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Algebraic Differentials Separability and Smooth Local Presentations

1 · Prerequisites

2 · Summary

Standard smoothness is characterised here by flatness together with geometrically regular fibres, and the criterion is reached through Kähler differentials: the universal property and its localisation, transitivity and conormal sequences, separating transcendence bases, the separable-residue cotangent sequence and the fact that a smooth presentation has an invertible Jacobian minor. Over a field, locally standard smooth maps are exactly the flat maps whose fibres stay regular under every field extension, which is geometric regularity; the page proves that equivalence, the descent from flatness plus a geometrically regular fibre to a local standard smooth chart, stability under base change and composition, and the submersion criterion identifying local standard smoothness of a finite-type morphism at k-rational points with injectivity of the pullback on cotangent spaces. The Axiom of Choice is carried explicitly where the local flatness criterion, regular parameters and geometric regularity require it, and the relative dimension of a presentation is kept distinct from the dimension of an individual fibre.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Universal algebraic differentials and A-derivations

Definition

Let $A \overset{φ}{\to} B$ be a ring homomorphism of commutative rings (Commutative ring), so that $B$ is an $A$ -algebra. Let $F$ be the free $B$ -module on the set underlying $B$ (Unital left and right modules over a ring; unqualified module means left module), with basis symbol $[b]$ for $b \in B$ , and let $R \subseteq F$ be the $B$ -submodule generated by all elements

$[b + b^{'}] - [b] - [b^{'}], [b b^{'}] - b [b^{'}] - b^{'} [b], [φ (a)]$

with $b, b^{'} \in B$ and $a \in A$ . The module of algebraic differentials of $B$ over $A$ , also called the module of Kähler differentials, is the quotient $B$ -module

$Ω_{B / A} : = F / R,$

and the map $d = d_{B / A} ⁣ : B \to Ω_{B / A}$ , $d b : = [b] + R$ , is the universal $A$ -derivation of $B$ over $A$ . By construction $d$ is additive, satisfies the Leibniz rule

$d (b b^{'}) = b d b^{'} + b^{'} d b (b, b^{'} \in B),$

and is $A$ -constant, meaning $d (φ (a)) = 0$ for every $a \in A$ ; these are exactly the three families of relators above.

Now let $M$ be a $B$ -module. An $A$ -derivation of $B$ into $M$ is a map $D ⁣ : B \to M$ that is additive, is $A$ -constant in the sense that $D (φ (a)) = 0$ for all $a \in A$ , and satisfies the Leibniz rule $D (b b^{'}) = b D (b^{'}) + b^{'} D (b)$ . The set of such maps is written ${Der}_{A} (B, M)$ ; it is a $B$ -module under the pointwise operations, since $M$ is a $B$ -module.

Three conventions are part of the definition. First, $Ω_{B / A}$ is presented by the whole set $B$ of generators, so no finiteness of $B$ over $A$ and no finite generation, finite presentation or Noetherian hypothesis is assumed anywhere in this definition. Second, $d 1 = 0$ , because $1_{B} = φ (1_{A})$ makes $[1_{B}]$ one of the relators; consequently $d (- b) = - d b$ , $d (b^{n}) = n b^{n - 1} d b$ for $n \geq 1$ by induction on the Leibniz rule, and $d (a b) = a d b$ for $a \in A$ , $b \in B$ . Third, the extreme case $B = A$ (with $φ$ the identity) gives $Ω_{A / A} = 0$ : every $[b]$ is the relator $[φ (b)]$ , so the quotient is the zero module and the only $A$ -derivation of $A$ into any $A$ -module is the zero map.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Universal property of algebraic differentials

Statement

Let $A \to B$ be a ring homomorphism of commutative rings and let $Ω_{B / A}$ with its universal derivation $d$ be as in Universal algebraic differentials and A-derivations. For every $B$ -module $M$ the assignment $g \mapsto g \circ d$ is a $B$ -module isomorphism

${Hom}_{B} (Ω_{B / A}, M) ≅ {Der}_{A} (B, M),$

natural in $M$ : for every $B$ -linear $h ⁣ : M \to N$ the diagram of the two assignment maps commutes, and the isomorphism is additive in $g$ and $D$ .

Facts & Assumptions

Given: A ring homomorphism $A \to B$ of commutative rings, the presented module $Ω_{B / A} = F / R$ with universal derivation $d$ , and a $B$ -module $M$ .

[F1]

Universal algebraic differentials and A-derivations: $Ω_{B / A} = F / R$ , where $F$ is the free $B$ -module on the basis symbols $[b]$ , $b \in B$ , and $R$ is generated by $[b + b^{'}] - [b] - [b^{'}]$ , $[b b^{'}] - b [b^{'}] - b^{'} [b]$ and $[φ (a)]$ ; $d b = [b] + R$ ; an $A$ -derivation into a $B$ -module $M$ is a map $D$ that is additive, $A$ -constant and satisfies the Leibniz rule.

Proof

1.1

Every $g \in {Hom}_{B} (Ω_{B / A}, M)$ gives $g \circ d \in {Der}_{A} (B, M)$ : additivity, $A$ -constancy and the Leibniz rule for $g \circ d$ are those of $d$ transported by the additive map $g$ , and $d$ has them by construction. The assignment is additive and $B$ -linear: $(g_{1} + g_{2}) \circ d = g_{1} \circ d + g_{2} \circ d$ and $(b g) \circ d = b (g \circ d)$ pointwise. It is injective, because the classes $d b$ generate $Ω_{B / A} = F / R$ as a $B$ -module, so $g \circ d = 0$ forces $g = 0$ .

F1given

2.1

Conversely let $D \in {Der}_{A} (B, M)$ . Since $F$ is free on the $[b]$ , there is a unique $B$ -linear $\tilde{D} ⁣ : F \to M$ with $\tilde{D} ([b]) = D (b)$ . Each relator is killed: $\tilde{D} ([b + b^{'}] - [b] - [b^{'}]) = D (b + b^{'}) - D (b) - D (b^{'}) = 0$ by additivity of $D$ , similarly the Leibniz relator, and $[φ (a)] \mapsto D (φ (a)) = 0$ by $A$ -constancy. Hence $\tilde{D}$ factors through $F / R = Ω_{B / A}$ , giving a $B$ -linear $g_{D} ⁣ : Ω_{B / A} \to M$ with $g_{D} (d b) = D (b)$ , that is, $g_{D} \circ d = D$ .

F1step 1.1algebra

3.1

The two assignments are mutually inverse: the derivation attached to $g_{D}$ is $g_{D} \circ d = D$ , and the homomorphism attached to $g \circ d$ agrees with $g$ on every generator $d b$ , hence equals $g$ because those generators span. Each assignment is additive, so the bijection is a $B$ -module isomorphism; and for $B$ -linear $h ⁣ : M \to N$ one has $h \circ (g \circ d) = (h \circ g) \circ d$ , which is exactly naturality in $M$ .

step 1.1step 2.1algebra∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Differentials of a polynomial quotient and the Jacobian cokernel

Statement

Let $A$ be a commutative ring, let $P = A [x_{1}, \dots, x_{n}]$ be the polynomial ring on finitely many variables $x_{1}, \dots, x_{n}$ , and let $B = P / I$ for an ideal $I \subseteq P$ . Then:

$Ω_{P / A}$ is a free $P$ -module with basis $d x_{1}, \dots, d x_{n}$ .
With $I / I^{2}$ regarded as a $B$ -module, the sequence of $B$ -modules $I / I^{2} ⟶ B \otimes_{P} Ω_{P / A} ⟶ Ω_{B / A} ⟶ 0$ is exact, where the first map sends the class of $f \in I$ to $1 \otimes d f$ and the second is induced by $P \to B$ .
If $I = (f_{1}, \dots, f_{c})$ , then $Ω_{B / A}$ is the cokernel of the $B$ -linear map $B^{c} \to B^{n}$ given by the Jacobian matrix $(\frac{\partial f_{j}}{\partial x_{i}})$ , that is, $Ω_{B / A} ≅ B^{n} / \sum_{j = 1}^{c} B \cdot (\partial_{i} f_{j})_{i}$ .

The first map of (2) need not be injective; it is not claimed to be.

Facts & Assumptions

Given: A commutative ring $A$ , the polynomial ring $P = A [x_{1}, \dots, x_{n}]$ , an ideal $I \subseteq P$ and the quotient $B = P / I$ .

[F1]

Universal property of algebraic differentials: for every $P$ -module $M$ , $g \mapsto g \circ d$ is an isomorphism ${Hom}_{P} (Ω_{P / A}, M) ≅ {Der}_{A} (P, M)$ , naturally in $M$ .

[F2]

Tensoring is right exact: for a commutative ring $R$ and an exact sequence $A^{'} \to B^{'} \to C^{'} \to 0$ of $R$ -modules, the sequence $A^{'} \otimes_{R} N \to B^{'} \otimes_{R} N \to C^{'} \otimes_{R} N \to 0$ is exact for every $R$ -module $N$ .

[F3]

The polynomial ring $R [x_{i} : i \in I]$ as finitely supported coefficient families on monomials: $A [x_{1}, \dots, x_{n}]$ is the set of finitely supported functions from monomials to $A$ , written as formal sums $\sum_{a} c_{a} x^{a}$ ; so each element of $P$ has a unique expression as a finite $A$ -linear combination of monomials $x^{a} = x_{1}^{a_{1}} \dots x_{n}^{a_{n}}$ , and the product of monomials is $x^{a} x^{a^{'}} = x^{a + a^{'}}$ .

