $Y_{r e g}$ is irreducible of dimension $n$ [step 1.3] this equals $\dim Y_{r e g} = n = \dim Y$ [F7, F8]. Hence $rank d_{x} f_{0} = n$ , and $d_{x} f_{0}$ is surjective.

4.1F2F3F20F27F28step 1.3step 2.1step 3.1given

From closed points to every scheme point. At each classical closed point $x \in X^{'}$ the morphism $f_{0} : X_{0} \to Y_{r e g}$ is between smooth classical varieties with their smooth scheme structures and is of finite type by step 1.3. Its differential is surjective by step 3.1, so [F20] gives smoothness at $x$ . Restricting over $U$ preserves this local property by [F3]; hence $g : X^{'} = f^{- 1} (U) \to U$ is smooth at every closed point. Let $S \subseteq X^{'}$ be the scheme smooth locus of $g$ . It is open: a standard smooth presentation after principal shrinking, as in [F3], witnesses smoothness at every prime of that shrinking, since its Jacobian minor is a unit there. If $X^{'} ∖ S$ were nonempty, intersect it with an affine chart $Spec B$ of the finite-type scheme $X^{'}$ . The intersection is a nonempty closed subset, and [F28] gives a closed point of that chart in it. By [F27] this is a classical point of $X^{'}$ , contrary to the closed-point conclusion just proved. Thus $S = X^{'}$ , and $g$ is smooth at every scheme point. This proves claim 1.

5.1F9F21F22F23F24F25F26F28step 2.1step 4.1given

The fibres over the further open set. Suppose $f$ is dominant and let $U$ be the dense open set of step 2.1, over which $f^{- 1} (U) \to U$ is smooth by step 4.1. By [F26] there is a nonempty open $V_{1} \subseteq Y$ , contained in $f (X)$ , such that for every closed point $y \in V_{1}$ the fibre $f^{- 1} (y)$ is nonempty and of pure dimension $r = m - n = \dim X - \dim Y$ ; put $V : = U \cap V_{1}$ , a nonempty open subset of the irreducible $Y$ , hence dense [F9]. For a closed point $y \in V$ , so that $k (y) = k$ , the scheme-theoretic fibre $X_{y} = X \times_{Y} Spec k (y)$ [F21] has underlying topological space $f^{- 1} (y)$ by [F22], so $X_{y}$ is nonempty. The classical fibre in [F26] is its closed-point space. Closed-point density [F28] identifies closed subsets and irreducible components of the scheme fibre with their classical traces, chart by chart, preserving strict chains and dimensions; nilpotents do not affect these spaces. Thus $X_{y}$ has pure dimension $r$ . For smoothness, the fibre product $X \times_{Y} U$ is represented by the open subscheme $f^{- 1} (U) \subseteq X$ by [F23], and the universal property of fibre products [F24] gives a canonical isomorphism $X_{y} ≅ f^{- 1} (U) \times_{U} Spec k (y)$ ; the projection on the right is the base change of the smooth morphism $f^{- 1} (U) \to U$ along $Spec k (y) \to U$ , hence is smooth over $k$ because locally standard smooth morphisms are stable under base change [F25]. Therefore every fibre $X_{y}$ with closed $y \in V$ is nonempty, smooth over $k$ , and of pure dimension $r = \dim X - \dim Y$ .

6.1F1F9F11F12F14F18F19F20F26step 1.2step 1.3step 2.1step 3.1step 4.1step 5.1given∎

Boundary and scope dispositions. Empty: in the non-dominant case $f^{- 1} (U) = \emptyset$ and the empty morphism $\emptyset \to U$ is smooth vacuously (step 1.2); in the dominant case $X$ and $Y$ are nonempty because irreducible [F9], the open sets $Y_{r e g}$ , $U$ and $V$ are nonempty by steps 1.3, 2.1 and 5.1, and the fibres over $V$ are nonempty by step 5.1, so no empty-fibre convention is invoked in claim 2. Zero: the target dimension $n = 0$ is admitted; then $C = Z = \emptyset$ , $U = Y_{r e g}$ and the rank computation of step 3.1 reads $0 \leq rank d_{x} f_{0} \leq 0$ , while the fibre clause gives a single fibre of pure dimension $r = m$ ; the relative dimension $r = 0$ is likewise admitted in step 5.1, where smooth fibres of pure dimension zero are finite reduced $k$ -schemes, and nothing in the argument divides by $r$ . One: no step divides by a natural number, selects a basis, or requires a positive dimension, codimension, or number of equations; the cases $n = 1$ with $r = 0$ or $r = 1$ are covered by the same steps 2.1 through 5.1. Degenerate: the smoothness of $X$ is essential for claim 1 and is used through the critical-locus bound [F18] in step 2.1 and the submersion criterion [F20] in step 4.1; the constant cusp family $X = Spec k [x, y, z] / (y^{2} - x^{3}) \to A_{k}^{1}$ , $(x, y, z) \mapsto z$ , is a dominant morphism of irreducible classical varieties over $k$ to a smooth target whose every fibre is the singular cusp, so that no nonempty open $U$ has $f^{- 1} (U) \to U$ smooth, as recorded on the counterexample page of this pair. The target $Y$ need not be smooth outside $Y_{r e g}$ : for the fold $f ⁣ : A_{k}^{1} \to A_{k}^{1}$ , $t \mapsto t^{2}$ , the differential vanishes at $0$ , the fibre over $0$ is the non-reduced $Spec k [ϵ] / (ϵ^{2})$ , and $U = A_{k}^{1} ∖ {0}$ is the best possible dense open, while the constant morphism $A_{k}^{1} \to A_{k}^{1}$ with value $0$ has $f^{- 1} (U) = \emptyset$ for $U = A_{k}^{1} ∖ {0}$ ; the fibres of step 5.1 are not asserted to be irreducible or connected. Endpoints: the statement has no interval parameter; the boundary $n = 0$ versus $n \geq 1$ is handled in step 2.1 through the convention $\dim \emptyset = - \infty \leq - 1$ of [F19] and the trivial lower bound in step 3.1, and the open sets $U, V$ are dense but need not be all of $Y$ , as the fold example shows; the generic fibre dimension is constant over $V$ by construction and not merely bounded. Nonempty-choice: AC is declared as [F1] and enters exactly through the AC-assuming suppliers [F11] (density of the regular locus), [F12] (the classical regular-point tangent test), [F14] (regularity versus smoothness), [F18] (the critical-locus dimension bound), [F20] (the submersion criterion) [F26] (generic fibre dimension), and [F28] (closed-point density), cited at steps 1.3, 2.1, 3.1, 4.1 and 5.1; the finite-type verification of step 1.1, the fibre-product pasting and base-change smoothing of step 5.1, and the remaining linear algebra are choice-free, and no family of nonempty sets is selected anywhere. Biconditional directions: the statement asserts no equivalence, so the forward and reverse directions of a biconditional are not applicable; the two equivalences used in the proof — the submersion criterion [F20], applied in the direction "surjective differential at a classical point implies smoothness there" in step 4.1, and regularity-versus-smoothness [F14], applied in the direction "regular implies smooth" in step 1.3 for $Y_{r e g}$ — are used only in those directions, and no converse of the theorem is claimed.

Source qualification

Vakil, Classes 51–52, §3.3, proves the corresponding target-open statement for a morphism $f ⁣ : X \to Y$ of $k$ -varieties with $char k = 0$ and $X$ smooth: there is a dense open subset of $Y$ over which the restricted morphism is smooth, with the explicit warning that the inverse image may be empty when $f$ is not dominant; the proof restricts to the smooth locus of $Y$ , removes the closure of the image of the rank- $(n - 1)$ locus using the §3.4 lemma, and then invokes the submersion criterion ("Hard Exercise 2.2") at every remaining closed point. The present theorem keeps the scaffold's hypotheses that $X$ and $Y$ are irreducible and makes the conclusion scheme-precise: the open set is produced by the authored critical-locus bound of this pair, the fibres in claim 2 are the scheme-theoretic fibres, and their smoothness is obtained from stability of locally standard smooth morphisms under base change rather than from a separate fibre-smoothness theorem. The second clause is the Bertini–Sard statement of Arapura, Theorem 5.4.2 (printed p. 34), which for a dominant morphism of nonsingular varieties over a field of characteristic $0$ produces a nonempty open set of the target over which the fibres are nonsingular with surjective differentials at every point; Arapura does not state nonemptiness of the fibres, pure dimension, or the non-dominant case, and refers for its proof to Hartshorne III 10.7, which is not used here. Vakil's "for pedants" remark generalizes the hypotheses to morphisms of locally Noetherian schemes over $Q$ ; the statement above keeps the algebraically closed characteristic- $0$ form. The characteristic- $0$ hypothesis is essential: the Frobenius morphism on $A_{k}^{1}$ in characteristic $p$ has vanishing differential everywhere, and the constant cusp family over $A_{k}^{1}$ shows that target-open generic smoothness fails without smoothness of the source; both are recorded on the counterexample page of this pair.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30Open item page →

A section of an invertible sheaf has a canonical zero subscheme

Statement

Let $X$ be a scheme, let $L$ be an invertible (locally free of rank one) $O_{X}$ -module, and let $s \in Γ (X, L)$ . Choose an affine open cover $U_{i} = Spec A_{i}$ on which $L$ has a generator $e_{i}$ , and write $s ∣_{U_{i}} = f_{i} e_{i}$ . The affine schemes $Spec (A_{i} / (f_{i}))$ glue, with their quotient maps, to a closed subscheme $Z (s) ↪ X$ that is canonical up to unique isomorphism over $X$ and independent of the chosen trivializations. The construction uses the ideal $(f_{i})$ itself, with no nonzerodivisor or reducedness hypothesis; it retains nilpotents and includes the empty and whole zero schemes.

Facts & Assumptions

Given: A scheme $X$ , an invertible $O_{X}$ -module $L$ , and a global section $s \in Γ (X, L)$ .

[F1]

An $O_{X}$ -module is a sheaf of modules compatible with restriction of scalars (Modules on a ringed space).

[F2]

A global section restricts along every open inclusion, and successive restrictions agree (Sections, restrictions, and global sections of a presheaf).

[F3]

Localizing a quotient by an ideal canonically gives the quotient by the localized ideal, including when the localization is zero (Localisation commutes with quotient rings: $S^{- 1} R / S^{- 1} I ≅ {\overset{ˉ}{S}}^{- 1} (R / I)$ ).

[F4]

For an ideal $I ⊴ R$ , $Spec (R / I)$ is homeomorphic to the closed subset $V (I) \subseteq Spec R$ (The spectrum of a quotient is a closed subspace).

[F5]

Affine schemes with compatible open-overlap isomorphisms satisfying the cocycle condition glue to a scheme, uniquely up to unique chart-compatible isomorphism (Gluing affine schemes along compatible open isomorphisms).

[F6]

Compatible scheme morphisms on an open cover glue uniquely (Morphisms of schemes are local on compatible open covers).

[F7]

A morphism is a closed immersion when its underlying map is a homeomorphism onto a closed subset and its structure-sheaf map is surjective (Closed immersions of schemes).

[F8]

Closed immersions can be checked on the inverse images of an open cover of the target (Closed immersions are local on the target).

Proof

Proof technique: construct the local quotients and glue them using the transition units of the line bundle.

1.1F1F2F3F4F7givenchoose

Since $L$ is locally free of rank one, choose an affine trivializing open cover $U_{i} = Spec A_{i}$ with generator $e_{i}$ . By [F1] and [F2], there is a unique $f_{i} \in A_{i}$ such that $s ∣_{U_{i}} = f_{i} e_{i}$ . Put $Z_{i} = Spec (A_{i} / (f_{i}))$ and let $q_{i} : Z_{i} \to U_{i}$ be the quotient morphism. Its image is $V (f_{i})$ by [F4], and it is a homeomorphism onto that image. On each distinguished open $D (g) \subseteq U_{i}$ , [F3] identifies the restricted quotient with the quotient map $A_{i, g} \to A_{i, g} / (f_{i})$ ; thus the map of structure sheaves is surjective locally. Hence $q_{i}$ is a closed immersion by [F7]. If $f_{i}$ is a unit then $A_{i} / (f_{i}) = 0$ and $Z_{i}$ is empty; if $f_{i} = 0$ then $Z_{i} = U_{i}$ .

2.1F1F2F3step 1.1algebra

On $U_{i} \cap U_{j}$ , the two generators differ by an invertible function: $e_{i} = u_{i j} e_{j}$ . Comparing the expressions for the same restricted section gives $f_{j} = u_{i j} f_{i}$ , so the generated ideals agree. Refine each overlap by affine opens $W = Spec R$ . On such a $W$ , both local zero schemes restrict to $Spec (R / (f_{i} ∣_{W}))$ and $Spec (R / (f_{j} ∣_{W}))$ ; equality of the ideals gives the canonical identity-on- $R$ isomorphism. On distinguished opens of $W$ , [F3] identifies the restrictions with the corresponding localized quotients.

3.1F5F6step 2.1

These overlap isomorphisms are compatible when further restricted: each acts on residue classes by the identity on functions from $O_{X}$ . They therefore satisfy the identity and cocycle conditions, including on triple overlaps. By [F5] the $Z_{i}$ glue to a scheme $Z$ , with each $Z_{i}$ an open subscheme of $Z$ . Their maps $q_{i}$ to the open subsets $U_{i} \subseteq X$ agree on overlaps, so [F6] glues them to a unique morphism $q : Z \to X$ .

4.1F5F6F8step 1.1step 2.1step 3.1

The restriction $q^{- 1} (U_{i}) \to U_{i}$ is $q_{i}$ , a closed immersion by step 1.1. The $U_{i}$ cover $X$ , so [F8] implies that $q$ is a closed immersion. For any other affine trivializing cover, refine both covers by affine opens. On each such open the two equations differ by a unit, hence define the same quotient ring and the same morphism to $X$ . The uniqueness in [F5] and [F6] then gives a unique isomorphism over $X$ between the two constructions.

5.1step 1.1step 2.1algebra∎

No radical or regularity condition entered the construction: it quotients by $(f_{i})$ , even when $f_{i}$ is a zero divisor or nilpotent. For example, on $X = Spec (k [ϵ] / (ϵ^{3}))$ with $L = O_{X}$ and $s = ϵ^{2}$ , the zero scheme is $Spec (k [ϵ] / (ϵ^{2}))$ , which is still nonreduced. Thus the construction retains precisely the nilpotents not killed by the section equation.