Proof

1.1

Define $\partial_{i} ⁣ : P \to P$ on the monomial basis of [F3] by $\partial_{i} (x^{a}) : = a_{i} x^{a - e_{i}}$ when $a_{i} \geq 1$ , and $\partial_{i} (x^{a}) : = 0$ when $a_{i} = 0$ , extended $A$ -linearly. Since exponents add under multiplication and $N$ -multiplication distributes, $\partial_{i} (f g) = f \partial_{i} (g) + g \partial_{i} (f)$ for monomials and hence, by $A$ -bilinearity of multiplication, for all $f, g \in P$ ; also $\partial_{i} (A) = 0$ . So each $\partial_{i}$ is an $A$ -derivation of $P$ . By [F1] there are $P$ -linear $α_{i} ⁣ : Ω_{P / A} \to P$ with $α_{i} (d x_{j}) = \partial_{i} (x_{j}) = δ_{i j}$ . The $P$ -linear map $Φ ⁣ : P^{n} \to Ω_{P / A}$ , $Φ (e_{i}) = d x_{i}$ , is surjective: by additivity and the Leibniz rule $d (x^{a}) = \sum_{i} a_{i} x^{a - e_{i}} d x_{i}$ , and an arbitrary element of $P$ is a finite $A$ -linear combination of monomials, so every $d f$ lies in $\sum_{i} P d x_{i}$ , and these elements generate $Ω_{P / A}$ by construction. The $P$ -linear endomorphism $Φ \circ (α_{1}, \dots, α_{n})$ of $Ω_{P / A}$ fixes each generator $d x_{j}$ , hence is the identity; therefore $Φ$ is injective as well, and $d x_{1}, \dots, d x_{n}$ are a basis.

F1F3algebra

2.1

Let $Q : = coker (B \otimes_{P} I \to B \otimes_{P} Ω_{P / A})$ where the map sends $1 \otimes f$ to $1 \otimes d f$ . This is well defined: the map $I \to B \otimes_{P} Ω_{P / A}$ , $f \mapsto 1 \otimes d f$ , is $P$ -linear, and the class of $1 \otimes f$ depends only on $f$ , so extension of scalars gives the displayed $B$ -linear map. There is a $B$ -linear surjection $B \otimes_{P} Ω_{P / A} \to Ω_{B / A}$ with $1 \otimes d f \mapsto d (f + I)$ , induced by the $A$ -derivation $P \to Ω_{B / A}$ , $f \mapsto d (f + I)$ ; it kills the image of $B \otimes_{P} I$ because $d f$ maps to the class of $d (f + I)$ with $f \in I$ . Hence it factors through a surjection $Q \to Ω_{B / A}$ . Conversely define $D ⁣ : B \to Q$ by $D (g + I) : =$ the class of $1 \otimes d g$ . This is well defined because $g \in I$ makes $1 \otimes d g$ lie in the image defining $Q$ ; it is additive, $A$ -constant, and satisfies Leibniz because $d$ does and the $B$ -module structure on $B \otimes_{P} Ω_{P / A}$ is that of $B$ . By [F1] applied to the $A$ -algebra $B$ and the $B$ -module $Q$ , $D$ induces a $B$ -linear map $Ω_{B / A} \to Q$ , and the two displayed maps are inverse on the generating classes of $1 \otimes d x_{i}$ and of $d x_{i}$ . Therefore $Ω_{B / A} ≅ Q$ , which is exactness of the sequence of (2); right exactness of $B \otimes_{P} -$ is the statement of [F2] applied to $I \to P \to B \to 0$ , and it is what makes $B \otimes_{P} Ω_{P / A}$ the receptacle of this cokernel presentation.

F1F2step 1.1algebra

3.1

Suppose $I = (f_{1}, \dots, f_{c})$ . Since $I = \sum_{j} P f_{j}$ , every class in $I / I^{2}$ is a $B$ -linear combination of the classes of $f_{1}, \dots, f_{c}$ : from $f = \sum_{j} g_{j} f_{j}$ and $I^{2} ∋$ (terms with two factors of $I$ ) one gets $f \equiv \sum_{j} (g_{j} + I) f_{j}$ modulo $I^{2}$ . Hence the image of the first map of (2) is generated by the classes of $1 \otimes d f_{j}$ , and $d f_{j} = \sum_{i} \partial_{i} f_{j} d x_{i}$ by step 1.1 and the Leibniz rule. Under the basis identification of step 1.1, $1 \otimes d f_{j}$ corresponds to the $j$ -th column $(\partial_{i} f_{j})_{i}$ of the Jacobian matrix, so the cokernel of $B^{c} \to B^{n}$ , $e_{j} \mapsto (\partial_{i} f_{j})_{i}$ , is exactly $Q ≅ Ω_{B / A}$ of step 2.1.

step 1.1step 2.1algebra

4.1

The first map of (2) is not injective in general: take $A = Z$ , $P = Z [x]$ and $I = (2)$ . Then $I / I^{2} = 2 Z [x] / 4 Z [x] ≅ F_{2} [x] \neq 0$ , while $d (2) = 0$ in $Ω_{P / A}$ because $2 = φ (2)$ for the structure map $φ ⁣ : Z \to Z [x]$ , so the first map has nonzero kernel. This proves the final clause and, with steps 2.1 and 3.1, the whole statement.

step 2.1step 3.1givenalgebra∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Localization, base change and functoriality of differentials

Statement

Let $A \to B$ be a homomorphism of commutative rings, with universal derivation $d$ of $Ω_{B / A}$ .

(Base change.) Let $A \to A^{'}$ be a ring homomorphism and put $B^{'} = B \otimes_{A} A^{'}$ . Then there is a $B^{'}$ -module isomorphism $B^{'} \otimes_{B} Ω_{B / A} ≅ Ω_{B^{'} / A^{'}}, (b \otimes a^{'}) \otimes d b^{''} ⟼ (b \otimes a^{'}) d (b^{''} \otimes 1),$ which is natural in the base-change data.
(Localization.) Let $U \subseteq B$ be multiplicative and let $V \subseteq A$ be multiplicative with the image of $V$ in $B$ contained in $U$ . Then there is a $(U^{- 1} B)$ -module isomorphism $U^{- 1} Ω_{B / A} ≅ Ω_{U^{- 1} B / V^{- 1} A} .$
(Functoriality.) For an arbitrary $A$ -algebra homomorphism $B \to C$ the $A$ -derivation $b \mapsto d (1 \otimes b)$ of $B$ into $Ω_{C / A}$ induces a canonical $C$ -linear map $C \otimes_{B} Ω_{B / A} ⟶ Ω_{C / A} .$ For a general algebra map $B \to C$ this map is neither asserted injective nor asserted an isomorphism.

Facts & Assumptions

Given: A ring homomorphism $A \to B$ of commutative rings with universal derivation $d$ of $Ω_{B / A}$ .

[F1]

Universal property of algebraic differentials: for every $B$ -module $M$ , the assignment $g \mapsto g \circ d$ is an isomorphism ${Hom}_{B} (Ω_{B / A}, M) ≅ {Der}_{A} (B, M)$ , natural in $M$ .

[F2]

Universal mapping property of the tensor product of commutative algebras: for commutative $R$ -algebras $A, B, C$ and $R$ -algebra maps $f ⁣ : A \to C$ , $g ⁣ : B \to C$ there is a unique $R$ -algebra map $h ⁣ : A \otimes_{R} B \to C$ with $h (a \otimes 1) = f (a)$ and $h (1 \otimes b) = g (b)$ ; so $B \otimes_{A} A^{'}$ carries the $A^{'}$ -algebra structure with structure maps $b \mapsto b \otimes 1$ , $a^{'} \mapsto 1 \otimes a^{'}$ .

[F3]

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

Proof

1.1

For (1), let $D ⁣ : B^{'} \to B^{'} \otimes_{B} Ω_{B / A}$ be given on the generating tensors of $B^{'} = B \otimes_{A} A^{'}$ by $D (b \otimes a^{'}) : = a^{'} \cdot (1 \otimes d b)$ . This is a well-defined additive map by the defining property of the tensor product of $A$ -modules, since $(b, a^{'}) \mapsto a^{'} (1 \otimes d b)$ is $A$ -balanced, and it satisfies Leibniz because $(b \otimes a^{'}) (b^{''} \otimes a^{''}) = b b^{''} \otimes a^{'} a^{''}$ and $d (b b^{''}) = b d b^{''} + b^{''} d b$ ; it is $A^{'}$ -constant since $D (1 \otimes a^{'}) = a^{'} (1 \otimes d 1) = 0$ and $A^{'}$ -linear as the written scalar action shows, the $A^{'}$ -algebra structure on $B^{'}$ being the one of [F2]. By [F1] for the $A^{'}$ -algebra $B^{'}$ there is a $B^{'}$ -linear $ψ ⁣ : Ω_{B^{'} / A^{'}} \to B^{'} \otimes_{B} Ω_{B / A}$ with $ψ (d (b \otimes a^{'})) = a^{'} (1 \otimes d b)$ . In the other direction $b \mapsto d (b \otimes 1)$ is an $A$ -derivation of $B$ into the $B$ -module $Ω_{B^{'} / A^{'}}$ , so [F1] gives a $B$ -linear $Ω_{B / A} \to Ω_{B^{'} / A^{'}}$ , and extension of scalars gives the $B^{'}$ -linear $φ ⁣ : B^{'} \otimes_{B} Ω_{B / A} \to Ω_{B^{'} / A^{'}}$ displayed in the statement. Both composites are $B^{'}$ -linear and fix the generators: $φ ψ (d (b \otimes a^{'})) = φ (a^{'} (1 \otimes d b)) = a^{'} d (b \otimes 1) = d (b \otimes a^{'})$ , using $d (1 \otimes a^{'}) = 0$ and Leibniz; and $ψ φ ((b \otimes a^{'}) \otimes d b^{''}) = ψ ((b \otimes a^{'}) d (b^{''} \otimes 1)) = (b \otimes a^{'}) (1 \otimes d b^{''}) = (b \otimes a^{'}) \otimes d b^{''}$ . Hence $φ$ and $ψ$ are mutually inverse and $φ$ is the isomorphism of (1).

F1F2algebra

1.2

For (2), write $B_{U} : = U^{- 1} B$ and let $D ⁣ : B_{U} \to U^{- 1} Ω_{B / A}$ be $D (b / u) : = (1 / u) d b - (b / u^{2}) d u$ , an element of $U^{- 1} Ω_{B / A}$ . This is well defined: if $b / u = b^{'} / u^{'}$ in $B_{U}$ , there is $w \in U$ with $w x = 0$ for $x : = b u^{'} - b^{'} u$ , and applying $d$ to $w x = 0$ gives $w d x = - x d w$ ; multiplying by $u^{2} u^{' 2}$ and writing $G : = u^{' 2} (u d b - b d u) - u^{2} (u^{'} d b^{'} - b^{'} d u^{'})$ , this yields $w G = - x (u u^{'} d w + w u d u^{'} + w u^{'} d u)$ . In $U^{- 1} Ω_{B / A}$ the class of $x$ is zero, because $w x = 0$ and $w$ becomes invertible, so the class of $w G$ is zero; since $w$ also becomes invertible as a scalar, the class of $G$ is zero, which is exactly $D (b / u) = D (b^{'} / u^{'})$ . The map $D$ is additive and kills $V^{- 1} A$ ; for the Leibniz rule one uses the relations implied by $d (u u^{- 1}) = 0$ , namely $d (u^{- 1}) = - u^{- 2} d u$ inside $U^{- 1} Ω_{B / A}$ , after which the product rule for fractions follows from the product rule for $d$ . Hence $D$ is a $V^{- 1} A$ -derivation and [F1] for the $V^{- 1} A$ -algebra $B_{U}$ gives a $B_{U}$ -linear $ψ ⁣ : Ω_{B_{U} / V^{- 1} A} \to U^{- 1} Ω_{B / A}$ with $ψ (d (b / u)) = D (b / u)$ . Conversely $b \mapsto d (b / 1)$ is an $A$ -derivation of $B$ into $Ω_{B_{U} / V^{- 1} A}$ , so [F1] gives a $B$ -linear $Ω_{B / A} \to Ω_{B_{U} / V^{- 1} A}$ with $d b \mapsto d (b / 1)$ , and under the identification $U^{- 1} Ω_{B / A} ≅ (U^{- 1} B) \otimes_{B} Ω_{B / A}$ of [F3] the balanced assignment $(c, ω) \mapsto c \cdot (image of ω)$ yields the $B_{U}$ -linear $φ ⁣ : U^{- 1} Ω_{B / A} \to Ω_{B_{U} / V^{- 1} A}$ with $φ ((b / u) d b^{'}) = (b / u) d (b^{'} / 1)$ . On generators, $φ (ψ (d (b / u))) = φ ((1 / u) d b - (b / u^{2}) d u) = (1 / u) d (b / 1) - (b / u^{2}) d (u / 1) = d (b / u)$ , the last equality being the Leibniz rule applied to $b / u = (b / 1) (u / 1)^{- 1}$ together with $d ((u / 1)^{- 1}) = - (u / 1)^{- 2} d (u / 1)$ ; and $ψ (φ ((1 / u) d b)) = ψ ((1 / u) d (b / 1)) = (1 / u) D (b / 1) = (1 / u) d b$ . Both composites are module maps fixing generating sets, so they are inverse and $φ$ is the isomorphism of (2).

F1F3algebra

2.1

For (3) let $B \to C$ be an $A$ -algebra map. The map $b \mapsto d (1 \otimes b)$ from $B$ to the $C$ -module $Ω_{C / A}$ is additive and $A$ -constant and satisfies Leibniz, since $b \mapsto 1 \otimes b$ is a ring homomorphism and $d$ is an $A$ -derivation; here $1 \otimes b$ denotes the image of $b$ in $C$ when $C$ is written as a $B$ -algebra. So it is an $A$ -derivation, [F1] turns it into a $B$ -linear map $Ω_{B / A} \to Ω_{C / A}$ , and extension of scalars along $B \to C$ gives the $C$ -linear map of (3). For $C = B \otimes_{A} A^{'}$ , composing this map with the canonical map $Ω_{C / A} \to Ω_{C / A^{'}}$ gives the base-change isomorphism of step 1.1: both send $1 \otimes d b$ to $d (b \otimes 1)$ . The map before this composition need not be an isomorphism. For $C = U^{- 1} B$ , step 1.2 with $V = {1}$ does identify the functorial map with a localization isomorphism. No injectivity or surjectivity is claimed for a general $B \to C$ .

step 1.1step 1.2F1given∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Transitivity sequence for differentials

Statement

Let $A \to B \to C$ be homomorphisms of commutative rings. Then the sequence of $C$ -modules

$C \otimes_{B} Ω_{B / A} ⟶ Ω_{C / A} ⟶ Ω_{C / B} ⟶ 0$

is exact, where the first map is the extension of scalars of $d_{B / A}$ along $B \to C$ (so that $c \otimes d b \mapsto c d (image of b)$ ) and the second is induced by the universal property of $Ω_{C / A}$ from the $B$ -derivation $d_{C / B}$ . The first arrow is not asserted injective, and in general it is not injective.

Facts & Assumptions

Given: Ring homomorphisms $A \to B \to C$ of commutative rings, with universal derivations $d_{B / A}$ of $Ω_{B / A}$ , $d_{C / A}$ of $Ω_{C / A}$ and $d_{C / B}$ of $Ω_{C / B}$ .

[F1]

Universal property of algebraic differentials: for a ring map $R \to S$ and an $S$ -module $M$ , $g \mapsto g \circ d_{S / R}$ is an isomorphism ${Hom}_{S} (Ω_{S / R}, M) ≅ {Der}_{R} (S, M)$ , for the pairs $(A, B)$ , $(A, C)$ and $(B, C)$ alike.

[F2]

Tensoring is right exact: if $A^{'} \to B^{'} \to C^{'} \to 0$ is exact, then $A^{'} \otimes_{R} N \to B^{'} \otimes_{R} N \to C^{'} \otimes_{R} N \to 0$ is exact; in particular an extension of scalars of a surjection is surjective and $C \otimes_{B} Ω_{B / A}$ is generated as a $C$ -module by the elements $c \otimes d b$ .

[F3]

Differentials of a polynomial quotient and the Jacobian cokernel: for $P = A [x]$ , $Ω_{P / A}$ is free on $d x$ , and for $B = P / I$ the quotient formula $Ω_{B / A} ≅ (B \otimes_{P} Ω_{P / A}) / ⟨ 1 \otimes d f : f \in I ⟩$ holds.

Proof

1.1

The second map exists by [F1]: $d_{C / B} ⁣ : C \to Ω_{C / B}$ is in particular an $A$ -derivation, so it induces a $C$ -linear $π ⁣ : Ω_{C / A} \to Ω_{C / B}$ with $π (d_{C / A} c) = d_{C / B} c$ . It is surjective because the elements $d_{C / B} c$ generate $Ω_{C / B}$ . The composite $π \circ (first map)$ is zero: on the generators $c \otimes d b$ of $C \otimes_{B} Ω_{B / A}$ the first map sends $c \otimes d b$ to $c d_{C / A} b$ , and $π$ sends that to $c d_{C / B} b = 0$ , since $b$ lies in the image of $B$ so that $b$ is killed by the universal $B$ -derivation of $C$ . Hence $im (first map) \subseteq \ker π$ .

F1F2given

2.1

For the reverse inclusion put $Q : = coker (C \otimes_{B} Ω_{B / A} \to Ω_{C / A})$ , with quotient map $q ⁣ : Ω_{C / A} \to Q$ . The $A$ -derivation $C \to Q$ , $c \mapsto q (d_{C / A} c)$ , kills $B$ , since $q$ kills the image of the first map; hence by [F1] it induces a $C$ -linear $β ⁣ : Ω_{C / B} \to Q$ with $β (d_{C / B} c) = q (d_{C / A} c)$ . The map $π$ of step 1.1 kills the image of the first map, so it factors as $π = γ \circ q$ for a $C$ -linear $γ ⁣ : Q \to Ω_{C / B}$ , and then $γ β$ is a $C$ -linear endomorphism of $Ω_{C / B}$ fixing the generators $d_{C / B} c$ , so $γ β = {i d}_{Ω_{C / B}}$ . Symmetrically $β γ$ is a $C$ -linear endomorphism of $Q$ , and the elements $q (d_{C / A} c)$ generate $Q$ because the elements $d_{C / A} c$ generate $Ω_{C / A}$ and $q$ is onto, so from $β γ (q (d_{C / A} c)) = β (π (d_{C / A} c)) = β (d_{C / B} c) = q (d_{C / A} c)$ we get $β γ = {i d}_{Q}$ . Thus $γ$ is injective, and for $ω \in \ker π$ we get $γ (q (ω)) = π (ω) = 0$ , hence $q (ω) = 0$ and $ω \in \ker q = im (first map)$ . Hence $\ker π \subseteq im (first map)$ and, with step 1.1, $\ker π = im (first map)$ ; moreover $Q ≅ Ω_{C / B}$ .

step 1.1F1algebra

3.1

The first arrow is not injective in general: take $A = k$ a field, $B = k [x]$ , $C = k$ with $x \mapsto 0$ . By [F3] the source is $C \otimes_{B} Ω_{B / A} ≅ (k [x] / (x)) \otimes_{k [x]} k [x] d x ≅ k d x \neq 0$ , while the target is $Ω_{C / A} = Ω_{k / k} = 0$ , the universal derivation of a ring over itself being zero. So the first map is zero on a nonzero module, and with steps 1.1 and 2.1 the asserted sequence is exact with a noninjective first arrow.

step 2.1F3givenalgebra∎

DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-27Open item page →

Separating transcendence basis and separably generated extensions

Definition

Let $k \subseteq K$ be a field extension and let $S \subseteq K$ be algebraically independent over $k$ (Algebraic and transcendental elements and algebraic extensions). If $K$ is algebraic over the subfield $k (S)$ generated by $S$ , then $S$ is a transcendence basis of $K$ over $k$ ; by A maximal algebraically independent set is a transcendence basis an algebraically independent subset that is maximal for inclusion among algebraically independent subsets automatically has this property, so that lemma and this definition describe the same notion.