Source note

Vakil, Foundations of Algebraic Geometry Classes 51–52, §3.9 Corollary 3.9, printed p. 10 (PDF page 10), states that a section of an invertible sheaf gives a closed subscheme and that general such sections are smooth; the subsequent Exercise 3.10 asks for the Bertini proof. The notes assert the zero-subscheme construction but do not give its local quotient-and-gluing proof. Steps 1.1–4.1 derive that construction from the local equations, localization, and scheme gluing; the source is context for the application, not a substitute for this argument.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-sol)audited 2026-09-30Open item page →

Linear systems, base loci, and general members

Definition

Fix an algebraically closed field $k$ . Let $X$ be a $k$ -scheme, let $L$ be an invertible (locally free of rank one) $O_{X}$ -module, and let $W \neq 0$ be a finite-dimensional $k$ -linear subspace of $Γ (X, L)$ . The subspace $W$ is a linear system on $X$ .

Its parameter space is

P (W) = (W ∖ {0}) / k^{\times},

the set of one-dimensional subspaces of $W$ . If $\dim_{k} W = r$ , a choice of basis identifies $P (W)$ with $P_{k}^{r - 1}$ ; give it the projective Zariski topology. A change of basis is an invertible linear coordinate change, which carries homogeneous zero sets to homogeneous zero sets, so this topology does not depend on the chosen basis. In particular, when $r = 1$ , $P (W)$ is the one-point space $P_{k}^{0}$ .

For $0 \neq s \in W$ , write $Z (s) ↪ X$ for its zero subscheme from A section of an invertible sheaf has a canonical zero subscheme. Replacing $s$ by $λ s$ for $λ \in k^{\times}$ multiplies each local equation by a unit, so it leaves the quotient ideals and the closed subscheme unchanged. Thus $Z (s)$ depends only on the parameter $[s] \in P (W)$ . The base locus of $W$ is the closed subset

Bs (W) = ⋂_{0 \neq s \in W} ∣ Z (s) ∣ = ⋂_{[s] \in P (W)} ∣ Z (s) ∣ \subseteq X .

It is closed because each $∣ Z (s) ∣$ is closed and arbitrary intersections of closed subsets are closed. It is base-point-free when this subset is empty.

A property holds for a general member of $W$ if there is a nonempty Zariski-open subset $U \subseteq P (W)$ such that every parameter in $U$ has that property.

For a fixed projective embedding $X \subseteq P_{k}^{N}$ , the hyperplane system is the system cut out by restrictions of degree-one homogeneous forms; the degree- $e$ hypersurface system, for $e \geq 1$ , is cut out by restrictions of homogeneous forms of degree $e$ . These are viewed as sections of the corresponding powers of the hyperplane line bundle, with forms giving the same section identified.

Source note

Vakil, Foundations of Algebraic Geometry Classes 51–52, §3.9 Corollary 3.9, printed p. 10 (PDF page 10, lines 437–441), describes a finite-dimensional base-point-free linear system as a vector space of sections of an invertible sheaf and treats a general section as a point of $P H^{0} (X, L)$ . It asserts that each section gives a closed subscheme, but leaves the Bertini proof to Exercise 3.10; the preceding item supplies the zero-scheme construction. Arapura, Notes on Basic Algebraic Geometry, §5.4, printed pp. 38–39 (PDF pages 38–39, lines 1689–1721), identifies hyperplanes defined by linear forms up to nonzero scalar with the dual projective space and states the smooth hyperplane conclusion on a nonempty open subset. These passages support the projective parameter and “general” conventions and the hyperplane example; the basis-independent topology and arbitrary-subspace wording are made explicit here.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

The universal member away from the base locus

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field, let $X$ be a smooth finite-type $k$ -scheme, let $L$ be an invertible $O_{X}$ -module, and let $W \subseteq Γ (X, L)$ be a nonzero finite-dimensional linear system with $\dim_{k} W = r + 1$ , base locus $Bs (W)$ and $X^{\circ} = X ∖ Bs (W)$ (Linear systems, base loci, and general members).

Then there is a finite-type $k$ -scheme $I$ , the incidence of the linear system, with $k$ -morphisms $π : I \to X^{\circ}$ and $p : I \to P (W) = P_{k}^{r}$ , determined by $L$ and $W$ up to canonical isomorphism, such that:

(Local product.) Let $s_{0}, \dots, s_{r}$ be a $k$ -basis of $W$ and let $X_{j}^{\circ} = X^{\circ} ∖ ∣ Z (s_{j}) ∣$ be the open subset on which $s_{j}$ is invertible. If $r \geq 1$ , then $π$ restricted over $X_{j}^{\circ}$ exhibits $π^{- 1} (X_{j}^{\circ})$ as isomorphic over $X_{j}^{\circ}$ to the projection $X_{j}^{\circ} \times_{k} P_{k}^{r - 1} \to X_{j}^{\circ}$ .
(Fibres.) For every $[s] \in P (W) (k)$ the fibre $p^{- 1} ([s])$ is isomorphic over $X^{\circ}$ to the zero subscheme $Z (s) \cap X^{\circ}$ of the section $s$ (A section of an invertible sheaf has a canonical zero subscheme).
(Smoothness.) The structure morphism $I \to Spec k$ is smooth in the local-standard-smooth sense (Smooth morphisms via local standard smooth presentations).
(Small parameter space.) If $r = 0$ , that is $\dim_{k} W = 1$ , then $I = \emptyset$ .

The construction uses the actual equations $\sum_{i} ξ_{i} g_{i}$ of the members, with no reducedness or nonzerodivisor hypothesis on the local equations.

Facts & Assumptions

Given: An algebraically closed field $k$ ; a smooth finite-type $k$ -scheme $X$ ; an invertible $O_{X}$ -module $L$ ; a nonzero finite-dimensional linear system $W \subseteq Γ (X, L)$ with $\dim_{k} W = r + 1$ , base locus $Bs (W)$ and $X^{\circ} = X ∖ Bs (W)$ ; and a $k$ -basis $s_{0}, \dots, s_{r}$ of $W$ .

[F1]

Linear systems, base loci, and general members: $Z (s)$ is the zero subscheme of a nonzero section and $Bs (W) = ⋂_{0 \neq s \in W} ∣ Z (s) ∣$ ; the base locus is closed.

[F2]

A section of an invertible sheaf has a canonical zero subscheme: on an affine open $V = Spec A$ with $L ∣_{V}$ generated by $e$ and $s ∣_{V} = f e$ , the zero subscheme $Z (s) \cap V$ is $Spec (A / (f))$ ; the construction is independent of the trivialization.

[F3]

Relative projective space from standard charts: the standard charts $U_{j} = Spec k [x_{ℓ}^{(j)} : ℓ \neq j]$ of $P_{k}^{r}$ are affine, cover $P_{k}^{r}$ , and on $U_{j} \cap U_{j^{'}}$ the coordinates satisfy $x_{ℓ}^{(j)} = x_{ℓ}^{(j^{'})} / x_{j}^{(j^{'})}$ for $ℓ \neq j, j^{'}$ and $x_{j^{'}}^{(j)} = 1 / x_{j}^{(j^{'})}$ .

[F4]

Gluing affine schemes along compatible open isomorphisms: compatible open immersions of affine schemes along principal opens glue to a scheme with the given affine cover.

[F5]

Existence of all scheme fibre products: fibre products of $k$ -schemes exist, and over affine charts the product has the affine chart $Spec (A \otimes_{k} B)$ .

[F6]

Closed immersions of schemes: a quotient $A \to A / J$ of a commutative ring presents a closed immersion $Spec (A / J) ↪ Spec A$ .

[F7]

Smooth morphisms via local standard smooth presentations: a finite-type morphism is smooth when at every source point there are affine neighbourhoods on which the ring map is standard smooth at the corresponding prime; the condition is local on the source.

[F8]

Standard smooth presentations and locally standard smooth maps: a polynomial algebra $k [x_{1}, \dots, x_{n}]$ is standard smooth over $k$ with no equations ( $c = 0$ ), and the case $c = 0$ is a localisation of a polynomial ring.

[F9]

Products preserve smoothness: the scheme-theoretic product of finite-type $k$ -schemes smooth over a field $k$ is smooth over $k$ .

[F10]

The Axiom of Choice: AC is assumed and is spent through the declared suppliers.

Proof

technique · direct

1.1F1F2F3step 1.2step 2.1step 4.1givenalgebra∎

Local affine models. Let $V = Spec A \subseteq X$ be an affine open on which $L$ has a generator $e$ , and write $s_{i} ∣_{V} = g_{i} e$ with $g_{i} \in A$ ; such charts exist because $L$ is invertible and $X$ is quasi-compact. For each chart $U_{j} = Spec k [x_{ℓ}^{(j)} : ℓ \neq j]$ of $P_{k}^{r}$ put $I_{V, j} : = Spec ⁣ (A [x_{ℓ}^{(j)} : ℓ \neq j] / (g_{j} + \sum_{ℓ \neq j} g_{ℓ} x_{ℓ}^{(j)})) \subseteq V \times_{k} U_{j},$ the closed subscheme of the product cut out by the equation $\sum_{i} ξ_{i} g_{i} = 0$ written in the chart $x_{j}^{(j)} = 1$ . This is a quotient presentation of a closed subscheme of the affine product [F5, F6], and the equation is the local equation of the general member in the sense of [F1] and [F2]. [F1, F2, F3, F5, F6, given, construct] 1.2 Independence of the choices. If $e^{'} = u e$ is a second generator on $V$ with $u \in A^{\times}$ and $s_{i} ∣_{V} = g_{i}^{'} e^{'}$ , then $g_{i}^{'} = u^{- 1} g_{i}$ , so the ideal $(g_{j}^{'} + \sum_{ℓ \neq j} g_{ℓ}^{'} x_{ℓ}) = u^{- 1} (g_{j} + \sum_{ℓ \neq j} g_{ℓ} x_{ℓ})$ is unchanged, and the two closed subschemes of $V \times_{k} U_{j}$ coincide. On the overlap $U_{j} \cap U_{j^{'}}$ of the two standard charts, the transition formulas of [F3] identify the coordinates $x_{ℓ}^{(j)} = ξ_{ℓ} / ξ_{j}$ and $x_{ℓ}^{(j^{'})} = ξ_{ℓ} / ξ_{j^{'}}$ for the same homogeneous coordinates $ξ$ , so substituting $x_{ℓ}^{(j)} = x_{ℓ}^{(j^{'})} / x_{j}^{(j^{'})}$ and $x_{j^{'}}^{(j)} = 1 / x_{j}^{(j^{'})}$ into the first equation and multiplying by the unit $x_{j}^{(j^{'})}$ gives exactly the second equation $g_{j^{'}} + \sum_{ℓ \neq j^{'}} g_{ℓ} x_{ℓ}^{(j^{'})} = x_{j}^{(j^{'})} (g_{j} + \sum_{ℓ \neq j} g_{ℓ} x_{ℓ}^{(j)}) .$ Hence the local models agree on all overlaps of base charts and projective charts. [F1, F2, F3, algebra] 2.1 Gluing. The affine schemes $I_{V, j}$ , indexed by a finite trivializing affine cover of $X$ and by $j = 0, \dots, r$ , have pairwise compatible open immersions on their overlaps by step 1.2, so they glue along the principal opens of [F4] to a $k$ -scheme $I_{X}$ together with a closed immersion $I_{X} ↪ X \times_{k} P_{k}^{r}$ commuting with the two projections $π_{X} : I_{X} \to X$ and $p : I_{X} \to P_{k}^{r}$ . The scheme $I_{X}$ is finite type over $k$ because it is covered by the finitely many affine charts $I_{V, j}$ , each a quotient of a finitely generated polynomial algebra over $k$ . Define the incidence of the linear system to be the open subscheme $I : = I_{X} \times_{X} X^{\circ}$ obtained by base change along the open immersion $X^{\circ} ↪ X$ , and keep $π : I \to X^{\circ}$ and $p : I \to P_{k}^{r}$ for the restrictions. [F3, F4, F5, F6, step 1.1, step 1.2, algebra] 3.1 Local product structure. Fix $j$ and let $V = Spec A \subseteq X_{j}^{\circ}$ be an affine chart trivializing $L$ by $e$ , with $s_{i} ∣_{V} = g_{i} e$ . Since $V \cap ∣ Z (s_{j}) ∣ = \emptyset$ and $Z (s_{j}) \cap V$ is cut out by $g_{j}$ by [F2], the function $g_{j}$ lies in no maximal ideal of $A$ , hence $g_{j} \in A^{\times}$ . Assume $r \geq 1$ . For an incidence point $[ξ_{0} : \dots : ξ_{r}]$ over $V$ , the equation $\sum_{i} g_{i} ξ_{i} = 0$ and invertibility of $g_{j}$ imply that some $ξ_{ℓ}$ with $ℓ \neq j$ is nonzero: otherwise also $ξ_{j} = 0$ . Thus projection to the other coordinates defines an everywhere-defined map to $P_{k}^{r - 1}$ , and its inverse is $[η_{ℓ}]_{ℓ \neq j} ⟼ [ξ_{j} = - g_{j}^{- 1} \sum_{ℓ \neq j} g_{ℓ} η_{ℓ} : ξ_{ℓ} = η_{ℓ} (ℓ \neq j)] .$ To check scheme morphisms rather than only point maps, choose $ℓ_{0} \neq j$ and work on the chart $η_{ℓ_{0}} \neq 0$ , equivalently the source chart $ξ_{ℓ_{0}} \neq 0$ . Normalize $ξ_{ℓ_{0}} = 1$ . Its incidence ring is $A [y_{m} : m \neq ℓ_{0}] / (g_{ℓ_{0}} + g_{j} y_{j} + \sum_{m \neq j, ℓ_{0}} g_{m} y_{m}) ≅ A [y_{m} : m \neq j, ℓ_{0}],$ where the isomorphism eliminates $y_{j}$ by $y_{j} = - g_{j}^{- 1} (g_{ℓ_{0}} + \sum_{m \neq j, ℓ_{0}} g_{m} y_{m})$ . These charts cover the incidence over $V$ , and their maps agree on overlaps because the displayed homogeneous formulas are scale invariant. They glue to $π^{- 1} (V) ≅ V \times_{k} P_{k}^{r - 1}$ over $V$ , and the identifications agree under a change of trivialization of $L$ since all $g_{i}$ acquire the same unit factor. The affine $V$ cover $X_{j}^{\circ}$ , giving the asserted product over $X_{j}^{\circ}$ . In particular, the chart $ξ_{j} \neq 0$ alone need not cover the incidence; the eliminated coordinate is $ξ_{j}$ , while the covering charts have $ξ_{ℓ_{0}} \neq 0$ for $ℓ_{0} \neq j$ . [F2, F3, F5, step 1.1, step 1.2, step 2.1, algebra] 3.2 Fibres. Let $[s] \in P (W) (k)$ , choose $0 \neq s \in W$ representing it and a $k$ -basis $s_{0}, \dots, s_{r}$ with $s = s_{0}$ . Over an affine chart $V = Spec A \subseteq X^{\circ}$ trivializing $L$ by $e$ with $s_{i} ∣_{V} = g_{i} e$ , the point $[s]$ lies in the chart $U_{0}$ of $P_{k}^{r}$ ; the fibre of $p$ over it is obtained by substituting the coordinates $(1, 0, \dots, 0)$ into $g_{0} + \sum_{ℓ \neq 0} g_{ℓ} x_{ℓ}^{(0)}$ , that is, it is $Spec (A / (g_{0})) = Z (s_{0}) \cap V$ by [F2]. Since the trivializing charts cover $X^{\circ}$ , this identifies the fibre with $Z (s) \cap X^{\circ}$ . [F1, F2, F3, step 1.1, step 2.1, algebra] 4.1 Smoothness. Let $x \in I$ . By step 3.1 some open neighbourhood of $x$ in $I$ is isomorphic over a smooth open subscheme of $X^{\circ}$ to $U \times_{k} P_{k}^{r - 1}$ when $r \geq 1$ , while for $r = 0$ the incidence is empty and the claim is vacuous. The open subscheme $X_{j}^{\circ}$ of the smooth $X$ is smooth over $k$ because smoothness is local on the source [F7]. The projective space $P_{k}^{r - 1}$ is smooth over $k$ : its standard charts are polynomial algebras $k [x_{1}, \dots, x_{r - 1}]$ , which are standard smooth with $c = 0$ by [F8], and smoothness is local on the source [F7]. By [F9] each product $X_{j}^{\circ} \times_{k} P_{k}^{r - 1}$ is smooth over $k$ , and these open pieces cover $I$ , so $I \to Spec k$ is smooth by [F7]. The Axiom of Choice enters only through the declared suppliers, notably [F9]; the finitely many charts and basis elements chosen here are finite choices and need no choice principle. [F7, F8, F9, F10, step 2.1, step 3.1, algebra] 5.1 The case $r = 0$ , and the boundary dispositions. If $r = 0$ then $W = k s_{0}$ for $0 \neq s_{0} \in W$ , so $Bs (W) = ∣ Z (s_{0}) ∣$ and $X^{\circ} = X ∖ ∣ Z (s_{0}) ∣$ . On every trivializing affine chart $V \subseteq X^{\circ}$ the coefficient $g_{0}$ is a unit by the argument of step 3.1, so the equation $g_{0} = 0$ cuts out the empty subscheme; as the charts cover $X^{\circ}$ , indeed $I = \emptyset$ , and the structural claims are vacuous. If $X = \emptyset$ then $Γ (X, L) = 0$ , so the hypothesis $W \neq 0$ has no instance. If $X^{\circ} = \emptyset$ , the incidence $I$ is empty by its definition, and the local product and fibre clauses are vacuous. The construction is canonical: a change of $k$ -basis of $W$ multiplies the vector of coefficients $(ξ_{i})$ by an invertible constant matrix and hence induces an automorphism of $P_{k}^{r}$ carrying the equation $\sum_{i} ξ_{i} g_{i} = 0$ to itself, and a change of trivialization multiplies all $g_{i}$ by a unit; so $I$ is determined up to unique isomorphism compatible with both $π$ and $p$ , equivalently over $X^{\circ} \times_{k} P (W)$ . Uniqueness follows because the glued quotient maps define a closed immersion into that product: a morphism over the product must be the identity on each quotient chart. This completes the proof.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