Now assume that $K / k$ is finitely generated (Finitely generated field extensions $F (a_{1}, \dots, a_{r})$ ), say $K = k (α_{1}, \dots, α_{n})$ . A finite tuple of pairwise distinct elements $t_{1}, \dots, t_{r} \in K$ is a separating transcendence basis of $K / k$ when ${t_{1}, \dots, t_{r}}$ is a transcendence basis of $K$ over $k$ and the extension $K / k (t_{1}, \dots, t_{r})$ is finite separable (Separable algebraic elements and separable extensions). The extension $K / k$ is separably generated over $k$ , or simply separably generated, when it admits a separating transcendence basis; in this terminology a finitely generated extension is separably generated precisely when it has a transcendence basis over which its residual extension is separable.

Finiteness of the residual extension is automatic: $K$ is finitely generated over $k$ , hence over $k (t_{1}, \dots, t_{r})$ , and algebraic there by the definition of a transcendence basis, so $K / k (t_{1}, \dots, t_{r})$ is finite by An extension generated by finitely many algebraic elements is finite. Three conventions are part of the definition.

The empty tuple is allowed: $r = 0$ is a separating transcendence basis exactly when $K / k$ is algebraic and separable, equivalently (under the finite-generation hypothesis) when $K / k$ is finite separable.
The notion concerns $K / k (t_{1}, \dots, t_{r})$ and says nothing about the intermediate field $k (t_{1}, \dots, t_{r})$ being perfect. If $k$ is perfect, the rational function field $k (t_{1}, \dots, t_{r})$ is nevertheless imperfect in characteristic $p$ as soon as $r \geq 1$ , so separability over it is a genuine restriction and cannot be replaced by separability of algebraic extensions of $k$ .
All transcendence bases of a finitely generated extension have the same finite cardinality $r$ , namely the transcendence degree, so the number of elements in a separating transcendence basis is determined by the extension.

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Finitely generated extensions of a perfect field are separably generated

Statement

Let $k$ be a perfect field and let $K / k$ be a finitely generated field extension. Then $K$ has a separating transcendence basis over $k$ (Separating transcendence basis and separably generated extensions): there are $t_{1}, \dots, t_{r} \in K$ , algebraically independent over $k$ , such that $K / k (t_{1}, \dots, t_{r})$ is finite separable.

The argument in characteristic $p$ uses $p$ -power linear independence and exchanges of finitely many generators; it does not infer that $k (t_{1}, \dots, t_{r})$ is perfect, and no perfectness of any intermediate field is asserted.

Facts & Assumptions

Given: A perfect field $k$ and a finitely generated field extension $K / k$ , say $K = k (α_{1}, \dots, α_{n})$ .

[F1]

Separating transcendence basis and separably generated extensions: a finite tuple is a separating transcendence basis when its entries are algebraically independent over $k$ and the residual extension is finite separable, and then $r$ is the common cardinality of all transcendence bases.

[F2]

Algebraic and transcendental elements and algebraic extensions: $a \in K$ is algebraic over a subfield $F$ when it satisfies a nonzero polynomial equation over $F$ , and transcendental otherwise; $K / F$ is algebraic when every element of $K$ is algebraic over $F$ .

[F3]

An extension generated by finitely many algebraic elements is finite: if $a_{1}, \dots, a_{s}$ are algebraic over $F$ , then $F (a_{1}, \dots, a_{s}) / F$ is finite.

[F4]

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

[F5]

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

[F6]

Separable algebraic elements and separable extensions: an extension is separable when each of its elements has separable minimal polynomial.

[F7]

The inseparable degree $[K : F]_{i} = [K : F] / [K : F]_{s}$ of a finite extension: for a finite extension $K / F$ , $[K : F]_{i} : = [K : F] / [K : F]_{s}$ .

[F8]

Separable degree is multiplicative in finite towers: $[L : F]_{s} = [L : K]_{s} [K : F]_{s}$ : $[L : F]_{s} = [L : K]_{s} [K : F]_{s}$ for a finite tower $F \subseteq K \subseteq L$ .

[F9]

Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ : $[L : F] = [L : K] [K : F]$ for a finite tower.

[F10]

A finite extension is separable if and only if $[K : F]_{s} = [K : F]$ : a finite extension is separable exactly when its separable degree equals its degree, so by [F7] a finite extension is separable exactly when its inseparable degree is $1$ .

[F11]

The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element: for algebraic $a$ over $F$ , $f (a) = 0$ implies $m_{a} ∣ f$ , where $m_{a}$ is the monic irreducible minimal polynomial; and $m_{a}$ generates the evaluation kernel.

[F12]

Gauss lemma over a UFD with For every field $F$ , $F [x]$ is a unique factorisation domain: a polynomial that is irreducible and of positive degree in one variable over a polynomial ring over a field remains irreducible over the fraction field, and a polynomial ring over a field is a unique factorisation domain.

[F13]

A nonzero polynomial over a field is separable exactly when its gcd with its derivative is $1$ : for $0 \neq f$ over a field, $f$ is separable if and only if $\gcd (f, f^{'}) = 1$ .

[F14]

An algebraic extension generated by separable elements is separable: an algebraic extension generated by separable elements is separable.

[F15]

One element of a transcendence basis can be exchanged for a suitable rival: given transcendence bases $S, T$ of $K / k$ and $s \in S$ there is $t \in T$ with $(T ∖ {t}) \cup {s}$ a transcendence basis; consequently two transcendence bases of a finitely generated extension have the same finite cardinality.

[F16]

A finite extension generated by elements all but possibly one of which are separable is simple: a finite extension generated by elements all but possibly one of which are separable is simple.

[F17]

The binomial theorem over an arbitrary commutative ring, A prime $p$ divides $(\binom{p}{k})$ for $0 < k < p$ : the binomial expansion holds in every commutative ring, and $p$ divides its intermediate coefficients for exponent $p$ . Hence $(a + b)^{p} = a^{p} + b^{p}$ in characteristic $p$ .

Proof

1.1

Running through the finite list $α_{1}, \dots, α_{n}$ and adjoining an element exactly when it is transcendental over the field generated by the elements already adjoined produces, after finitely many steps, an algebraically independent $S \subseteq K$ over which every $α_{i}$ is algebraic; then $K$ is algebraic over $k (S)$ and finitely generated over it, hence finite over $k (S)$ by [F3], so $S$ is a transcendence basis and $[K : k (S)]_{i}$ is defined by [F7]. The set of values $[K : k (T)]_{i}$ , taken over the finite, algebraically independent $T \subseteq K$ with $K / k (T)$ finite, is a nonempty set of natural numbers and therefore has a least element; fix $T = {x_{1}, \dots, x_{d}}$ attaining it, and note that each such $T$ is a transcendence basis of $K / k$ by [F2]. If $char k = 0$ , then $k (T)$ also has characteristic zero, so is perfect by [F4], and $K / k (T)$ is separable by [F5], so $T$ is already a separating transcendence basis. Henceforth assume $char k = p > 0$ . Then Frobenius is surjective on $k$ by [F4], and $k$ -linearly independent elements $a_{1}, \dots, a_{m} \in K$ have $k$ -linearly independent $p$ -th powers: from $\sum_{i} c_{i} a_{i}^{p} = 0$ with $c_{i} \in k$ and $b_{i}^{p} = c_{i}$ one gets $\sum_{i} (b_{i} a_{i})^{p} = 0$ , hence $\sum_{i} b_{i} a_{i} = 0$ because $x \mapsto x^{p}$ is additive by [F17] and injective on the field $K$ , so all $b_{i}$ , hence all $c_{i}$ , vanish.

F2F3F4F5F7F17givenF1

2.1

Suppose $K / k (T)$ is not separable. By [F6] there is $x_{d + 1} : = x \in K$ that is not separable over $K^{'} : = k (x_{1}, \dots, x_{d})$ , and $x$ is algebraic over $K^{'}$ by [F2] since $K / K^{'}$ is algebraic. Choose $F \in k [X_{1}, \dots, X_{d + 1}]$ nonzero of least total degree with $F (x_{1}, \dots, x_{d + 1}) = 0$ . Then $F$ is irreducible: a factorisation $F = G H$ with $G, H$ nonconstant satisfies $\deg F = \deg G + \deg H$ , and either $G$ or $H$ vanishes at $(x_{1}, \dots, x_{d + 1})$ with strictly smaller total degree, contradicting minimality.

F2F6step 1.1

3.1

Suppose every monomial exponent of $F$ were divisible by $p$ . Since $k$ is perfect, write each coefficient $λ_{α} = b_{α}^{p}$ using [F4]. Then $F = G^{p}$ for the nonzero polynomial $G = \sum_{α} b_{α} X^{α / p}$ , by Frobenius additivity [F17]. Evaluation gives $G (x_{1}, \dots, x_{d + 1})^{p} = 0$ in the field $K$ , hence $G (x_{1}, \dots, x_{d + 1}) = 0$ , contradicting minimality because $\deg G = \deg F / p < \deg F$ . Thus some variable $X_{j}$ occurs with an exponent not divisible by $p$ ; fix such $j$ .