Bertini smoothness away from the base locus

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field of characteristic $0$ . Let $X$ be a smooth $k$ -scheme of finite type that admits a locally closed immersion into some projective space over $k$ (that is, $X$ is smooth and quasi-projective; Immersion of schemes), let $L$ be an invertible $O_{X}$ -module, and let $W \subseteq Γ (X, L)$ be a nonzero finite-dimensional linear system, with $\dim_{k} W = r + 1$ , base locus $Bs (W)$ and $X^{\circ} = X ∖ Bs (W)$ (Linear systems, base loci, and general members); thus $P (W) = P_{k}^{r}$ .

Then there is a nonempty Zariski-open subset $U \subseteq P (W)$ such that for every closed point $[s] \in U$ of the $k$ -scheme $P (W)$ (equivalently, by Irreducible classical varieties and integral separated finite-type schemes, every classical parameter $[s]$ lying in $U$ ), and every representative $0 \neq s \in W$ , the closed subscheme $Z (s) \cap X^{\circ} \subseteq X^{\circ}$ is smooth over $k$ . Thus the property "the member $Z (s) \cap X^{\circ}$ is smooth over $k$ " holds for general members of $W$ in the sense of Linear systems, base loci, and general members, with generalizing open set $U$ ; the members are closed subschemes of the open subscheme $X^{\circ}$ , which may be empty, and $U$ contains classical parameters.

In particular, suppose $X \neq \emptyset$ , fix a locally closed immersion $X ↪ P_{k}^{N}$ , let $L = O_{X} (1)$ be the hyperplane line bundle of that immersion, and let $W_{h} \subseteq Γ (X, O_{X} (1))$ be the hyperplane system, the span of the restrictions of the degree-one forms (Linear systems, base loci, and general members). Then $Bs (W_{h}) = \emptyset$ , and there is a nonempty Zariski-open subset $U \subseteq P (W_{h})$ such that for every closed point $[s] \in U$ the scheme-theoretic hyperplane section $X \times_{P_{k}^{N}} V_{+} (F)$ of $X$ — for any degree-one form $F$ with $F ∣_{X} = s$ , equivalently the zero scheme $Z (s)$ (A section of an invertible sheaf has a canonical zero subscheme) — is smooth over $k$ .

No irreducibility or connectedness of $X$ or of the members is asserted, and no statement is made about the dimension or the nonemptiness of the members.

Facts & Assumptions

Given: The Axiom of Choice; an algebraically closed field $k$ of characteristic $0$ ; a smooth finite-type $k$ -scheme $X$ admitting a locally closed immersion into a projective space; an invertible $O_{X}$ -module $L$ ; a nonzero finite-dimensional linear system $W \subseteq Γ (X, L)$ with $\dim_{k} W = r + 1$ ; the associated incidence $I$ with morphisms $π, p$ ; the open subscheme $X^{\circ} = X ∖ Bs (W)$ ; and, for the final clause, a fixed locally closed immersion $X ↪ P_{k}^{N}$ with hyperplane system $W_{h}$ .

[F1]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function.

[F2]

Linear systems, base loci, and general members: for a $k$ -scheme $X$ , an invertible $O_{X}$ -module $L$ and a nonzero finite-dimensional $k$ -subspace $W \subseteq Γ (X, L)$ , the parameter space is $P (W) = (W ∖ {0}) / k^{\times}$ with the projective Zariski topology, independent of a basis; $Z (s)$ depends only on $[s]$ ; the base locus $Bs (W) = ⋂_{0 \neq s \in W} ∣ Z (s) ∣$ is closed; a property holds for a general member if there is a nonempty Zariski-open $U \subseteq P (W)$ such that every parameter in $U$ has it; and for a fixed embedding $X \subseteq P_{k}^{N}$ the hyperplane system is the span of the restrictions of the degree-one forms, viewed as sections of the hyperplane line bundle with forms giving the same section identified.

[F3]

The universal member away from the base locus: under AC, for an algebraically closed field $k$ , a smooth finite-type $k$ -scheme $X$ , an invertible $O_{X}$ -module $L$ , and a nonzero finite-dimensional linear system $W \subseteq Γ (X, L)$ with $\dim_{k} W = r + 1$ , base locus $Bs (W)$ and $X^{\circ} = X ∖ Bs (W)$ , there is a finite-type $k$ -scheme $I$ with $k$ -morphisms $π ⁣ : I \to X^{\circ}$ and $p ⁣ : I \to P (W) = P_{k}^{r}$ , determined by $L$ and $W$ up to canonical isomorphism, such that: (1) over $X_{j}^{\circ} = X^{\circ} ∖ ∣ Z (s_{j}) ∣$ , for a $k$ -basis $s_{0}, \dots, s_{r}$ of $W$ and $r \geq 1$ , the map $π$ exhibits $π^{- 1} (X_{j}^{\circ})$ as isomorphic over $X_{j}^{\circ}$ to $X_{j}^{\circ} \times_{k} P_{k}^{r - 1}$ ; (2) for every $[s] \in P (W) (k)$ the fibre $p^{- 1} ([s])$ is isomorphic over $X^{\circ}$ to the zero subscheme $Z (s) \cap X^{\circ}$ ; (3) $I \to Spec k$ is smooth in the local-standard-smooth sense; (4) if $r = 0$ then $I = \emptyset$ . Moreover the construction in its proof glues the local models $I_{V, j} \subseteq V \times_{k} U_{j}$ to a closed subscheme $I_{X} ↪ X \times_{k} P_{k}^{r}$ and defines $I = I_{X} \times_{X} X^{\circ}$ (its step 2.1), so $I$ is a locally closed subscheme of $X^{\circ} \times_{k} P_{k}^{r}$ .

[F4]

Generic smoothness over a dense target open: under AC, for $k$ algebraically closed of characteristic $0$ , irreducible classical varieties $X, Y$ over $k$ and a morphism $f ⁣ : X \to Y$ of classical varieties with $X$ smooth over $k$ : (1) there is a dense open $U \subseteq Y$ such that $f^{- 1} (U) \to U$ is a smooth morphism of finite-type $k$ -schemes, with $f^{- 1} (U) = \emptyset$ allowed when $f$ is not dominant; (2) if $f$ is dominant there is a nonempty open $V \subseteq U$ such that for every closed point $y \in V$ the scheme-theoretic fibre $X_{y} = X \times_{Y} Spec k (y)$ is nonempty, smooth over $k$ , and of pure dimension $r = \dim X - \dim Y$ .

[F5]

A regular point lies on one irreducible component: under AC, a regular point of a reduced Noetherian scheme lies on exactly one irreducible component.

[F6]

regular local rings are domains and cohen macaulay: under AC, a regular local ring is a domain (and Cohen-Macaulay).

[F7]

Regular equals smooth over a perfect field: under AC, for a perfect field $k$ and a finite-type $k$ -scheme $X$ , $X$ is regular (every local ring is regular local) if and only if $X \to Spec k$ is smooth in the local-standard-smooth sense.

[F8]

Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: every field of characteristic zero is perfect, and every algebraically closed field is perfect.

[F9]

The reduction of a scheme: for a scheme $X$ the nilradical ideal sheaf $N_{X}$ has nilpotent germs, and the reduction $X_{r e d}$ is the closed subscheme with structure sheaf $O_{X} / N_{X}$ ; on $Spec A$ it is $Spec (A / \sqrt{(0)})$ . Thus $X$ is reduced exactly when $N_{X} = 0$ , equivalently when every local ring of $X$ is reduced.

[F10]

Fields and $Z$ are Noetherian, and so are their polynomial rings in finitely many variables and Every algebra of finite type over a Noetherian ring is a Noetherian ring: every field is a Noetherian ring, and a commutative algebra of finite type over a Noetherian ring is a Noetherian ring.

[F11]

Locally Noetherian and Noetherian schemes: a scheme is locally Noetherian if it has an affine open cover by spectra of Noetherian rings, and Noetherian if it is locally Noetherian and quasi-compact; equivalently, it has a finite affine open cover by spectra of Noetherian rings.

[F12]

Locally finite type and finite type morphisms: a morphism is locally of finite type if locally on source and target it is given by a finitely generated algebra map, and of finite type if it is locally of finite type and quasi-compact.

[F13]

A Noetherian space is a finite union of irreducible closed subsets: under AC, a Noetherian topological space is a finite union of irreducible closed subsets and has only finitely many irreducible components.

[F14]

Existence and basic properties of irreducible components: irreducible components are closed, and every irreducible subset is contained in an irreducible component; in particular every point lies on some component.

[F15]

Integral schemes: an integral scheme is a nonempty scheme that is reduced and whose underlying topological space is irreducible.

[F16]

Affine-overlap separation condition: an $S$ -scheme $X$ satisfies the affine-overlap separation condition if for every pair of affine opens $U, V \subseteq X$ over a common affine open of $S$ the intersection $U \cap V$ is affine and $Γ (U, O_{X}) \otimes_{R} Γ (V, O_{X}) \to Γ (U \cap V, O_{X})$ is surjective.

[F17]

Affine-overlap criterion for separatedness: a morphism $f ⁣ : X \to S$ is separated if and only if it satisfies the affine-overlap separation condition of [F16].

[F18]

The relative projective-space diagonal is closed: for every scheme $S$ and $n \geq 0$ the diagonal of $P_{S}^{n} / S$ is a closed immersion; hence $P_{S}^{n} \to S$ is separated.

[F19]

Open and closed immersions are separated: every open immersion, every closed immersion and every immersion (locally closed immersion) of schemes is separated as a morphism.

[F20]

Separated morphisms compose: a composite of separated morphisms is separated.

[F21]

Separatedness survives base change: a base change of a separated morphism is separated.

[F22]

Immersion of schemes: a morphism is an immersion (locally closed immersion) if it factors as an open immersion followed by a closed immersion.

[F23]

Classical algebraic prevarieties, regular maps, and varieties: a classical algebraic prevariety over $k$ is a quasi-compact locally ringed space with a structure sheaf of $k$ -algebras covered by open subspaces isomorphic to affine models; it is separated when the equalizer of every pair of regular maps into it is closed, and a classical algebraic variety is a separated prevariety; varieties may be reducible or empty, and an irreducible classical variety is nonempty and irreducible.

[F24]

Irreducible classical varieties and integral separated finite-type schemes: under AC, the closed-point construction and its inverse give an equivalence between irreducible classical $k$ -varieties and integral finite-type $k$ -schemes satisfying the affine-overlap separation condition; classical points correspond to closed points and classical regular maps to scheme $k$ -morphisms.

[F25]

projective algebraic set and projective space points: for homogeneous $T \subseteq k [x_{0}, \dots, x_{n}]$ , $V_{+} (T) = {[a] \in P_{k}^{n} : F (a) = 0 for all F \in T}$ is a projective algebraic set, with $V_{+} (\emptyset)$ conventionally equal to $P_{k}^{n}$ ; and $P_{k}^{n} = (k^{n + 1} ∖ {0}) / \sim$ with $a \sim b$ exactly when $b = λ a$ for some $λ \in k^{\times}$ , so $P_{k}^{n} \neq \emptyset$ for every $n \geq 0$ .

[F26]

homogeneous polynomial and homogeneous ideal: a polynomial is homogeneous of degree $d$ if every occurring monomial has total degree $d$ , and an ideal is homogeneous if it contains all homogeneous components of its elements.

[F27]

projective irreducibility homogeneous prime: over algebraically closed $k$ , a nonempty projective algebraic set $X$ is irreducible if and only if its homogeneous ideal $I_{+} (X)$ of forms vanishing on $X$ is prime.

[F28]

A polynomial ring in finitely many indeterminates over an integral domain is an integral domain: a polynomial ring in finitely many variables over a domain is a domain; in particular $k [x_{0}, \dots, x_{r}]$ is a domain for the field $k$ .

[F29]

A polynomial vanishing at every tuple from an infinite subdomain is the zero polynomial: if $R \subseteq S$ is a subring whose underlying set is infinite inside an integral domain $S$ and $f \in S [x_{1}, \dots, x_{m}]$ , $m \geq 1$ , vanishes at all $R$ -points, then $f = 0$ .

[F30]

projective variety classical: a classical projective variety over $k$ is a nonempty irreducible projective algebraic set, understood with its standard affine charts.

[F31]

Irreducibility via nonempty open subsets, connectedness and open subspaces: a space is irreducible if and only if it is nonempty and every two nonempty open subsets meet; equivalently, if and only if it is nonempty and every nonempty open subset is dense; a nonempty open subspace of an irreducible space is irreducible.

[F32]

Smooth morphisms via local standard smooth presentations: a morphism of finite-type $k$ -schemes is smooth if every source point has affine neighbourhoods on which the induced ring map is standard smooth at that prime; the condition is local on the source and on the target, and it is imposed at every source point.

[F33]

Restricting fibre products to open subschemes: fibre products commute with restriction to open subschemes; for an open immersion $Z ↪ X$ the base change $Z \times_{X} Y \to Y$ is an open immersion with image the open subscheme $Y \cap Z$ (scheme intersection along $X$ ).

[F34]

Scheme-theoretic fibre: for a morphism $f ⁣ : X \to S$ and a point $s \in S$ with residue field $k (s)$ , the scheme-theoretic fibre is $X_{s} = X \times_{S} Spec k (s)$ .

[F35]

Intersections of subschemes: the scheme-theoretic intersection of closed subschemes of a scheme is their fibre product over that scheme.

[F36]

A section of an invertible sheaf has a canonical zero subscheme: for a section $s$ of an invertible sheaf on $X$ and a trivializing affine cover $U_{i} = Spec A_{i}$ with $s ∣_{U_{i}} = f_{i} e_{i}$ , the affine schemes $Spec (A_{i} / (f_{i}))$ glue to a closed subscheme $Z (s) ↪ X$ , canonical up to unique isomorphism over $X$ , using the ideal $(f_{i})$ itself with no reducedness or nonzerodivisor hypothesis; on a trivializing chart $Z (s)$ is cut out by the local equation $f_{i}$ .

[F37]

Dominant classical morphisms and rational maps: a morphism of classical varieties is dominant when its image is dense.

[F38]

In a finite-type algebra over a field, closed points are dense in every closed subset of the spectrum, standard projective opens are affine spaces and Classical affine points are maximal ideals: for a finite-type $k$ -algebra $A$ , every nonempty open subset of a closed $Z \subseteq Spec A$ contains a closed point of $Spec A$ ; the standard opens $D_{+} (x_{i}) \subseteq P_{k}^{r}$ are affine spaces $A_{k}^{r}$ ; and for an affine algebraic set over algebraically closed $k$ the classical points correspond bijectively to maximal ideals, with residue field $k$ .

[F39]

The spectrum of a Noetherian ring is a Noetherian topological space: under AC, for a Noetherian commutative ring $R$ the space $Spec R$ is a Noetherian topological space.

Proof

technique · direct

1.1F2F3given

Setup, indexing, and the parameter space. Put $r = \dim_{k} W - 1 \geq 0$ , so that $P (W) = P_{k}^{r}$ by [F2] and [F3], and let $I$ , $π ⁣ : I \to X^{\circ}$ , $p ⁣ : I \to P (W)$ be the incidence of [F3]; by [F3] clause (3) the morphism $I \to Spec k$ is smooth, so $I$ is a finite-type $k$ -scheme, and by the construction recorded in [F3] the scheme $I$ is a locally closed subscheme of $X^{\circ} \times_{k} P_{k}^{r}$ .

1.2F9F12F23F25F26F27F28F29F30F38

The k-rational parameter space is an irreducible classical projective variety. By [F25] the space $P_{k}^{r}$ is nonempty, and $P_{k}^{r} = V_{+} ((0))$ is a projective algebraic set. Its homogeneous vanishing ideal is $I_{+} (P_{k}^{r}) = (0)$ : if $0 \neq F \in k [x_{0}, \dots, x_{r}]$ is homogeneous of positive degree and vanished at every point of $P_{k}^{r}$ , then the polynomial $F \in k [x_{0}, \dots, x_{r}]$ would vanish at every point of $k^{r + 1}$ (a nonzero point $a$ gives $[a]$ , and $F (0) = 0$ in positive degree), so $F = 0$ by [F29] applied with $R = S = k$ (the algebraically closed field $k$ is infinite) and $m = r + 1 \geq 1$ , a contradiction. Since $(0)$ is prime by [F28] and $k [x_{0}, \dots, x_{r}]$ is the homogeneous coordinate ring of [F26], [F27] shows that $P_{k}^{r}$ is irreducible; by [F30] it is a classical projective variety. Moreover $P_{k}^{r}$ is an integral finite-type $k$ -scheme: its standard affine charts are spectra of polynomial rings over $k$ by [F38], which are domains by [F28], so the nilradical ideal sheaf of [F9] vanishes on a chart cover and $P_{k}^{r}$ is reduced, and it is finite type over $k$ because the charts of [F38] give a finite affine cover by finitely generated $k$ -algebras [F12].

1.3F3F6F7F8F9F10F11F12F39

The incidence is regular, reduced and Noetherian. By [F3] clause (3) and [F8], the finite-type $k$ -scheme $I$ is smooth over the perfect field $k$ , so [F7] makes $I$ regular: every local ring $O_{I, x}$ is a regular local ring. Each such ring is a domain by [F6], hence reduced; therefore the nilradical ideal sheaf $N_{I}$ of [F9] has zero stalks, $I = I_{r e d}$ , and $I$ is a reduced scheme. By [F12] the finite-type morphism $I \to Spec k$ is quasi-compact and locally of finite type, so $I$ has a finite affine open cover by spectra $Spec A_{j}$ of finitely generated $k$ -algebras $A_{j}$ ; each $A_{j}$ is Noetherian by [F10] since the field $k$ is Noetherian, so $I$ is a Noetherian scheme by [F11]. The underlying space $∣ I ∣$ is a Noetherian topological space: each $Spec A_{j}$ is Noetherian by [F39], and a descending chain of closed subsets of $I$ restricts to descending chains in the finitely many charts, each of which stabilizes, whence the chain itself stabilizes.

1.4F3F18F19F20F21F22given

Separatedness of the components. Since $X$ admits a locally closed immersion into a projective space [F22], $X \to Spec k$ is separated: the immersion is separated by [F19], the projective space is separated over $Spec k$ by [F18], and separated morphisms compose by [F20]. The open subscheme $X^{\circ} \subseteq X$ is separated over $Spec k$ by [F19] and [F20], $P_{k}^{r} \to Spec k$ is separated by [F18], so $X^{\circ} \times_{k} P_{k}^{r} \to Spec k$ is separated by [F21] and [F20]; the locally closed subscheme $I$ of [F3] is therefore separated over $Spec k$ by [F19] and [F20].

1.5F2F25F38given

The hyperplane system and its base locus. Suppose now that $X \neq \emptyset$ , fix the locally closed immersion $X ↪ P_{k}^{N}$ , let $L = O_{X} (1)$ and let $W_{h} \subseteq Γ (X, O_{X} (1))$ be the hyperplane system of [F2]. Then $W_{h} \neq 0$ : if the restriction of every degree-one form vanished on $X$ , then $x_{0}, \dots, x_{N}$ would all vanish on $X$ , whence $X \subseteq V_{+} (x_{0}, \dots, x_{N}) = \emptyset$ by [F25], contradicting $X \neq \emptyset$ . Also $Bs (W_{h}) = \emptyset$ : for every point $x \in X$ some standard chart $D_{+} (x_{j})$ of $P_{k}^{N}$ contains $x$ by [F38], and on that chart the restricted linear form $x_{j} ∣_{X}$ is a unit at $x$ , so its zero subscheme does not contain $x$ and $x \notin Bs (W_{h})$ by [F2].

2.1F5F13F14F15F19F20F32step 1.3step 1.4

The finite component decomposition. By [F13] and step 1.3 the scheme $I$ has only finitely many irreducible components $Z_{1}, \dots, Z_{m}$ ; each $Z_{i}$ is closed by [F14], and every point of $I$ lies on at least one $Z_{i}$ by [F14]. By [F5] and step 1.3 every point of $I$ lies on exactly one irreducible component, so the $Z_{i}$ are pairwise disjoint; since they are finitely many closed pairwise disjoint subsets, the complement of $Z_{i}$ is the union of the remaining closed $Z_{j}$ , hence $Z_{i}$ is also open in $I$ . Give $Z_{i}$ the open subscheme structure. Then each $Z_{i}$ is irreducible and, as an open subscheme of the reduced scheme $I$ , reduced, hence integral by [F15]; it is finite type over $k$ as an open subscheme of the finite-type $k$ -scheme $I$ , smooth over $k$ because smoothness is local on the source [F32], and separated over $k$ because it is an open subscheme of the separated scheme $I$ of step 1.4, using [F19] and [F20].

3.1F3F16F17F24step 1.2step 2.1

The components and the parameter space as classical varieties. Each $Z_{i}$ of step 2.1 is an integral finite-type $k$ -scheme, and by [F17] and [F16] the separatedness of $Z_{i} \to Spec k$ from step 2.1 is exactly the affine-overlap separation condition; hence by [F24] and [F23] $Z_{i}$ corresponds to an irreducible classical variety over $k$ , with classical points the closed points and with scheme $k$ -morphisms corresponding to regular maps. Similarly $P (W) = P_{k}^{r}$ is an integral finite-type $k$ -scheme by step 1.2 and separated over $k$ by [F18], so by [F24] and [F23] it is an irreducible classical variety whose classical points are its closed points, and the restriction $p_{i} = p ∣_{Z_{i}} ⁣ : Z_{i} \to P (W)$ of [F3] is a $k$ -morphism of schemes, hence a morphism of classical varieties under [F24].

4.1F4F24F34F37step 3.1

Target generic smoothness on the dominant components. Let $i \in {1, \dots, m}$ be such that $p_{i}$ is dominant in the sense of [F37]. By step 3.1 the source $Z_{i}$ and the target $P (W)$ are irreducible classical varieties, $p_{i}$ is a morphism of classical varieties, and $Z_{i}$ is smooth over $k$ ; so [F4] clause (2) applies and produces a nonempty open subvariety $V_{i} \subseteq P (W)$ such that for every closed point $y \in V_{i}$ , equivalently every classical point of $V_{i}$ by [F24], the scheme-theoretic fibre $p_{i}^{- 1} (y) = Z_{i} \times_{P (W)} Spec k (y)$ of [F34] is nonempty, smooth over $k$ , and of pure dimension $\dim Z_{i} - r$ .

4.2F34F37step 3.1

The non-dominant components. For the component morphism $p_{i}$ of step 3.1, if it is not dominant, then by [F37] the image $p_{i} (Z_{i})$ is not dense in $P (W)$ , so its closure is a proper closed subset and $W_{i} : = P (W) ∖ \overline{p_{i} (Z_{i})}$ is a nonempty open subset of $P (W)$ ; by definition of the image, every point $y \in W_{i}$ has empty fibre $p_{i}^{- 1} (y) = \emptyset$ .

5.1F31step 1.2step 2.1step 4.1step 4.2

The common parameter open set. There are finitely many components, so the family of nonempty open sets consisting of the $V_{i}$ of step 4.1 for the dominant components and the $W_{i}$ of step 4.2 for the non-dominant components is finite; let $U$ be their intersection, an open subset of $P (W)$ . By steps 1.2 and [F31], $P (W)$ is irreducible, so any two of these nonempty open sets meet and, by induction on the finite list, $U \neq \emptyset$ ; if $I = \emptyset$ , so that there are no components, take $U = P (W)$ . In either case $U$ is a nonempty open subset of $P (W)$ . Distinct $Z_{i}$ are disjoint by step 2.1, so for every point $[s] \in U$ exactly one alternative of steps 4.1 and 4.2 applies to each component.

6.1F24F32F33F34step 2.1step 4.1step 4.2step 5.1

Smoothness of the incidence fibres over $U$ . Fix a closed point $[s] \in U$ of $P (W)$ and write $F = p^{- 1} ([s]) = I \times_{P (W)} Spec k ([s])$ for the scheme-theoretic fibre of [F34]; here $k ([s]) = k$ by [F24], since classical points correspond to closed points and all classical points have residue field $k$ . Because the pairwise disjoint open subschemes $Z_{i}$ cover $I$ by step 2.1, the open subschemes $F \times_{I} Z_{i}$ cover $F$ , and by [F33] each $F \times_{I} Z_{i}$ is canonically identified with the fibre $p_{i}^{- 1} ([s]) = Z_{i} \times_{P (W)} Spec k$ of $p_{i}$ . For a dominant $i$ , with $[s] \in V_{i}$ , this fibre is nonempty and smooth over $k$ by step 4.1; for a non-dominant $i$ , with $[s] \in W_{i}$ , it is empty by step 4.2, and the empty scheme is smooth over $k$ . Smoothness is local on the source by [F32], so $F$ is smooth over $k$ .

7.1F2F3F36step 6.1

Identification with the general member. By [F3] clause (2), the fibre $F = p^{- 1} ([s])$ of step 6.1 is isomorphic over $X^{\circ}$ to the zero subscheme $Z (s) \cap X^{\circ}$ of the section $s$ restricted to $X^{\circ}$ ([F36] and [F2]); hence $Z (s) \cap X^{\circ}$ is smooth over $k$ . This holds for every closed point $[s] \in U$ and every representative $0 \neq s \in W$ , since $Z (s)$ depends only on $[s]$ by [F2]. That is the first assertion of the statement.