F4F17step 1.1step 2.1algebra

4.1

Set $L : = k (x_{i} : 1 \leq i \leq d + 1, i \neq j)$ and $T^{''} : = {x_{i} : i \neq j}$ . Write $F = \sum_{ν = 0}^{e} C_{ν} (X_{i} : i \neq j) X_{j}^{ν}$ with $e \geq 1$ and $C_{e} \neq 0$ . The total degree of $C_{e}$ is strictly less than that of $F$ , so $C_{e} (x_{i} : i \neq j) \neq 0$ by the minimality in step 2.1. Thus $P (T) : = F (x_{i} : i \neq j; T)$ is a nonconstant polynomial in $L [T]$ vanishing at $x_{j}$ , which proves $x_{j}$ algebraic over $L$ . This makes $T^{''}$ a transcendence basis of $K / k$ : if $T^{''}$ were dependent, a maximal independent subset $S ⊊ T^{''}$ would be a transcendence basis of $L$ over $k$ , and since $L (x_{j}) = k (x_{1}, \dots, x_{d + 1})$ is algebraic over $L$ by the preceding coefficient argument, the same finite set $S$ would be a transcendence basis of $k (x_{1}, \dots, x_{d + 1})$ over $k$ with $∣ S ∣ < d = ∣ {x_{1}, \dots, x_{d}} ∣$ , contradicting that ${x_{1}, \dots, x_{d}}$ is a transcendence basis of that field and that all transcendence bases of a finitely generated extension have the same cardinality [F15]; so $T^{''}$ is independent, $K / L$ is algebraic, and $L$ is a rational function field over $k$ in the variables $x_{i}$ , $i \neq j$ . Consequently $P (T) : = F (x_{i} : i \neq j; T) \in L [T]$ is the image of the polynomial $F \in k [X_{i} : i \neq j] [X_{j}]$ , which has positive $X_{j}$ -degree and is irreducible in $k [X_{i} : i \neq j] [X_{j}] = k [X_{1}, \dots, X_{d + 1}]$ ; its coefficients are primitive, since a nonunit common factor would factor the irreducible $F$ of positive $X_{j}$ -degree. Thus Gauss' lemma [F12] makes its image irreducible in $L [T]$ . Since some $X_{j}$ -exponent of $F$ is not divisible by $p$ , the same exponent occurs in $P$ , so $P^{'} \neq 0$ and $\gcd (P, P^{'}) = 1$ because $P$ is irreducible and $\deg P^{'} < \deg P$ ; thus $P$ is separable by [F13]. As $P (x_{j}) = 0$ and $P$ is irreducible, $P$ is a nonzero scalar multiple of the monic minimal polynomial of $x_{j}$ over $L$ by [F11], so $x_{j}$ is separable over $L$ and $L (x_{j}) / L$ is separable by [F14].

F11F12F13F14F15step 2.1step 3.1

5.1

By [F8] and [F9] the inseparable degree is multiplicative, $[E : F]_{i} = [E : M]_{i} [M : F]_{i}$ for finite $F \subseteq M \subseteq E$ ; and $[L (x_{j}) : L]_{i} = 1$ by [F10]. With $K^{'} = k (T)$ and $L (x_{j}) = K^{'} (x_{d + 1})$ we get $[K : L]_{i} = [K : L (x_{j})]_{i} \cdot [L (x_{j}) : L]_{i} = [K : L (x_{j})]_{i}$ and $[K : K^{'}]_{i} = [K : L (x_{j})]_{i} \cdot [L (x_{j}) : K^{'}]_{i}$ , while $[L (x_{j}) : K^{'}]_{i} > 1$ by [F10] because $K^{'} (x_{d + 1}) / K^{'}$ is not separable. Therefore $[K : L]_{i} < [K : K^{'}]_{i}$ , and $T^{''}$ is a transcendence basis of $K / k$ by step 4.1 with a strictly smaller inseparable degree than $T$ , contradicting the minimality of step 1.1. Hence $K / k (T)$ is separable and $T$ is a separating transcendence basis, which proves the theorem; moreover, by [F16] the residual finite separable extension is simple, so $K = k (T) (γ)$ for a single element $γ$ separable over $k (T)$ .

F8F9F10F16step 1.1step 4.1∎

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Differentials of a separably generated field extension

Statement

Let $k \subseteq K$ be a finitely generated field extension that is separably generated over $k$ by $t_{1}, \dots, t_{r}$ (Separating transcendence basis and separably generated extensions). Then $d t_{1}, \dots, d t_{r}$ are a $K$ -basis of $Ω_{K / k}$ ; in particular $Ω_{K / k}$ is a free $K$ -module of rank $r$ .

Assume moreover the Axiom of Choice. Let $char k = 0$ and let $k \subseteq K \subseteq L$ be a tower of fields with $K / k$ finitely generated. Then the natural map $L \otimes_{K} Ω_{K / k} ⟶ Ω_{L / k}$ induced by $K \to L$ is injective. The Axiom of Choice is used exactly to choose maximal algebraically independent subsets, that is transcendence bases, of $L$ over $K$ and of $K$ over $k$ ; every subsequent step is choice-free. The assumption is declared as The Axiom of Choice and is inherited by the consumers of this theorem.

Facts & Assumptions

Given: A finitely generated field extension $k \subseteq K$ separably generated by $t_{1}, \dots, t_{r}$ , so that $K / k (T)$ is finite separable for $T = {t_{1}, \dots, t_{r}}$ ; and, for the second part, an extension $k \subseteq K \subseteq L$ with $char k = 0$ , $K / k$ finitely generated, together with the Axiom of Choice.

[F1]

Separating transcendence basis and separably generated extensions: $T = (t_{1}, \dots, t_{r})$ is algebraically independent over $k$ and the residual extension $K / k (T)$ is finite separable; every finitely generated extension is separably generated exactly when it possesses such a tuple.

[F2]

A finite extension generated by elements all but possibly one of which are separable is simple: a finite extension generated by elements all but possibly one of which are separable over the base is simple; in particular every finite separable extension is simple, so $K = k (T) (α)$ for an element $α$ separable over $k (T)$ .

[F3]

The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element: for $α$ algebraic over a field $F$ there is a unique monic irreducible minimal polynomial $m_{α} \in F [x]$ with $\ker ({ev}_{α}) = (m_{α})$ , and $f (α) = 0$ holds if and only if $m_{α} ∣ f$ .

[F4]

A nonzero polynomial over a field is separable exactly when its gcd with its derivative is $1$ : for $0 \neq f$ over a field, $f$ is separable over that field if and only if $\gcd (f, f^{'}) = 1$ .

[F5]

Differentials of a polynomial quotient and the Jacobian cokernel: for a commutative ring $A$ and $n \geq 0$ the module $Ω_{A [x_{1}, \dots, x_{n}] / A}$ is free with basis $d x_{1}, \dots, d x_{n}$ .

[F6]

Localization, base change and functoriality of differentials: for a multiplicative set $U$ inside an $A$ -algebra $B$ the canonical map $U^{- 1} Ω_{B / A} \to Ω_{U^{- 1} B / A}$ is an isomorphism, with inverse sending $d (b / u)$ to $u^{- 1} d b - b u^{- 2} d u$ ; and for every $A$ -algebra map $B \to C$ there is a natural $C$ -linear map $C \otimes_{B} Ω_{B / A} \to Ω_{C / A}$ .

[F7]

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

[F8]

Universal property of algebraic differentials: for every $B$ -module $M$ , composition with $d$ is an isomorphism ${Hom}_{B} (Ω_{B / A}, M) ≅ {Der}_{A} (B, M)$ , naturally in $M$ .

[F9]

A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective: a field is perfect if and only if its characteristic is $0$ , or its characteristic is $p > 0$ and its Frobenius map is surjective.

[F10]

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

[F11]

A maximal algebraically independent set is a transcendence basis: if $S \subseteq L$ is maximal for inclusion among the subsets of $L$ algebraically independent over a subfield $K$ , then every element of $L$ is algebraic over $K (S)$ .

[F12]

An extension generated by finitely many algebraic elements is finite: if $a_{1}, \dots, a_{s}$ are algebraic over a field $F$ , then $F (a_{1}, \dots, a_{s}) / F$ is finite.

[F13]

The Axiom of Choice: every family of nonempty sets has a choice function; this is what licenses the maximal algebraically independent subsets chosen below.

Proof

1.1

Set $K^{'} : = k (T) = k (t_{1}, \dots, t_{r})$ and let $α \in K$ with $K = K^{'} (α)$ be separable over $K^{'}$ , as supplied by [F2]. Let $P \in K^{'} [X]$ be the minimal polynomial of $α$ over $K^{'}$ ; by [F3] it is monic and irreducible, and by [F1] and the definition of a separable element $P$ is separable over $K^{'}$ . Hence $\gcd (P, P^{'}) = 1$ by [F4], $P^{'} \neq 0$ , and the class of $P^{'}$ is invertible modulo $P$ , so $P^{'} (α) \neq 0$ in the field $K = K^{'} [X] / (P)$ .

F1F2F3F4given

1.2

$Ω_{K^{'} / k}$ has $K^{'}$ -basis $d t_{1}, \dots, d t_{r}$ . Indeed $k [T] \to K^{'}$ , $T_{i} \mapsto t_{i}$ , identifies $k [T]$ with the polynomial ring on the algebraically independent elements $t_{i}$ by [F1], so by [F5] (with $A = k$ and $n = r$ ) the module $Ω_{k [T] / k}$ is free on $d T_{1}, \dots, d T_{r}$ ; since a nonzero element of $k [T]$ maps to a nonzero element of $K^{'} = k (T)$ , the localisation isomorphism of [F6] applies with $U = k [T] ∖ {0}$ and exhibits $Ω_{K^{'} / k} ≅ K^{'} \otimes_{k [T]} Ω_{k [T] / k}$ with the images of $d T_{1}, \dots, d T_{r}$ as a $K^{'}$ -basis, and those images are exactly $d t_{1}, \dots, d t_{r}$ .

F1F5F6

2.1

$Ω_{K / K^{'}} = 0$ : for every $K$ -module $M$ and every $K^{'}$ -derivation $D ⁣ : K \to M$ we have $D (α) = 0$ , because $0 = D (P (α)) = \sum_{i} D (c_{i}) α^{i} + P^{'} (α) D (α) = P^{'} (α) D (α)$ with $c_{i} \in K^{'}$ the coefficients of $P$ and $D (c_{i}) = 0$ , and $P^{'} (α) \neq 0$ by step 1.1; then $D = 0$ on all of $K = K^{'} [α]$ , since a derivation vanishing on $K^{'}$ and on $α$ vanishes on every polynomial in $α$ . By [F8] this says ${Hom}_{K} (Ω_{K / K^{'}}, M) = 0$ for all $M$ , hence $Ω_{K / K^{'}} = 0$ . Applying the transitivity sequence of [F7] to $k \to K^{'} \to K$ , the first map $τ ⁣ : K \otimes_{K^{'}} Ω_{K^{'} / k} \to Ω_{K / k}$ is therefore surjective, and by step 1.2 the elements $τ (1 \otimes d t_{i}) = d t_{i}$ generate $Ω_{K / k}$ as a $K$ -module.

step 1.1step 1.2F7F8

3.1

Independence of the generators. For each $j$ the $K^{'}$ -linear functional on $Ω_{K^{'} / k}$ with $d t_{i} \mapsto δ_{i j}$ exists by step 1.2 and corresponds by [F8] to a $k$ -derivation $\partial_{j} ⁣ : K^{'} \to K^{'}$ with $\partial_{j} (t_{i}) = δ_{i j}$ . Write $P = \sum_{i} c_{i} X^{i}$ and put $β_{j} : = - (\sum_{i} \partial_{j} (c_{i}) α^{i}) P^{'} (α)^{- 1} \in K$ , which is defined by step 1.1. On the polynomial ring $K^{'} [X]$ define $T_{j} (F) : = \sum_{i} \partial_{j} (b_{i}) α^{i} + F^{'} (α) β_{j}$ for $F = \sum_{i} b_{i} X^{i}$ , so that $T_{j} (c F) = c T_{j} (F) + \partial_{j} (c) F (α)$ and $T_{j} (F G) = F (α) T_{j} (G) + G (α) T_{j} (F)$ for all $c \in K^{'}$ and $F, G \in K^{'} [X]$ : these identities follow from additivity of $\partial_{j}$ and the product rules for $\partial_{j}$ and for the formal derivative, and they say that $T_{j}$ is a derivation along the evaluation $K^{'} [X] \to K$ , $X \mapsto α$ , with $T_{j} (X) = β_{j}$ . Moreover $T_{j} (P) = \sum_{i} \partial_{j} (c_{i}) α^{i} + P^{'} (α) β_{j} = 0$ , so $T_{j}$ vanishes on the ideal $(P)$ ; since $K = K^{'} [X] / (P)$ by [F3], $T_{j}$ descends to a well-defined $k$ -derivation $D_{j} ⁣ : K \to K$ extending $\partial_{j}$ , and by [F8] to a $K$ -linear map $Ω_{K / k} \to K$ sending $d t_{i}$ to $D_{j} (t_{i}) = \partial_{j} (t_{i}) = δ_{i j}$ . If now $\sum_{i} c_{i} d t_{i} = 0$ with $c_{i} \in K$ , applying that map gives $c_{j} = 0$ for each $j$ , so $d t_{1}, \dots, d t_{r}$ are linearly independent over $K$ and, with step 2.1, form a $K$ -basis of $Ω_{K / k}$ .

step 1.1step 1.2step 2.1F3F8

4.1

The second part. Assume $char k = 0$ and choose, using [F13], a maximal algebraically independent subset $T_{0} \subseteq K$ over $k$ and a maximal algebraically independent subset $B \subseteq L$ over $K$ . By [F11] the extensions $K / k (T_{0})$ and $L / K (B)$ are algebraic; by [F9] every field of characteristic $0$ is perfect and by [F10] every algebraic extension of a perfect field is separable, so $K / k (T_{0})$ and $L / K (B)$ are separable algebraic. Because $K / k$ is finitely generated, [F12] makes $K / k (T_{0})$ finite, so $T_{0}$ is a separating transcendence basis of $K / k$ in the sense of [F1], and step 3.1 exhibits a finite $K$ -basis $d t_{1}, \dots, d t_{r}$ of $Ω_{K / k}$ .

F1F9F10F11F12F13step 3.1

5.1

Extension of derivations. Let $D ⁣ : K \to L$ be a $k$ -derivation. First extend $D$ to $K (B)$ : each element of $K (B)$ lies in $K (B_{0})$ for some finite $B_{0} \subseteq B$ , and on the polynomial ring $K [B_{0}]$ , for which $Ω_{K [B_{0}] / K}$ is free on the $d b$ , $b \in B_{0}$ , by [F5], the prescription $b \mapsto 0$ for $b \in B_{0}$ , together with $D$ on $K$ , defines a unique $k$ -derivation of $K [B_{0}]$ extending $D$ ; it extends uniquely to the fraction field $K (B_{0})$ by the quotient rule $d (c / d) = (d d c - c d d) / d^{2}$ of [F6], and for $B_{0} \subseteq B_{1}$ the extension on $K (B_{1})$ restricts to the one on $K (B_{0})$ by uniqueness, so a derivation $D_{1} ⁣ : K (B) \to L$ extending $D$ is well defined on the union of the fields $K (B_{0})$ . Next let $x \in L$ ; then $K (B) (x) / K (B)$ is finite separable by step 4.1, hence simple, with minimal polynomial $Q$ of $x$ over $K (B)$ that is separable by [F10], so $Q^{'} (x) \neq 0$ by [F3] and [F4]. Substituting $K (B)$ for $K^{'}$ , $K (B) (x)$ for $K$ and $Q$ for $P$ in the construction of step 3.1 produces an extension of $D_{1}$ to $K (B) (x)$ , and any two extensions of $D_{1}$ to $K (B) (x)$ agree, because their difference vanishes on $K (B)$ and takes at $x$ a value killed by $Q^{'} (x) \neq 0$ . Declaring the value at $x$ to be that unique value defines $D_{2} (x)$ for every $x \in L$ ; it is a derivation because for $x, y \in L$ the field $K (B) (x, y)$ is finite separable over $K (B)$ by [F10] and [F12], carries an extension of $D_{1}$ by the same construction, and on it the derivation laws hold while its restrictions to $K (B) (x)$ and $K (B) (y)$ agree with the unique extensions, so the values $D_{2}$ assigns are additive and satisfy Leibniz.

step 3.1step 4.1F3F4F5F6F8F10F12

6.1

Injectivity. Keep the notation of step 4.1, let $ρ ⁣ : L \otimes_{K} Ω_{K / k} \to Ω_{L / k}$ be the natural map of [F6], and let $x = \sum_{i} c_{i} (1 \otimes d t_{i}) \in \ker ρ$ , the elements $d t_{i}$ being a $K$ -basis of $Ω_{K / k}$ by step 4.1. For each $j$ the functional $d t_{i} \mapsto δ_{i j}$ is a $K$ -linear map $Ω_{K / k} \to L$ , hence by [F8] equals $E_{j} \circ d$ for a $k$ -derivation $E_{j} ⁣ : K \to L$ ; by step 5.1 there is a $k$ -derivation ${\tilde{E}}_{j} ⁣ : L \to L$ extending $E_{j}$ , and by [F8] it induces an $L$ -linear $φ_{j} ⁣ : Ω_{L / k} \to L$ with $φ_{j} (ρ (1 \otimes d t_{i})) = {\tilde{E}}_{j} (t_{i}) = E_{j} (t_{i}) = δ_{i j}$ . Then $0 = φ_{j} (ρ (x)) = c_{j}$ for every $j$ , so $x = 0$ . Hence $ρ$ is injective, which is the second assertion, and the theorem is proved.

step 4.1step 5.1F6F8∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Separable residue and the cotangent sequence of a local algebra

Statement

Let $k$ be a field and let $R$ be a Noetherian local $k$ -algebra with maximal ideal $m$ and residue field $κ = R / m$ . Assume that $κ$ is finitely generated and separably generated over $k$ in the sense of Separating transcendence basis and separably generated extensions. Then $0 ⟶ m / m^{2} ⟶ Ω_{R / k} \otimes_{R} κ ⟶ Ω_{κ / k} ⟶ 0$ is a short exact sequence, the first map sending the class of $x \in m$ to $d x \otimes 1$ . If in addition $κ / k$ is finite separable, then $Ω_{κ / k} = 0$ , so the first map is an isomorphism $m / m^{2} ≅ Ω_{R / k} \otimes_{R} κ$ ; this applies in particular at a closed point of a finite-type $k$ -algebra whose residue field is a finite separable extension of $k$ .

Facts & Assumptions

Given: A field $k$ , a Noetherian local $k$ -algebra $(R, m, κ)$ whose residue field is finitely generated and separably generated over $k$ , and, for the final clause, $κ / k$ finite separable.

[F1]

Universal property of algebraic differentials: for a ring map $A \to B$ and every $B$ -module $M$ , composition with $d$ is an isomorphism ${Hom}_{B} (Ω_{B / A}, M) ≅ {Der}_{A} (B, M)$ , naturally in $M$ .

[F2]

Separating transcendence basis and separably generated extensions: a finitely generated extension that is separably generated has algebraically independent elements $t_{1}, \dots, t_{r}$ over the base whose residual extension is finite separable; so $κ = k (t_{1}, \dots, t_{r}) (α) = k (T) (α)$ with $κ / k (T)$ finite separable.

[F3]

A finite extension generated by elements all but possibly one of which are separable is simple: every finite separable extension is simple; this is applied both to $κ / k (T)$ and, in the final clause, to $κ / k$ .