8.1F24F38step 5.1

Non-vacuity of the parameter set in the classical reading. The set $U$ of step 5.1 is a nonempty open subset of the projective space $P (W)$ over the algebraically closed field $k$ , and the standard charts $D_{+} (x_{j})$ of [F38] cover it, so $U \cap D_{+} (x_{j})$ is a nonempty open subset of an affine space $A_{k}^{r}$ for some $j$ ; by [F38] it contains a closed point of that affine spectrum, which by [F38] is a classical point of $P_{k}^{r}$ , hence by [F24] a closed point of the scheme $P (W)$ . Thus $U$ contains closed points and the general-member statement of step 7.1 is not vacuous; in the classical dictionary of [F24] these are exactly the parameters $[s] \in U$ .

8.2F2F35F36step 1.5step 7.1

Hyperplane sections of the fixed embedding. Here $X^{\circ} = X ∖ Bs (W_{h}) = X$ , so the first assertion, which is established by the argument of steps 1.1-7.1 applied with $W = W_{h}$ and $X^{\circ} = X$ , gives a nonempty open $U \subseteq P (W_{h})$ such that for every closed point $[s] \in U$ the zero scheme $Z (s) \subseteq X$ is smooth over $k$ . Let $F$ be any degree-one form with $F ∣_{X} = s \neq 0$ ; on a standard affine chart $V = Spec A \subseteq X$ trivializing $O_{X} (1)$ , the zero scheme $Z (s)$ is cut out by the local equation $f$ of $s$ ([F36]) and the scheme-theoretic intersection $X \times_{P_{k}^{N}} V_{+} (F)$ is cut out by the same equation, because $V_{+} (F)$ is defined by the dehomogenized form $F$ and $f$ is its restriction to the chart; so the two closed subschemes of $X$ agree by [F35] and [F36]. Hence the scheme-theoretic hyperplane sections of $X$ for parameters in $U$ are smooth over $k$ , which is the final assertion.

9.1F1F3F4F5F7F10F13F14F17F18F19F20F21F24F27F38F39step 2.1step 6.1step 7.1step 1.5step 8.2∎

Boundary, choice, and scope dispositions. Empty: if $X = \emptyset$ then $Γ (X, L) = 0$ and no nonzero $W$ exists, so the theorem is vacuous; if $X \neq \emptyset$ but $X^{\circ} = \emptyset$ , then by [F3] clause (2) every fibre $p^{- 1} ([s]) = Z (s) \cap X^{\circ}$ is empty and smooth, and one may take $U = P (W)$ in step 5.1, so the statement holds; [F3] clause (4) and the same fibre identification give the parallel empty-member conclusion when $r = 0$ . Zero: the parameter space is $P_{k}^{0}$ when $\dim_{k} W = 1$ , and its single member is empty on $X^{\circ}$ by the previous sentence; conversely, the incidence $I$ itself can be empty exactly when $X^{\circ} = \emptyset$ or $r = 0$ , and in both cases the argument of step 5.1 uses the empty family of components. One: the case $\dim_{k} W = 2$ , $r = 1$ , is included; nothing in the proof requires $r \geq 2$ . Degenerate: $X$ is assumed neither irreducible nor connected nor of pure dimension, and the argument decomposes $I$ rather than $X$ ; the members $Z (s) \cap X^{\circ}$ may be reducible, empty, or non-reduced as ambient data, and no smoothness of $X^{\circ}$ outside $X$ is used. Endpoints: the proof covers $N = 0$ in the final clause (where $X = P_{k}^{0}$ and $W_{h}$ is one-dimensional, $Bs (W_{h}) = \emptyset$ ) and imposes no upper bound on $N$ or on $r$ ; the claimed open set may be all of the base-locus-free parameter space, and no density or dimension of the good locus beyond nonemptiness openness is asserted. Nonempty-choice: AC is declared in [F1] and is used exactly through the AC-assuming suppliers [F3] (incidence), [F4] (generic smoothness), [F5] (one-component lemma), [F7] (regularity versus smoothness), [F10] (Noetherianity routes), [F13]-[F14] (finitely many components), [F24] (the classical-scheme dictionary), [F38]-[F39] (closed points and Noetherian spectra), and [F18]-[F21] (separatedness); the finite choices of charts, bases and component indices and the fibre computations of steps 2.1, 6.1, 7.1, 1.5 and 8.2 are finite and add no choice principle. Both iff cases: the only biconditional invoked as a supplier is [F7] (regular if and only if smooth over the perfect field $k$ ), used in step 1.3 in the direction "smooth over $k$ implies regular"; the criterion [F17] is used in the direction "separated implies the affine-overlap condition" in step 3.1; and [F27] is used in the direction "the vanishing ideal is prime implies irreducibility" in step 1.2. No irreducibility, connectedness, dimension, or nonemptiness of the members is asserted, in accordance with the statement. This completes the proof.

Source qualification

Vakil, Classes 51-52, §3.9 Corollary (with §3.10-3.11) states Bertini for a finite-dimensional base-point-free linear system on a smooth $k$ -variety over an algebraically closed field of characteristic $0$ : almost every element, as a closed subscheme, is nonsingular over $k$ . Arapura, §5.4, Theorem 5.4.5, proves the hyperplane version on a nonempty open subset of the dual projective space by the incidence correspondence and notes that the statement is valid in every characteristic although the proof given works only in characteristic $0$ . The present item generalizes the base-point-free hypothesis by removing the base locus from the ambient scheme: the conclusion is smoothness of the whole zero scheme inside $X^{\circ} = X ∖ Bs (W)$ , for a nonempty open set of parameters, and the hyperplane case for a fixed immersion is recovered because the hyperplane system of an embedding has empty base locus. The proof is not copied from either source: it decomposes the incidence of The universal member away from the base locus into its finitely many irreducible components and applies the in-run target-side generic smoothness theorem Generic smoothness over a dense target open componentwise, which also delivers the statement that no dense part of a general member (rather than the whole base-locus-free part) is singular. Neither source asserts anything about the size of the good locus beyond open nonemptiness, about irreducibility or connectedness of the members, or about their dimension or nonemptiness, and neither claim is made here. The characteristic- $0$ hypothesis is used only through perfectness of $k$ and generic smoothness; the failure of the arbitrary base-point-free form of Bertini in positive characteristic is recorded on the examples page of this pair.

CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-30Open item page →

General hypersurfaces give smooth complete intersections

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be an algebraically closed field of characteristic $0$ and let $X \subseteq P_{k}^{N}$ be a nonempty smooth projective classical variety over $k$ of pure dimension $d$ (projective variety classical, Global and local dimension of classical varieties). Fix an integer $r \geq 0$ and positive degrees $e_{1}, \dots, e_{r}$ . For each $i$ let $S_{e_{i}} = k [x_{0}, \dots, x_{N}]_{e_{i}}$ be the space of degree- $e_{i}$ forms (homogeneous polynomial and homogeneous ideal) and let $P (S_{e_{i}})$ be its projective space of lines, the parameter space of degree- $e_{i}$ hypersurfaces (Linear systems, base loci, and general members); put $Π_{r} = P (S_{e_{1}}) \times_{k} \dots \times_{k} P (S_{e_{r}}),$ the product of hypersurface parameter spaces, and $Π_{0} = Spec k$ for the empty tuple. For a tuple $(F_{1}, \dots, F_{r})$ of nonzero forms $F_{i} \in S_{e_{i}}$ let $Z (F_{1}, \dots, F_{r}) = X \cap V_{+} (F_{1}) \cap \dots \cap V_{+} (F_{r})$ be the scheme-theoretic intersection inside $P_{k}^{N}$ (Intersections of subschemes); it depends only on the parameter point $([F_{1}], \dots, [F_{r}]) \in Π_{r}$ .

Then:

(nonempty intersections) if $0 \leq r \leq d$ there is a nonempty Zariski-open subset $U \subseteq Π_{r}$ such that for every closed point of $U$ (equivalently, by Irreducible classical varieties and integral separated finite-type schemes, every classical parameter), with representatives $F_{1}, \dots, F_{r}$ , the closed subscheme $Z (F_{1}, \dots, F_{r}) \subseteq X$ is nonempty, smooth over $k$ (Smooth morphisms via local standard smooth presentations) and of pure dimension $d - r$ . For $r = 0$ this says that $X$ itself is nonempty, smooth over $k$ and of pure dimension $d$ , with $Π_{0} = Spec k$ ;
(empty intersections) if $r > d$ there is a nonempty Zariski-open subset $U \subseteq Π_{r}$ such that for every closed point of $U$ , with representatives $F_{1}, \dots, F_{r}$ , one has $Z (F_{1}, \dots, F_{r}) = \emptyset$ .

No claim is made about the tuples outside $U$ , about the size or density of $U$ , about the irreducibility or connectedness of the members, or about singular $X$ .

Facts & Assumptions

Given: The Axiom of Choice; an algebraically closed field $k$ of characteristic $0$ ; a nonempty smooth projective classical variety $X \subseteq P_{k}^{N}$ of pure dimension $d$ ; an integer $r \geq 0$ ; positive degrees $e_{1}, \dots, e_{r}$ ; the spaces $S_{e_{i}}$ of degree- $e_{i}$ forms, the parameter spaces $P (S_{e_{i}})$ , the product $Π_{r}$ , and the scheme-theoretic intersections $Z (F_{1}, \dots, F_{r})$ .

[F1]

The Axiom of Choice: AC asserts that every family of nonempty sets has a choice function.

[F2]

projective variety classical and Global and local dimension of classical varieties: a classical projective variety over $k$ is a nonempty irreducible projective algebraic set with its standard affine charts; for a classical variety $X$ with irreducible components $X_{1}, \dots, X_{m}$ and a closed point $x$ , $\dim_{x} X = \max_{x \in X_{i}} \dim X_{i}$ , and $X$ has pure dimension $d$ if every component has dimension $d$ . An open subvariety of a variety is a variety, and the local dimension at a closed point of an irreducible variety of dimension $d$ equals $d$ .

[F3]

projective algebraic set and projective space points: for homogeneous $T \subseteq k [x_{0}, \dots, x_{n}]$ , $V_{+} (T) = {[a] \in P_{k}^{n} : F (a) = 0 for all F \in T}$ , with $V_{+} (\emptyset) = P_{k}^{n}$ and $V_{+} ((x_{0}, \dots, x_{n})) = \emptyset$ ; and $P_{k}^{n} = (k^{n + 1} ∖ {0}) / \sim$ with $a \sim b$ exactly when $b = λ a$ for some $λ \in k^{\times}$ . By An algebraically closed field: every nonconstant polynomial has a root in the field, $k$ is infinite and has no nontrivial finite extensions.

[F4]

homogeneous polynomial and homogeneous ideal: a polynomial is homogeneous of degree $e$ when every occurring monomial has total degree $e$ ; the degree- $e$ part of the polynomial ring is denoted $k [x_{0}, \dots, x_{N}]_{e}$ , and it is a $k$ -vector space of finite dimension. If $F$ is homogeneous of degree $e$ and $λ \in k^{\times}$ , then the ideal $(λ F)$ equals $(F)$ .

[F5]

standard projective opens are affine spaces: for every $i$ , normalization of the $i$ -th coordinate identifies $D_{+} (x_{i}) \subseteq P_{k}^{n}$ with $A_{k}^{n} = k^{n}$ , and transporting polynomial functions gives compatible regular-function structures; the opens $D_{+} (x_{i})$ cover $P_{k}^{n}$ .

[F6]

Classical algebraic prevarieties, regular maps, and varieties: a classical algebraic prevariety over $k$ is a quasi-compact locally ringed space covered by open subspaces isomorphic to polynomial zero sets, its regular maps are the morphisms of locally ringed spaces, and a classical algebraic variety is a prevariety whose "equalizer of regular maps" separation condition holds; varieties may be reducible or empty, and closed subvarieties and nonempty open subvarieties of varieties are varieties.

[F7]

Irreducible classical varieties and integral separated finite-type schemes: the closed-point construction and its inverse give an equivalence between irreducible classical $k$ -varieties and integral finite-type $k$ -schemes satisfying the affine-overlap separation condition; classical points correspond to closed points and classical regular maps to scheme $k$ -morphisms.

[F8]

In a finite-type algebra over a field, closed points are dense in every closed subset of the spectrum, The closed points of the prime spectrum are exactly the maximal ideals, A maximal ideal of an affine algebra has finite residue field over the base field and Classical affine points are maximal ideals: for a finite-type $k$ -algebra $A$ and a closed $Z \subseteq Spec A$ , every nonempty open subset of $Z$ contains a closed point of $Spec A$ ; a prime $p$ of a commutative ring is closed in $Spec R$ if and only if it is maximal; a maximal ideal of a finite-type $k$ -algebra has finite residue field, equal to $k$ when $k$ is algebraically closed; and for a classical affine algebraic set the classical points are the maximal ideals.

[F9]

[F10]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation $B ≅ (A [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ has a $c \times c$ minor of the Jacobian matrix invertible in $B$ and relative dimension $n - c$ ; a polynomial algebra $A [x_{1}, \dots, x_{n}]_{g}$ (the case $c = 0$ ) is standard smooth of relative dimension $n$ .

[F11]

Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: every field of characteristic zero is perfect, and every algebraically closed field is perfect.

[F12]

Regular equals smooth over a perfect field: for a perfect field $k$ and a finite-type $k$ -scheme $X$ , $X$ is regular (every local ring $O_{X, x}$ is regular local) if and only if $X \to Spec k$ is smooth under the convention of [F9].

[F13]

Openness of the regular locus over a perfect field: for a perfect field $k$ and a finite-type $k$ -scheme $X$ , the regular locus $X_{r e g} = {x : O_{X, x} is a regular local ring}$ is open in $X$ .

[F14]

regular local rings are domains and cohen macaulay: a regular local ring is a domain (and Cohen-Macaulay).

[F15]

Regular points of locally Noetherian schemes: for a point $x$ of a locally Noetherian scheme, $x$ is regular exactly when $\dim_{κ (x)} T_{x} X = \dim O_{X, x}$ , where $T_{x} X = {Hom}_{κ (x)} (m_{x} / m_{x}^{2}, κ (x))$ is the intrinsic Zariski tangent space of The intrinsic Zariski tangent space.

[F16]

Local dimension for a reducible classical algebraic set: under AC, for a reduced classical finite-type space $Z$ over algebraically closed $k$ and a closed point $x$ , $\dim O_{Z, x}$ equals the maximum of $\dim Z_{i}$ over the irreducible components $Z_{i}$ of $Z$ containing $x$ .

[F17]

A Noetherian space is a finite union of irreducible closed subsets and Existence and basic properties of irreducible components: a Noetherian topological space has only finitely many irreducible components; every irreducible component is closed; every irreducible subset is contained in an irreducible component; and every point lies on an irreducible component.