[F4]

The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element: for $α$ algebraic over a field $F$ the evaluation map $F [X] \to F (α)$ has kernel generated by a monic irreducible $P$ , so $F (α) ≅ F [X] / (P)$ and $f (α) = 0$ implies $P ∣ f$ .

[F5]

Separable algebraic elements and separable extensions: $α$ is separable over $F$ when it is algebraic over $F$ and its minimal polynomial over $F$ is separable.

[F6]

A nonzero polynomial over a field is separable exactly when its gcd with its derivative is $1$ : for $0 \neq f$ over a field, $f$ is separable if and only if $\gcd (f, f^{'}) = 1$ .

Proof

1.1

A conormal computation. Let $S$ be any commutative $k$ -algebra, let $I \subseteq S$ be an ideal and $B = S / I$ . Then $I / I^{2} \to B \otimes_{S} Ω_{S / k} \to Ω_{B / k} \to 0$ is exact, the first map sending the class of $x \in I$ to $1 \otimes d x$ . The second map exists by [F1] applied to $S \to B$ , and it is surjective because the elements $d b$ generate $Ω_{B / k}$ ; the composite is zero because $x \in I$ maps to $d (x + I) = 0$ . The first map is well defined because for $x, y \in I$ one has $d (x y) = x d y + y d x$ , which lies in the image of $I \otimes_{S} Ω_{S / k}$ in $Ω_{S / k}$ , so the classes of $x y$ and of $0$ in $I / I^{2}$ have the same image. Let $Q$ be the cokernel of the first map. A $B$ -linear map $Q \to M$ is the same thing as a $k$ -derivation $D ⁣ : S \to M$ with $D (x) = 0$ for all $x \in I$ , because $B$ -linear maps out of $B \otimes_{S} Ω_{S / k}$ correspond by [F1] to $k$ -derivations of $S$ , and the quotient imposes exactly the vanishing on the classes $d x$ , $x \in I$ . Such a $D$ factors through a $k$ -derivation $B \to M$ : it is constant on cosets, since $D (x + b) = D (b)$ for $x \in I$ , and it satisfies Leibniz on classes, since for $x \in I$ the identity $D ((x + b) b^{'}) = x D (b^{'}) + b^{'} D (x) + b D (b^{'}) = b D (b^{'}) + b^{'} D (b)$ holds because $x M = 0$ for the $B$ -module $M$ . Conversely every $k$ -derivation of $B$ pulled back along $S \to B$ is such a $D$ . By [F1] the functor $M \mapsto {Hom}_{B} (Q, M)$ is therefore isomorphic to $M \mapsto {Hom}_{B} (Ω_{B / k}, M)$ , so the canonical map $Q \to Ω_{B / k}$ is an isomorphism.

F1algebra

1.2

Preparation of the section. Write $κ = k (T) (α)$ with $T = (t_{1}, \dots, t_{r})$ as in [F2] and [F3], and let $P \in k (T) [X]$ be the minimal polynomial of $α$ over $k (T)$ ; by [F4] and [F5] the polynomial $P$ is monic, irreducible and separable, so $P^{'} (α) \neq 0$ by [F6], since $P^{'}$ is nonzero and $\gcd (P, P^{'}) = 1$ makes its class a unit of $κ = k (T) [X] / (P)$ . Put $\overline{R} : = R / m^{2}$ , a local ring with maximal ideal $m / m^{2}$ and residue field $κ$ , and write $π ⁣ : \overline{R} \to κ$ for the quotient map. Choose $y_{1}, \dots, y_{r}, a \in \overline{R}$ with $π (y_{i}) = t_{i}$ and $π (a) = α$ , and let $φ_{0} ⁣ : k [T] \to \overline{R}$ be the $k$ -algebra map with $T_{i} \mapsto y_{i}$ . It is injective because the $t_{i}$ are algebraically independent over $k$ , so $k [y_{1}, \dots, y_{r}]$ is a polynomial ring and every nonzero element $q (y)$ of it is a unit of the local ring $\overline{R}$ , since $π (q (y)) = q (t) \neq 0$ places it outside the maximal ideal. By the universal property of localisation, $φ_{0}$ extends to a $k$ -algebra map $φ ⁣ : k (T) \to \overline{R}$ , and $π \circ φ$ is the inclusion $k (T) ↪ κ$ because they agree on the generators $T_{i}$ .

F2F3F4F5F6

1.3

Finite separable residue. If $κ / k$ is finite separable, then by [F3] there is $α \in κ$ with $κ = k (α)$ ; its minimal polynomial $P \in k [X]$ is separable by [F5], so $P^{'} (α) \neq 0$ by [F4] and [F6]. Every $k$ -derivation $D ⁣ : κ \to M$ into a $κ$ -module satisfies $0 = D (P (α)) = \sum_{i} D (c_{i}) α^{i} + P^{'} (α) D (α) = P^{'} (α) D (α)$ with $c_{i} \in k$ and $D (c_{i}) = 0$ , so $D (α) = 0$ because $κ$ is a field, and then $D = 0$ on $κ = k [α]$ . By [F1] this forces $Ω_{κ / k} = 0$ .

F1F3F4F5F6

2.1

Applying step 1.1 with $S = R$ , $I = m$ and $B = κ$ gives exactness of $m / m^{2} \to Ω_{R / k} \otimes_{R} κ \to Ω_{κ / k} \to 0$ ; it remains to prove that the first map is injective.

step 1.1

2.2

Correction of the lift. With the notation of step 1.2 form $δ : = φ (P) (a) \in \overline{R}$ , where $φ (P) \in \overline{R} [X]$ is $P$ with coefficients transported by $φ$ . Then $π (δ) = P (α) = 0$ , so $δ \in m / m^{2}$ , and $(m / m^{2})^{2} = 0$ in $\overline{R}$ because $m \cdot m \subseteq m^{2}$ . Also $π (φ (P^{'}) (a)) = P^{'} (α) \neq 0$ , so $φ (P^{'}) (a)$ is a unit of $\overline{R}$ . Set $a^{'} : = a - φ (P^{'}) (a)^{- 1} δ$ , an element of $\overline{R}$ with the same residue $π (a^{'}) = π (a) = α$ . Taylor expansion in the commutative ring $\overline{R}$ terminates after the linear term because $a^{'} - a \in m / m^{2}$ has square zero, so $φ (P) (a^{'}) = φ (P) (a) + φ (P^{'}) (a) (a^{'} - a) = δ - δ = 0$ . Hence the $k$ -algebra map $k (T) [X] \to \overline{R}$ with $T_{i} \mapsto y_{i}$ and $X \mapsto a^{'}$ kills $P$ , and by [F4] it descends along $κ ≅ k (T) [X] / (P)$ to a $k$ -algebra map $s ⁣ : κ \to \overline{R}$ with $π \circ s = {id}_{κ}$ .

step 1.2F4algebra

3.1

A derivation inverse to $d$ . Define $D : = {id}_{\overline{R}} - s \circ π ⁣ : \overline{R} \to \overline{R}$ ; its image lies in $\ker π = m / m^{2}$ , and $D (x) = x$ for $x \in m / m^{2}$ because $π (x) = 0$ . For $u, v \in \overline{R}$ write $u = s (π (u)) + D (u)$ and $v = s (π (v)) + D (v)$ ; since $π$ is a ring map, $D (u)$ and $D (v)$ have square zero and their product with any element of $m / m^{2}$ vanishes, so expanding gives $u v = s (π (u v)) + u D (v) + v D (u)$ and hence $D (u v) = u D (v) + v D (u)$ . Thus $D$ is a $k$ -derivation of $\overline{R}$ into the $\overline{R}$ -module $m / m^{2}$ , and composing with $R \to \overline{R}$ gives a $k$ -derivation $R \to m / m^{2}$ whose value at $x \in m$ is the class of $x$ modulo $m^{2}$ .

step 2.2algebra

4.1

Conclusion. By [F1] the derivation of step 3.1 corresponds to an $R$ -linear map $Ω_{R / k} \to m / m^{2}$ , which factors through $Ω_{R / k} \otimes_{R} κ \to m / m^{2}$ because the target is annihilated by $m$ , giving a $κ$ -linear $θ$ with $θ (1 \otimes d x) = x m o d m^{2}$ for $x \in R$ . The first map of step 2.1 sends the class of $x \in m$ to $1 \otimes d x$ , so $θ$ is a left inverse of it and that map is injective; with step 2.1, the displayed sequence is exact.

step 2.1step 3.1F1

5.1

Finite separable residue concluded. By step 1.3 the hypothesis of the final clause gives $Ω_{κ / k} = 0$ , so the exact sequence of step 4.1 reads $0 \to m / m^{2} \to Ω_{R / k} \otimes_{R} κ \to 0$ , that is, $m / m^{2} ≅ Ω_{R / k} \otimes_{R} κ$ via the first map.

step 1.3step 4.1∎

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Extension of scalars of a scheme along a field extension

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field, let $X$ be a scheme (Schemes) with a morphism $X \to Spec k$ , and let $K / k$ be a field extension. Then:

Construction. For every affine open $U = Spec A \subseteq X$ the ring $A$ is a $k$ -algebra, the $k$ -algebra map $A \to A \otimes_{k} K$ gives a morphism $Spec (A \otimes_{k} K) \to U$ (Affine schemes are contravariantly equivalent to commutative rings), and for a principal open $Spec A_{f} = Spec B_{g}$ of $X$ the canonical identifications $A_{f} = B_{g} = Γ (W, O_{X})$ (Sections and restrictions on distinguished opens of an affine scheme) identify the corresponding open subschemes of $Spec (A \otimes_{k} K)$ and $Spec (B \otimes_{k} K)$ . These data satisfy the identity and cocycle conditions, so Gluing affine schemes along compatible open isomorphisms glues the affine schemes $Spec (A \otimes_{k} K)$ , over all affine opens $U = Spec A$ of $X$ , to a $K$ -scheme $X_{K}$ with a morphism $X_{K} \to X$ over $k$ .
Independence of the cover. If $X_{K}$ and $X_{K}^{'}$ arise from two affine covers of $X$ by this construction, there is a unique isomorphism $X_{K} \to X_{K}^{'}$ commuting with both the morphisms to $X$ and the structure morphisms to $Spec K$ .
Affine restrictions and fibres. For every affine open $U = Spec A \subseteq X$ the restriction of $X_{K} \to X$ over $U$ is canonically $Spec (A \otimes_{k} K) \to Spec A$ . For $x \in X$ with residue field $κ (x)$ (The residue field at a point of an affine scheme) and any affine open $U = Spec A$ containing $x$ , the fibre of $X_{K} \to X$ over $x$ , computed as $Spec ((A \otimes_{k} K) \otimes_{A} κ (x))$ , is independent of $U$ up to canonical isomorphism and is canonically $Spec (κ (x) \otimes_{k} K)$ .
Transitivity. For a tower of fields $K \subseteq L$ the canonical morphism $(X_{K})_{L} \to X_{L}$ is an isomorphism, canonically over $X$ .