[F18]

Irreducibility via nonempty open subsets, connectedness and open subspaces: a space is irreducible if and only if it is nonempty and every two nonempty open subsets meet; a nonempty open subspace of an irreducible space is irreducible; and an irreducible subset contained in a finite union of closed subsets is contained in one of them (if $C \subseteq F_{1} \cup \dots \cup F_{t}$ with each $F_{j}$ closed and $C$ irreducible, then $C \subseteq F_{j}$ for some $j$ ).

[F19]

Nonempty opens preserve irreducible dimension: a nonempty open subset of an irreducible classical variety has the same dimension as the variety, and a proper closed subvariety has strictly smaller dimension.

[F20]

Affine and projective n-space have dimension n: for every $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ .

[F21]

Intersections of subschemes: the scheme-theoretic intersection of finitely many closed subschemes of a scheme is their iterated fibre product and is cut out by the sum of their ideal sheaves; the empty intersection is the whole scheme.

[F22]

Restricting fibre products to open subschemes: fibre products commute with restriction to open subschemes, so the restriction of a scheme-theoretic intersection to an open subscheme is computed there.

[F23]

Linear systems, base loci, and general members: for a $k$ -scheme $X$ , an invertible $O_{X}$ -module $L$ and a nonzero finite-dimensional linear system $W \subseteq Γ (X, L)$ , the parameter space is $P (W) = (W ∖ {0}) / k^{\times}$ with the projective Zariski topology, independent of a basis; the base locus $Bs (W)$ is closed; and for a fixed projective embedding the hyperplane system is the span of the restrictions of the degree-one forms.

[F24]

projective irreducibility homogeneous prime: over algebraically closed $k$ , a nonempty projective algebraic set is irreducible if and only if its homogeneous vanishing ideal is prime. In particular $P_{k}^{n}$ is irreducible, since the vanishing ideal of $P_{k}^{n}$ is $(0)$ .

[F25]

A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces: for a finite-dimensional vector space $V$ over an infinite field $F$ , no finite family of proper linear subspaces of $V$ has union $V$ .

[F26]

Nontrivial projective hypersurface sections: under AC, if $Y \subseteq P_{k}^{N}$ is irreducible of dimension $m \geq 1$ and $f$ is homogeneous of positive degree not vanishing identically on $Y$ , then $Y \cap V_{+} (f)$ is nonempty and every irreducible component of it has dimension $m - 1$ .

[F27]

A degree-d homogeneous equation becomes a hyperplane section under Veronese: if $n \geq 1$ and $F$ is a nonzero homogeneous polynomial of degree $d \geq 1$ on $P^{n}$ , then the linear form $L$ whose coefficients are those of $F$ in the ordered Veronese coordinates satisfies $V_{+} (F) = ν_{n, d}^{- 1} (H)$ for $H = V_{+} (L)$ on underlying sets, and the proof of the item records the pointwise identity $L (ν_{n, d} ([x])) = F (x)$ .

[F28]

The degree-d Veronese map and The Veronese map is a well-defined closed immersion: for $d \geq 1$ the map $ν_{n, d} ⁣ : P_{k}^{n} \to P_{k}^{N}$ , $[x] \mapsto [M_{0} (x) : \dots : M_{N} (x)]$ over all degree- $d$ monomials, is a well-defined closed immersion of projective varieties.

[F29]

Bertini smoothness away from the base locus: under AC, for $k$ algebraically closed of characteristic $0$ , a smooth finite-type quasi-projective $k$ -scheme $X$ (locally closed immersion into a projective space), an invertible $O_{X}$ -module $L$ and a nonzero finite-dimensional linear system $W \subseteq Γ (X, L)$ with $\dim_{k} W = r + 1$ , base locus $Bs (W)$ and $X^{\circ} = X ∖ Bs (W)$ , there is a nonempty Zariski-open $U \subseteq P (W)$ such that for every closed point $[s] \in U$ the zero scheme $Z (s) \cap X^{\circ}$ is smooth over $k$ ; in particular, for a fixed locally closed immersion $X ↪ P_{k}^{N}$ with $X \neq \emptyset$ , the hyperplane system $W_{h} \subseteq Γ (X, O_{X} (1))$ has empty base locus and there is a nonempty Zariski-open $U \subseteq P (W_{h})$ such that for every closed point $[s] \in U$ the scheme-theoretic hyperplane section $X \times_{P_{k}^{N}} V_{+} (F)$ , for any degree-one form $F$ with $F ∣_{X} = s$ , is smooth over $k$ .

[F30]

Products of nonempty projective varieties exist as projective varieties: nonempty projective varieties $X \subseteq P_{k}^{m}$ and $Y \subseteq P_{k}^{n}$ have a product, realized as their Segre image, and that product is a projective variety.

[F31]

Projection from projective space over a variety is closed: under AC, for every classical variety $Y$ and $N \geq 0$ the projection $p ⁣ : Y \times P_{k}^{N} \to Y$ is a closed map.

[F32]

Equation rows and coordinate columns in an affine Jacobian and The Jacobian kernel computes the tangent space: for a finite-type affine $k$ -scheme $Spec (k [t_{1}, \dots, t_{n}] / I)$ and a $k$ -rational point $a$ with equation-row Jacobian $J (a)$ of a chosen finite generating list of $I$ , the tangent space $T_{a}$ is canonically $\ker J (a)$ in $k^{n}$ , and the kernel is independent of the chosen generating list of the actual ideal.

[F33]

Jacobian rank detects regularity at closed points: under AC, for $A = P / I$ with $P = k [t_{1}, \dots, t_{n}]$ and a specified finite generating list of the actual ideal $I$ , and for a maximal ideal $m$ with $L = A / m$ : if $k$ is perfect then ${rank}_{L} J (m) = n - \dim A_{m}$ if and only if $A_{m}$ is regular local; at a $k$ -rational point the same equivalence holds for every field $k$ ; and if $I = I (X)$ for a reduced classical affine algebraic set $X$ over algebraically closed $k$ and $m$ corresponds to a closed point $x$ , then $\dim A_{m} = \dim_{x} X$ . The generating list need not be minimal and $I$ need not be radical in the first two assertions.

[F34]

Differentials of a polynomial quotient and the Jacobian cokernel and Tensoring is right exact: for $B = P / I$ with $I = (f_{1}, \dots, f_{c})$ , the module $Ω_{B / A}$ is the cokernel of the transpose of the row-oriented Jacobian matrix of $f_{1}, \dots, f_{c}$ ; tensoring a cokernel presentation with a module preserves the cokernel, so the fibre dimension of $Ω$ at a point equals the source rank minus the rank of the Jacobian matrix over the residue field.

[F35]

Relative differential-rank condition and Fibres of standard smooth algebras are regular of relative dimension: a standard smooth presentation of relative dimension $n$ presents $Ω_{B / A}$ as a free module of rank $n$ ; conversely the differential rank alone is not smoothness; and for a standard smooth $R$ -algebra $S$ with presentation of relative dimension $n - c$ , every irreducible component of the base-changed spectrum $S \otimes_{R} κ (p) \otimes_{κ} K$ has dimension $n - c$ .

[F36]

The submersion criterion between smooth varieties: under AC, for algebraically closed $k$ , smooth classical varieties $X, Y$ over $k$ with their finite-type $k$ -scheme structures, a finite-type morphism $f ⁣ : X \to Y$ and a classical closed point $x \in X$ with $y = f (x)$ : $f$ is smooth at $x$ if and only if $d_{x} f ⁣ : T_{x} X \to T_{y} Y$ is surjective; if these conditions hold, the scheme-theoretic fibre $X_{y} = X \times_{Y} Spec k$ has a regular local ring at $x$ of dimension $\dim_{x} X - \dim_{y} Y$ ; and for every such $f$ , whether or not it is smooth at $x$ , the fibre tangent space is canonically $T_{x} (X_{y}) = \ker (d_{x} f)$ .

[F37]

Differentials, open restriction, and the chain rule: a $k$ -open immersion induces a tangent-space isomorphism at every rational point; no finite-type, reducedness, or smoothness hypothesis is needed.

[F38]

Zero-dimensional varieties are finite sets: a classical variety $X$ has $\dim X \leq 0$ if and only if its underlying set is finite; the empty set is included, and a nonempty irreducible variety of dimension zero is a point.

[F39]

Locally finite type and finite type morphisms, Subalgebra generated by a subset, algebras of finite type, and module-finite algebras and Finite-variable polynomial algebras over fields are Noetherian by finite generators: finite-type $k$ -schemes are quasi-compact locally of finite type, their affine charts have finitely generated coordinate rings, and polynomial algebras in finitely many variables over a field are Noetherian, so ideals in the affine charts admit finite generating lists.

[F40]

A section of an invertible sheaf has a canonical zero subscheme: a section of an invertible sheaf has a canonical closed zero subscheme, cut out on each affine trivializing chart by its local equation and independent of the chosen trivializations. By Closed immersions of schemes, a closed immersion is a homeomorphism onto its closed image with a surjective structure-sheaf map; composing two closed immersions again has both properties, since the direct image of the second surjection is surjective on stalks.

[F41]

A polynomial vanishing at every tuple from an infinite subdomain is the zero polynomial: over an infinite field, a nonzero polynomial in finitely many variables cannot vanish at every field-valued tuple.

Proof

technique · direct

1.1F1F2F3F6F7F9F11F12F16F19

Setup and the dictionary. By [F1] AC is available. The field $k$ is algebraically closed of characteristic $0$ , hence perfect and infinite by [F3] and [F11]. By [F2] the variety $X$ is a nonempty irreducible projective algebraic set of pure dimension $d$ , so it is reduced by the definition in [F6]; under the equivalence [F7] the associated finite-type $k$ -scheme $X$ is integral and separated, with closed points corresponding to classical points and with residue field $k$ at every closed point. The structure morphism $X \to Spec k$ is smooth in the sense of [F9], so by [F12] every local ring of $X$ is regular, and since $X$ is irreducible with component $X$ of dimension $d$ , [F16] gives $\dim O_{X, x} = d$ for every closed point $x \in X$ ; by [F9] the nonempty open subvarieties $X_{j} = X \cap D_{+} (x_{j})$ are smooth over $k$ of pure dimension $d$ , and by [F19] they are irreducible of dimension $d$ .

1.2F2F3F4F23F24F30F41

The parameter spaces. For $e \geq 1$ put $S_{e} = k [x_{0}, \dots, x_{N}]_{e}$ , a finite-dimensional $k$ -vector space by [F4], and let $P (S_{e}) = (S_{e} ∖ {0}) / k^{\times}$ be the space of lines with the projective Zariski topology of [F23]; choosing a basis identifies it with a projective space $P_{k}^{M_{e}}$ , whose dimension is $M_{e}$ . Its homogeneous vanishing ideal is $(0)$ : if a nonzero homogeneous polynomial vanished at every point of $P_{k}^{M_{e}}$ , it would vanish at every nonzero tuple of $k^{M_{e} + 1}$ and also at the zero tuple when its degree is positive, contradicting the polynomial nonvanishing theorem [F41] over the infinite field $k$ ; a nonzero constant cannot vanish anywhere. The zero ideal is prime because the coordinate polynomial ring over $k$ is a domain [F4], so $P (S_{e})$ is a nonempty irreducible projective variety by [F2] and [F24]. Consequently $Π_{r} = P (S_{e_{1}}) \times_{k} \dots \times_{k} P (S_{e_{r}})$ is a nonempty classical projective variety for every $r \geq 1$ , by [F30] applied iteratively, and $Π_{0} = Spec k$ is a one-point classical variety; the classical points of $P (S_{e})$ are exactly the lines $[F]$ with $0 \neq F \in S_{e}$ by [F23], and those of $Π_{r}$ are the tuples of lines.

1.3F3F4F5F21F22F23F39F40

The intersection subschemes and their local equations. For $0 \neq F \in S_{e}$ let $V_{+} (F)$ be the zero subscheme of the section of $O_{P_{k}^{N}} (e)$ represented by $F$ , supplied by [F40]; on the standard chart $D_{+} (x_{j})$ is cut out by the dehomogenized form $f = (F / x_{j}^{e})$ regarded as a polynomial in the coordinates $y_{1}, \dots, y_{N}$ of [F5]; the two dehomogenizations of $F$ on an overlap $D_{+} (x_{j}) \cap D_{+} (x_{k})$ differ by the unit $(x_{k} / x_{j})^{e}$ (with $x_{j} = 1$ on the first chart), so the principal ideals agree and [F40] gives the well-defined closed subscheme with underlying set the classical hypersurface $V_{+} (F)$ of [F3]; clearly $(F) = (λ F)$ for $λ \neq 0$ by [F4], so $V_{+} (F)$ depends only on the line $[F]$ . For a tuple $F_{1}, \dots, F_{r}$ put $Z (F_{1}, \dots, F_{r}) = X \cap V_{+} (F_{1}) \cap \dots \cap V_{+} (F_{r})$ , the scheme-theoretic intersection of closed subschemes of $P_{k}^{N}$ in the sense of [F21], a closed subscheme of $X$ depending only on the parameter point of [F23] in $Π_{r}$ ; for $r = 0$ the intersection is $X$ by the empty-family clause of [F21]. On the chart $D_{+} (x_{j})$ one has $X \cap D_{+} (x_{j}) = X_{j} = Spec A_{j}$ with $A_{j} = k [y_{1}, \dots, y_{N}] / I_{j}$ a finite-type $k$ -algebra [F39], and by [F21] and [F22] the restriction of $Z (F_{1}, \dots, F_{r})$ to $D_{+} (x_{j})$ is the closed subscheme $Spec (A_{j} / (f_{1}, \dots, f_{r}))$ , where $f_{i}$ is the dehomogenization of $F_{i}$ .

1.4F3F5F8F39

Closed points of the intersections. Let $Z \subseteq P_{k}^{N}$ be any closed subscheme of finite type over $k$ , for instance $Z (F_{1}, \dots, F_{r})$ , with the induced reduced projective algebraic set as its underlying space. A point $x \in Z$ is a closed point of $Z$ if and only if $κ (x) = k$ : closedness of the singleton is local on the finite affine chart cover $Z \cap D_{+} (x_{j}) = Spec B_{j}$ with $B_{j}$ finite type over $k$ [F5, F39], and on an affine finite-type $k$ -algebra a prime is maximal if and only if its residue field is finite over $k$ , hence equal to $k$ because $k$ is algebraically closed [F8]; moreover every nonempty open subset of $Z$ contains a closed point of $Z$ , because it meets some chart and [F8] supplies a closed point of that chart's spectrum, which has residue field $k$ and is closed in $Z$ by the first assertion.

2.1F2F9step 1.2step 1.3

The case of no forms. If $r = 0$ then $Π_{0} = Spec k$ by 1.2, the intersection is $Z = X$ by 1.3, and $X$ is nonempty, smooth over $k$ and of pure dimension $d$ by hypothesis and [F2]; so $U = Π_{0}$ exhibits claim 1 in this case.

2.2F8F27F28F29F40step 1.3step 1.4

The Veronese transfer of Bertini. Let $Y \subseteq P_{k}^{N}$ be a nonempty closed subscheme which is smooth over $k$ of pure dimension $m \geq 1$ (for instance an intersection produced below), let $e \geq 1$ , and consider the degree- $e$ Veronese map $ν = ν_{e, N} ⁣ : P_{k}^{N} \to P_{k}^{M}$ of [F28]; the composite of the closed immersions $Y ↪ P_{k}^{N} \overset{ν}{\to} P_{k}^{M}$ is again a closed immersion by [F40]; write $Y^{'}$ for its closed scheme image, which is isomorphic to $Y$ by that composite, so $Y^{'}$ is nonempty, smooth over $k$ of pure dimension $m$ , and it is the image of a closed immersion into $P_{k}^{M}$ . Applying the "in particular" clause of [F29] to the closed immersion $Y^{'} ↪ P_{k}^{M}$ produces a nonempty Zariski-open subset $U \subseteq P (W_{h})$ of the hyperplane parameter space of the embedding, such that for every closed point $[s] \in U$ and every degree-one form $L$ with $L ∣_{Y^{'}} = s$ , the scheme-theoretic hyperplane section $Y^{'} \times_{P_{k}^{M}} V_{+} (L)$ is smooth over $k$ . The coefficient assignment $L \mapsto L \circ ν$ is a linear isomorphism from the space of linear forms on $P_{k}^{M}$ onto $S_{e}$ (both are $k$ -vector spaces with basis indexed by the degree- $e$ monomials), and by [F27] (applicable with $n = N \geq 1$ , since $m \geq 1$ ) the associated linear form $L_{F}$ of $0 \neq F \in S_{e}$ satisfies $L_{F} (ν ([x])) = F (x)$ and $ν^{- 1} (V_{+} (L_{F})) = V_{+} (F)$ ; comparing dehomogenized equations on the standard charts as in 1.3, the local equations identify $Y \cap V_{+} (F)$ with the fibre product $Y^{'} \times_{P_{k}^{M}} V_{+} (L_{F})$ over the isomorphism $Y \to Y^{'}$ from [F40]; let $K \subseteq S_{e}$ be the subspace of forms restricting to zero on $Y^{'}$ , the kernel of the $k$ -linear map $F \mapsto L_{F} ∣_{Y^{'}}$ , which is proper because $Y \neq \emptyset$ , and let $π ⁣ : P (S_{e}) ∖ P (K) \to P (W_{h})$ be the induced morphism. Then $π$ is defined on a nonempty open subset, it carries $k$ -rational points to $k$ -rational points because it is induced by a $k$ -linear map, and it is surjective because the composite $S_{e} \to Γ (P_{k}^{M}, O (1)) \to W_{h}$ is onto by definition of the hyperplane system $W_{h}$ as the span of the restricted coordinate forms. Hence $V : = π^{- 1} (U)$ is a nonempty open subset of $P (S_{e})$ , and every closed point $[F] \in V$ is good: by the criterion of 1.4 the point $[F]$ has residue field $k$ , hence so does its image $π ([F])$ , so $π ([F])$ is a closed point of $P (W_{h})$ lying in $U$ , and the identification above together with [F29] makes $Y \cap V_{+} (F) ≅ Y^{'} \times_{P_{k}^{M}} V_{+} (L_{F})$ smooth over $k$ .

2.3F2F3F4F17F18F19F25F26step 1.2step 1.4step 2.2

The universal intersection and the nonemptiness locus. In the universal-intersection and defect-locus constructions all parameter-space and incidence loci are classical loci of $k$ -points; [F31] is applied only in that category. The open-set correspondence of [F7] on the irreducible parameter variety $Π_{r}$ gives a scheme open with exactly the same closed points for each classical open constructed here. Every fibre assertion in the remainder of the proof is for a closed parameter $t$ , so $κ (t) = k$ . For $1 \leq i \leq r$ let $W_{i} \subseteq P_{k}^{N} \times_{k} Π_{r}$ be the set of pairs $(x, [F_{1}], \dots, [F_{r}])$ with $F_{i} (x) = 0$ ; on a product of a standard chart of $P_{k}^{N}$ [F5] and affine charts of the factors $P (S_{e_{i}})$ (normalizing one coefficient of each form to $1$ ), the condition is cut out by the polynomial obtained by dehomogenizing $F_{i}$ , so $W_{i}$ is closed and so is the intersection $W_{r} = ⋂_{i} W_{i}$ . Put $W : = W_{r} \cap (X \times_{k} Π_{r})$ , a closed subset of $X \times_{k} Π_{r}$ : its fibre over a classical parameter $t \in Π_{r}$ is exactly the classical closed-point set of $Z (t) = X \cap V_{+} (F_{1}) \cap \dots \cap V_{+} (F_{r})$ by [F21] and the local description of 1.3. The projection $q ⁣ : X \times_{k} Π_{r} \to Π_{r}$ is a closed map: $X$ is closed in $P_{k}^{N}$ and nonempty [F2], $Π_{r}$ is a classical variety [F30], so by [F31] the projection $Π_{r} \times_{k} P_{k}^{N} \to Π_{r}$ is closed, and a closed subset of the closed subset $X \times_{k} Π_{r}$ has closed image under its restriction; since $W \subseteq X \times_{k} Π_{r}$ is closed in $P_{k}^{N} \times_{k} Π_{r}$ , the image $N_{r} : = q (W) = {t \in Π_{r} : Z (t) \neq \emptyset}$ is closed, and its complement ${t : Z (t) = \emptyset}$ is open. Moreover $N_{r} = Π_{r}$ whenever $0 \leq r \leq d$ . For $r = 0$ this is $Z = X \neq \emptyset$ by the hypothesis on $X$ . Given any tuple of $r \geq 1$ forms, begin with the irreducible closed set $C_{0} = X$ of dimension $d$ . Inductively, if $i \leq r \leq d$ and an irreducible closed set $C_{i - 1} \subseteq Z (F_{1}, \dots, F_{i - 1})$ has dimension $m \geq d - i + 1 \geq 1$ , then either $F_{i}$ vanishes identically on $C_{i - 1}$ , in which case take $C_{i} = C_{i - 1}$ , or [F26] gives a nonempty irreducible component $C_{i}$ of $C_{i - 1} \cap V_{+} (F_{i})$ of dimension $m - 1$ . In both cases $C_{i} \subseteq Z (F_{1}, \dots, F_{i})$ and $\dim C_{i} \geq d - i$ . Thus the final intersection is nonempty for every closed parameter tuple. Thus $N_{r} = Π_{r}$ as classical loci for $r \leq d$ , which proves the required nonemptiness for every closed parameter. The emptiness locus for any $r$ is a classical open, hence corresponds to a scheme open by [F7]. [F2, F5, F8, F21, F26, F30, F31, F7, step 1.3, induction] 3.1 The dimension and nonemptiness step. Let $Y \subseteq P_{k}^{N}$ and $e \geq 1$ be as in 2.2, with $m \geq 1$ , and let $V = π^{- 1} (U) \subseteq P (S_{e})$ be the nonempty open set of 2.2, whose closed points are the forms $F$ with $Y \cap V_{+} (F)$ smooth over $k$ . The irreducible components $Y_{1}, \dots, Y_{t}$ of $Y$ are finite in number and closed by [F17], each is a nonempty closed subvariety of $P_{k}^{N}$ of dimension $m$ [F2, F19] (pure dimension $m$ means every component has dimension $m$ ), and $F$ vanishes on $Y_{l}$ exactly when $F \in I (Y_{l})_{e}$ , a proper linear subspace of $S_{e}$ : since $Y_{l} \neq \emptyset$ , some coordinate function $x_{m}$ is nonzero at a point of $Y_{l}$ , and then $x_{m}^{e} \notin I (Y_{l})_{e}$ [F3, F4]; thus the set $W$ of forms not vanishing on any component of $Y$ is the complement of finitely many proper closed subsets, hence open and nonempty by [F25]. Both $V$ and $W$ are nonempty open in the irreducible space $P (S_{e})$ of 1.2, so $V \cap W$ is nonempty and open by [F18], and by 1.4 (applied to the projective space $P (S_{e})$ ) it contains a closed point $[F]$ ; by 2.2 this $F$ satisfies that $Y \cap V_{+} (F)$ is smooth over $k$ , and in particular $F \neq 0$ . For each $l$ , $F$ does not vanish on $Y_{l}$ and $\dim Y_{l} = m \geq 1$ , so [F26] gives that $Y_{l} \cap V_{+} (F)$ is nonempty and has all components of dimension $m - 1$ ; every component of the finite union $Y \cap V_{+} (F) = ⋃_{l} (Y_{l} \cap V_{+} (F))$ is contained in one of the closed pieces and contains a component of one of them, so by [F18] every component of $Y \cap V_{+} (F)$ has dimension exactly $m - 1$ .

3.2F5F7F31F34F39step 1.3step 2.3

The rank defect locus is closed, so the full-rank locus is open. For $0 \leq r \leq d$ , fix once and for all a finite generating list $g_{1}, \dots, g_{s}$ of the ideal $I_{j}$ of $X_{j}$ in each chart $D_{+} (x_{j})$ , possible by [F39]. On the product of such a chart with affine charts of all factors of $Π_{r}$ , the dehomogenized forms $f_{1}, \dots, f_{r}$ and the $g_{l}$ are polynomials, and we differentiate only in the $N$ ambient coordinates, holding parameter coefficients constant, to obtain the rows of the combined Jacobian matrix $J_{c o m b}$ of $g_{1}, \dots, g_{s}, f_{1}, \dots, f_{r}$ ; define $B_{j} \subseteq D_{+} (x_{j}) \times_{k} Π_{r}$ to be the common zero locus of all $g_{l}$ , all $f_{i}$ and all $(N - d + r) \times (N - d + r)$ minors of $J_{c o m b}$ . This locus is closed in the product chart, since all displayed functions are polynomial there. At every classical pair $(x, t)$ , both residue fields equal $k$ . By [F34] the module of differentials of the chart ring of this fixed $k$ -fibre $Z (t)$ over $k$ is the cokernel of the transpose of $J_{c o m b}$ , so its fibre dimension over $x$ equals $N - {rank}_{κ (x)} J_{c o m b}$ (the rank of a matrix is unchanged by transposition); this number depends only on the point $x$ and the tuple $t$ , not on the chart or the chosen finite generating list, because $Ω$ and its base change do not. Hence the closed loci $B_{j}$ agree on overlaps and, closedness being local on an open cover, they glue to one closed subset $B \subseteq X \times_{k} Π_{r}$ : the locus of pairs $(x, t)$ with $x \in Z (t)$ and $\dim_{κ (x)} (Ω_{Z (t) / k} \otimes κ (x)) > d - r$ . By the classical closed projection of step 2.3, $q (B)$ is classically closed. Consequently its classical complement corresponds under [F7] to a scheme open $U_{r}^{'} \subseteq Π_{r}$ , whose closed parameters are exactly those with no classical rank-defect pair. No assertion about $Ω_{Z (t) / k}$ for nonclosed parameters is used.

4.1F12F14step 1.1step 2.1step 2.2step 3.1

Existence of a good tuple for $0 \leq r \leq d$ . We claim that for every $0 \leq i \leq \min (r, d)$ there are forms $F_{1}, \dots, F_{i}$ , $0 \neq F_{l} \in S_{e_{l}}$ , such that $Z_{i} = Z (F_{1}, \dots, F_{i})$ is nonempty, smooth over $k$ and of pure dimension $d - i$ . For $i = 0$ this is 1.1 and 2.1. For the induction step, let $1 \leq i \leq \min (r, d)$ , so that $Y = Z_{i - 1}$ is a nonempty closed subscheme of $P_{k}^{N}$ which is smooth over $k$ of pure dimension $d - i + 1 \geq 1$ and reduced (regular by [F12], hence a domain at each local ring by [F14]); applying 2.2 and 3.1 with $e = e_{i}$ and $m = d - i + 1$ produces $F_{i} \in S_{e_{i}}$ such that $Z_{i} = Y \cap V_{+} (F_{i})$ is nonempty, smooth over $k$ and of pure dimension $d - i$ . In particular, for $0 \leq r \leq d$ there is a tuple $t_{0} = (F_{1}, \dots, F_{r})$ with $Z (t_{0})$ nonempty, smooth over $k$ and of pure dimension $d - r$ .

4.2F9F12F15F20F32F33F36F37step 1.1step 1.3step 1.4step 3.2

Closed points of full-rank tuples are regular of dimension $d - r$ . Let $t$ be a closed point of $U_{r}^{'}$ , let $x \in Z (t)$ be a closed point, and work in a chart $D_{+} (x_{j})$ containing $x$ with the notation of 3.2. Since $t \notin q (B)$ , the rank of $J_{c o m b} (x)$ over $κ (x) = k$ (step 1.4) is at least $N - d + r$ ; on the other hand the $g$ -block has rank $N - d$ , because $X_{j}$ is smooth over $k$ hence regular at the rational point $x$ [F9, F12] and [F33] (rational-point clause together with the classical dimension clause, since $\dim O_{X, x} = d$ by 1.1) gives $rank J (g) (x) = N - \dim (A_{j})_{m_{x}} = N - d$ , where $m_{x}$ is the maximal ideal of $A_{j}$ corresponding to $x$ , while the $f$ -block adds at most its $r$ rows; hence $rank J_{c o m b} (x) = N - d + r$ . By [F32] applied to the actual ideal of $Z (t)$ in the chart, whose finite generating list is $g_{1}, \dots, g_{s}, f_{1}, \dots, f_{r}$ , we get $\dim_{k} T_{x} Z (t) = N - rank J_{c o m b} (x) = d - r$ . Consider the morphism of classical varieties $g ⁣ : X_{j} \to A_{k}^{r}$ whose components are the dehomogenized forms $f_{1}, \dots, f_{r}$ ; its source and target are smooth over $k$ [F9], and its scheme-theoretic fibre over the origin is $Z (t) \cap X_{j}$ by 1.3. By [F36] the fibre tangent space at $x$ is $\ker d_{x} g$ , and by [F37] the open immersion $Z (t) \cap X_{j} \subseteq Z (t)$ induces an isomorphism of tangent spaces, so $\dim_{k} \ker d_{x} g = d - r$ ; rank-nullity together with $\dim_{k} T_{x} X_{j} = d$ (from $\dim O_{X_{j}, x} = d$ and [F15]) gives that $d_{x} g$ is surjective, of rank $r = \dim_{0} A_{k}^{r}$ [F20]. But then [F36] applies and shows that the fibre $Z (t) \cap X_{j}$ has a regular local ring at $x$ of dimension $\dim_{x} X_{j} - \dim_{0} A_{k}^{r} = d - r$ ; since $Z (t) \cap X_{j}$ is an open subscheme of $Z (t)$ , the local ring $O_{Z (t), x}$ is regular of dimension $d - r$ .

5.1F9F10F19F35step 4.1step 3.2

The good tuple lies in the full-rank locus. Since $r \leq d$ , 4.1 provides a tuple $t_{0} = (F_{1}, \dots, F_{r})$ with $Z = Z (t_{0})$ nonempty, smooth over $k$ and of pure dimension $d - r$ . By [F9] and [F10] each point $x \in Z$ has an affine neighbourhood on which $Z \to Spec k$ is standard smooth at $x$ of some relative dimension $n$ ; by [F35] the module $Ω$ is free of rank $n$ there, and every component of that standard smooth affine neighbourhood has dimension $n$ , while those components are nonempty open pieces of the components of $Z$ , all of dimension $d - r$ [F19]; hence $n = d - r$ and $\dim_{κ (x)} (Ω_{Z / k} \otimes κ (x)) = d - r$ for every $x \in Z$ . Therefore no point of $Z$ has the defect of 3.2, and $t_{0} \in U_{r}^{'}$ .

5.2F11F12F13F14step 1.4step 4.2

Full-rank tuples with nonempty intersection are smooth. Let $t$ be a closed point of $U_{r}^{'}$ and suppose $Z (t) \neq \emptyset$ . By 4.2 every closed point of the finite-type $k$ -scheme $Z (t)$ is a regular point. The regular locus of $Z (t)$ is open by [F13]; if its complement $S$ were nonempty, then $S$ with its reduced closed-subscheme structure would be a nonempty closed subscheme of $P_{k}^{N}$ of finite type over $k$ , so 1.4 applied to $S$ would produce a point closed in $S$ , hence in $Z (t)$ because $S$ is closed in $Z (t)$ , a contradiction. Hence $Z (t)$ is regular, so $Z (t) \to Spec k$ is smooth by [F12] since $k$ is perfect [F11]; and $Z (t)$ is reduced because its local rings are regular, hence domains, by [F14].

5.3F3F4F21F25F38step 2.1step 4.1step 2.3

Claim 2. Suppose $r > d$ . By 4.1 with $r = d$ (and 2.1 when $d = 0$ ) there is a tuple $F_{1}, \dots, F_{d}$ with $Z_{d} = Z (F_{1}, \dots, F_{d})$ nonempty, smooth over $k$ and of pure dimension $0$ ; by [F38] the underlying set of $Z_{d}$ is finite, say ${p_{1}, \dots, p_{q}}$ . For each $l$ the forms of $S_{e_{d + 1}}$ vanishing at $p_{l}$ form a proper linear subspace: some coordinate function $x_{m}$ is nonzero at the closed point $p_{l}$ [F3], and then $x_{m}^{e_{d + 1}}$ does not vanish there [F4]. Since the field $k$ is infinite [F3], [F25] provides $F_{d + 1} \in S_{e_{d + 1}}$ vanishing at none of $p_{1}, \dots, p_{q}$ , and we choose arbitrary nonzero forms $F_{d + 2}, \dots, F_{r}$ (for instance powers of coordinates), which exist because $S_{e} \neq 0$ for $e \geq 1$ ; then the underlying set of $Z (F_{1}, \dots, F_{r})$ , being contained in $Z_{d} \cap V_{+} (F_{d + 1})$ , is empty, so $Z (F_{1}, \dots, F_{r}) = \emptyset$ . Therefore the open set ${t \in Π_{r} : Z (t) = \emptyset}$ of 2.3 is nonempty, which is claim 2.

6.1F16F17F18step 1.4step 4.2step 5.2

Full-rank tuples with nonempty intersection have pure dimension $d - r$ . Let $t$ be a closed point of $U_{r}^{'}$ with $Z (t) \neq \emptyset$ ; then $Z (t)$ is reduced by 5.2, so [F16] applies at every closed point $x$ of $Z (t)$ and, together with 4.2, gives that the maximum of $\dim W$ over the irreducible components $W$ of $Z (t)$ containing $x$ equals $\dim O_{Z (t), x} = d - r$ . Let $W$ be any irreducible component of $Z (t)$ (finitely many exist by [F17]): the open subset $W ∖ ⋃_{W^{'} \neq W} W^{'}$ of $W$ is nonempty, because otherwise the irreducible $W$ would be contained in the finite union of the closed sets $W^{'}$ and hence in one of them by [F18], contradicting that components are maximal; by 1.4 applied in a chart meeting it, it contains a closed point $x$ of $Z (t)$ , which then lies on no component other than $W$ , so the maximum above is $\dim W$ and $\dim W = d - r$ . Hence every irreducible component of $Z (t)$ has dimension $d - r$ , i.e. $Z (t)$ is of pure dimension $d - r$ .

7.1step 2.1step 2.3step 3.2step 4.1step 5.1step 5.2step 6.1

Claim 1. Let $0 \leq r \leq d$ and put $U_{r} = U_{r}^{'}$ , the open full-rank locus of 3.2. It is nonempty because it contains the tuple $t_{0}$ of 4.1 by 5.1. For every closed point of $U_{r}$ the intersection is nonempty by 2.3, smooth over $k$ by 5.2 and of pure dimension $d - r$ by 6.1; this is claim 1, and for $r = 0$ it is also the statement of 2.1.

8.1F1F7F8F11F12F13F16F24F26F29F31F33F35F36F38step 1.2step 1.4step 4.1step 4.2step 5.2∎

Boundary, choice, and iff dispositions. Empty: the statement has $X \neq \emptyset$ ; for $r \leq d$ every member over $U_{r}$ is nonempty by 7.1, and for $r > d$ the members over the open set of 5.3 are empty, the empty scheme being allowed there. Zero: the case $r = 0$ is the empty-tuple case of 2.1 with $Π_{0} = Spec k$ , and $d = 0$ is covered by 4.1 and 5.3; for $r = d$ the conclusion is pure dimension $0$ , i.e. a finite nonempty set of closed points, consistent with [F38]. One: $r = 1$ , $1 \leq d$ , is the first induction step of 4.1, and the parabolas/hypersurface computations of the companion page are instances; no step requires $r \geq 2$ . Degenerate: $X$ is irreducible of pure dimension $d$ and smooth, so no singular-source case arises; the members $Z (t)$ are allowed to be reducible or non-reduced as subschemes of $X$ , and no irreducibility, connectedness, or nonemptiness is asserted for tuples outside $U$ . Endpoints: the degrees $e_{i} \geq 1$ and $N \geq 0$ are arbitrary; $d = 0$ , $r = 0$ , $r = d$ and $r > d$ are all covered, and for $r > d$ the statement covers every $r$ , not merely $d + 1$ . Nonempty-choice: AC is declared as [F1] and is used exactly through the AC-assuming suppliers [F7] (dictionary), [F8] (closed-point density and Nullstellensatz), [F12]-[F13] (regular versus smooth, openness of the regular locus), [F16] (componentwise local dimension), [F26] (hypersurface dimension drop), [F29] (Bertini), [F31] (closedness of the projection), [F33] (Jacobian criterion), [F35]-[F36] (standard smooth fibres and the tangent criterion), and [F38] (zero-dimensional varieties are finite); the finite choices of charts, generating lists, components, coefficients and forms in steps 1.3, 2.3, 3.1, 3.2, 5.1, 5.3 and 6.1 are finite and add no choice principle, and the linear algebra and differential computations are choice-free. Both iff cases: the biconditional [F12] is used in the direction "smooth implies regular" in 1.1, 4.1 and 4.2 and in the direction "regular implies smooth" in 5.2; the criterion [F36] is used in the direction "surjective differential implies smooth at $x$ " in 4.2 after the converse direction is only used through the kernel identification $T_{x} (fibre) = \ker d_{x} f$ , which [F36] supplies for every such morphism; the Jacobian criterion [F33] is used in the rational-point direction "regular implies the rank formula" in 4.2 and in the perfect-field direction only through the same equivalence; the irreducibility criterion [F24] is used in the direction "vanishing ideal $(0)$ prime implies $P_{k}^{n}$ irreducible" in 1.2; and the Nullstellensatz facts [F8] are used in both directions in 1.4 to identify closed points with residue field $k$ . No claim is made about the size or density of the open sets $U$ , and the characteristic- $0$ hypothesis enters only through Bertini [F29] and the perfectness of $k$ [F11]. This completes the proof.

Source qualification

Vakil, Classes 51-52, §3.9 Corollary (with §3.10-3.11), proves Bertini for a single general member of a base-point-free linear system on a smooth variety over an algebraically closed field of characteristic $0$ , and Arapura, §5.4, states the complete-intersection version for hypersurfaces of prescribed degrees on a smooth projective variety. The present corollary is not copied from either source. Its first claim is proved here by the induction of steps 2.2, 3.1 and 4.1, which applies the in-run Bertini theorem Bertini smoothness away from the base locus to the Veronese image of the current intersection — this is the only way degree- $e$ forms enter, avoiding any use of the cohomology of twisting sheaves — and combines it with the componentwise dimension drop of Nontrivial projective hypersurface sections and the finite-union-of-subspaces lemma to keep every intersection nonempty and pure. The second and harder point, openness of the property in the full product of parameter spaces, is proved in steps 2.3, 3.2, 4.2, 5.1, 5.2 and 6.1 by a rank-defect argument: the locus where the Jacobian of the tuple fails to have the expected rank is closed, its image under the projection from the projective $X$ is closed, and on the complement the smooth-map criterion produces regular local rings of dimension $d - r$ , which openness of the regular locus and the local dimension formula upgrade to smoothness and purity. Neither source states openness in the product, and neither source makes any statement about the size of the good locus, about nonemptiness of members for $r > d$ being detectable on an open set, or about the characteristic-zero hypothesis beyond Bertini. The characteristic- $0$ assumption is used only through Bertini smoothness away from the base locus and perfectness of $k$ ; the positive-characteristic failure of the general-member statement is recorded on the companion examples page of this pair. The Veronese transfer in step 2.2 uses A degree-d homogeneous equation becomes a hyperplane section under Veronese only for the coefficient identity $L_{F} (ν ([x])) = F (x)$ and the set equality $V_{+} (F) = ν^{- 1} (V_{+} (L_{F}))$ ; the scheme-theoretic identification of $Y \cap V_{+} (F)$ with the fibre product $Y^{'} \times_{P_{k}^{M}} V_{+} (L_{F})$ is proved there by comparing local equations, since the library records the Veronese corollary only as a statement about underlying sets.

RemarkRemark: Literature-sourcedProof: Not applicableaudited 2026-09-30Open item page →

Conventions and hypotheses carried by this pair

Choice conventions. Every item of this pair that needs the Axiom of Choice declares it (The Axiom of Choice) and passes the assumption on through the cited suppliers; an item that does not name AC uses none of the choice-dependent results. No incompatible-axiom branch is opened anywhere on the pair.

Jacobians have equation rows, and the presentation does not matter. The Jacobian of Equation rows and coordinate columns in an affine Jacobian is read with one row per defining equation and one column per coordinate, for the actual defining ideal of the scheme, not for the ideal of its reduction. The kernel statement The Jacobian kernel computes the tangent space is proved for every finite generating list of that ideal, so no result of this pair depends on the chosen presentation; a proper subset is also covered if it still generates the same ideal; if it does not, the theorem does not identify its kernel with the tangent space of the original scheme, and the scheme-theoretic tangent space is not computed from a reduced ideal.

Dual numbers are used only at rational points. The identification of the intrinsic tangent space with the fibre of the dual-number points Tangent vectors at rational points are dual-number points is asserted at $k$ -rational points and at those points only. The intrinsic definition The intrinsic Zariski tangent space is the one used at a general scheme point, where no such identification is claimed; every statement of the pair names the kind of point it uses.

Tangent cones retain all initial forms. The tangent cone The scheme-theoretic tangent cone at a point is the spectrum of the full associated graded ring, without quotienting by nilpotents, and therefore remembers every initial form of the local equation. The scheme-theoretic linear span of the tangent cone shows that no proper linear closed subscheme contains this cone scheme-theoretically. The qualification is not cosmetic: the reduced cone can span strictly less than the tangent space. At the origin of the doubled line $Spec k [x, y] / (y^{2})$ the reduced cone is the line $y = 0$ , of dimension one, while the tangent space is two-dimensional. The examples page of this pair records the corresponding cone computations for plane curves, and no computation there replaces a scheme-theoretic cone by its reduced support.

Regularity is absolute; smoothness is relative. Regularity is a property of the local ring of a scheme at a point (Regular points of locally Noetherian schemes), while smoothness is a property of a morphism, here of the structure morphism to $Spec k$ (Smooth morphisms via local standard smooth presentations). Purely inseparable field algebras separate regularity from smoothness shows that the two notions diverge over imperfect fields: the spectrum of $L = k [t] / (t^{p} - a)$ , $a \notin k^{p}$ , is regular at its only point but not smooth over $k$ , and no equivalence between regularity and smoothness may be quoted without the perfectness hypothesis that the pair's perfect-field items carry.

Target-open generic smoothness needs a smooth source. The theorem Generic smoothness over a dense target open assumes the source smooth over $k$ ; that hypothesis is not decoration. The examples page of this pair records a dominant morphism of irreducible classical varieties over a smooth target whose source has a singular point in every fibre, so that no nonempty target open has smooth restriction. In positive characteristic the Frobenius phenomenon defeats the arbitrary base-point-free form of Bertini; the counterexample recorded on the examples page is stated for general linear systems and deliberately makes no claim about the embedded hyperplane-section case, so no item of this pair quotes it as such a claim.

Source notes

The Jacobian row and column convention, the tangent-cone construction from the associated graded ring, and the treatment of regularity as an absolute local condition follow Milne, Algebraic Geometry, Ch. 4 §§d–i. The positive- characteristic divergence between regularity and smoothness, and the Frobenius failure of Bertini for general linear systems, follow the source accounts read for the individual items; the exact locators are recorded on those items. This remark asserts no theorem of its own: it fixes which conventions and hypotheses the page's items actually use, and it points to the items that carry each claim.

5 · Examples, counterexamples and false statements

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