The Axiom of Choice is used only to present the affine cover as a family of affine opens indexed by the points of $X$ ; the same construction runs on the set of all affine opens of $X$ , which needs no choice. The assumption is declared and is inherited by consumers, since the statement promises it.

Facts & Assumptions

Given: A field $k$ , a scheme $X$ with a morphism to $Spec k$ , a field extension $K / k$ (and a tower $K \subseteq L$ for clause 4), and the Axiom of Choice.

[F1]

Gluing affine schemes along compatible open isomorphisms: affine schemes equipped with open subschemes and isomorphisms on overlaps satisfying the identity and cocycle conditions glue to a scheme, uniquely up to unique isomorphism respecting the chart identifications, and the given affine schemes become an open affine cover.

[F2]

Affine schemes are contravariantly equivalent to commutative rings: ${Hom}_{C R i n g} (A, B) ≅ {Hom}_{L R S} (Spec B, Spec A)$ naturally, and $Spec$ is a contravariant equivalence with quasi-inverse global sections.

[F3]

Sections and restrictions on distinguished opens of an affine scheme: for $f \in A$ , $Γ (D (f), O) = A_{f}$ , and if $D (g) \subseteq D (f)$ the restriction is the canonical localisation $A_{f} \to A_{g}$ .

[F4]

Intersections of affine opens admit principal affine covers: if $U, V$ are affine open subschemes of a scheme, then $U \cap V$ is covered by open subschemes that are principal opens in $U$ and principal opens in affine open charts of $V$ .

[F5]

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

[F6]

Universal property of localisation: maps that invert $S$ factor uniquely through $S^{- 1} R$ : a unital ring homomorphism $R \to A$ carrying every element of $S$ to a unit factors uniquely through $S^{- 1} R$ .

[F7]

Universal mapping property of the tensor product of commutative algebras: $A \otimes_{R} B$ is the coproduct of the commutative $R$ -algebras $A$ and $B$ , with the universal map $h (a \otimes b) = f (a) g (b)$ for each pair of $R$ -algebra maps $f, g$ into a common $R$ -algebra $C$ .

[F8]

Associativity of tensor products for compatible bimodules: there is a canonical isomorphism $(M \otimes_{R} N) \otimes_{S} P \to M \otimes_{R} (N \otimes_{S} P)$ , $((m \otimes n) \otimes p) \mapsto m \otimes (n \otimes p)$ , respecting compatible outer module actions.

[F9]

The residue field at a point of an affine scheme: $κ (x) = O_{X, x} / m_{x}$ , and for $x = p$ in an affine spectrum the canonical isomorphisms $κ (p) ≅ A_{p} / p A_{p} ≅ Frac (A / p)$ hold.

[F10]

Schemes: a scheme is a locally ringed space every point of which has an open neighbourhood that is an affine scheme with the restricted structure sheaf.

[F11]

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

Proof

1.1

Principal opens under base change. Let $A$ be a $k$ -algebra and $f \in A$ . The $k$ -algebra map $A_{f} \to (A \otimes_{k} K)_{f \otimes 1}$ , $a / f^{n} \mapsto (a \otimes 1) (f \otimes 1)^{- n}$ , is well defined by [F6], and after tensoring with $K$ gives a $(A \otimes_{k} K)_{f \otimes 1}$ -linear map $A_{f} \otimes_{k} K \to (A \otimes_{k} K)_{f \otimes 1}$ , because $(A \otimes_{k} K)_{f \otimes 1}$ is an $A_{f}$ -algebra by [F7]. Conversely $a \otimes λ \mapsto (a / 1) \otimes λ$ is a $k$ -algebra map $A \otimes_{k} K \to A_{f} \otimes_{k} K$ carrying $f \otimes 1$ to a unit, so by [F6] it factors through a map $(A \otimes_{k} K)_{f \otimes 1} \to A_{f} \otimes_{k} K$ . The two maps are inverse on the generators $a \otimes λ$ and $1 / (f \otimes 1)$ ; composing with $Spec$ by [F2], the principal open $D (f \otimes 1)$ of $Spec (A \otimes_{k} K)$ is canonically $Spec (A_{f} \otimes_{k} K)$ , compatibly with further principal localisations $f \mapsto f^{n}$ and with the restriction maps of [F3].

F2F3F6F7F10F11

1.2

Fibre rings. Let $A$ be a $k$ -algebra and $p \in Spec A$ with residue field $κ (p)$ as in [F9]; then $κ (p) = A_{p} / p A_{p} = A / p \otimes_{A} A_{p}$ , and the composite of the canonical isomorphisms $(A \otimes_{k} K) \otimes_{A} κ (p) ≅ κ (p) \otimes_{A} (A \otimes_{k} K) ≅ (κ (p) \otimes_{A} A) \otimes_{k} K ≅ κ (p) \otimes_{k} K$ of [F5] and [F8] sends $(a \otimes λ) \otimes c$ to $(c a) \otimes λ$ . Hence there is a canonical $κ (p)$ -linear isomorphism $(A \otimes_{k} K) \otimes_{A} κ (p) ≅ κ (p) \otimes_{k} K$ .

F5F8F9

1.3

Common principal neighbourhoods. For affine opens $U = Spec A$ , $V = Spec B$ and $x \in U \cap V$ , first take $x \in D_{U} (f) \subseteq V$ using the principal-open basis. Then take $x \in D_{V} (g) \subseteq D_{U} (f)$ . The restriction of $g$ to $D_{U} (f)$ is $a / f^{r} \in A_{f}$ by [F3], and its nonvanishing locus there is both $D_{V} (g)$ and $D_{U} (f a)$ : the equality follows by applying the residue-field maps of the open immersion to this section. Thus $D_{V} (g) = D_{U} (f a)$ is principal in both original affines. This supplies the common principal refinements needed below, with independent denominators in the two coordinate rings.

F2F3F4F9algebra

2.1

Gluing. Let $X$ be a scheme over $k$ and let ${U_{i} = Spec A_{i}}$ be a family of affine opens covering $X$ ; each $A_{i}$ is a $k$ -algebra because $X \to Spec k$ restricts to $U_{i}$ . For each $i$ the $k$ -algebra map $A_{i} \to A_{i} \otimes_{k} K$ gives by [F2] a morphism $Spec (A_{i} \otimes_{k} K) \to U_{i} \subseteq X$ , and the affine pieces over distinct $i$ are to be identified over the principal opens. Whenever $W$ is an open subscheme of $X$ which is principal in $U_{i}$ and in $U_{j}$ , say $W = Spec (A_{i})_{f} = Spec (A_{j})_{g}$ , the rings $(A_{i})_{f}$ and $(A_{j})_{g}$ are both $Γ (W, O_{X})$ by [F3] and hence canonically equal, and the identity of rings induces by step 1.1 an identification of the corresponding open subschemes $Spec ((A_{i})_{f} \otimes_{k} K)$ and $Spec ((A_{j})_{g} \otimes_{k} K)$ . These identifications are induced by identities of section rings and are therefore compatible: the identity and cocycle conditions hold on triple overlaps because all the identifications are the canonical comparison of $Γ (W, O_{X}) \otimes_{k} K$ with itself. Since step 1.3 covers every overlap by such common principal opens, the data satisfy the hypotheses of [F1], which glues the schemes $Spec (A_{i} \otimes_{k} K)$ to a scheme $X_{K}$ with open affine cover ${Spec (A_{i} \otimes_{k} K)}$ , and the morphisms to $X$ glue to $X_{K} \to X$ . The maps to $Spec K$ induced by $λ \mapsto 1 \otimes λ$ also agree on overlaps, so they glue to the $K$ -scheme structure on $X_{K}$ .

F1F2F3F4step 1.1step 1.3

3.1

Affine restriction. Let $U = Spec A$ be any affine open of $X$ . By step 1.3 cover each $U_{i} \cap U$ by common principal opens $W = D_{U_{i}} (f) = D_{U} (h)$ . Their section rings $(A_{i})_{f}$ and $A_{h}$ are canonically identified by [F3]. Step 1.1 identifies the corresponding base-changed opens with $Spec (Γ (W, O_{X}) \otimes_{k} K)$ on either side. These opens cover the inverse image of $U$ in $X_{K}$ and cover $Spec (A \otimes_{k} K)$ : a principal cover remains a cover under inverse image, since $D (h \otimes 1)$ is precisely the inverse image of $D (h)$ . The identifications agree on common refinements by step 1.1, so glue to an isomorphism over both $U$ and $Spec K$ by [F1]. Hence the restriction is the asserted affine base change.

F1F2F3step 1.1step 1.3step 2.1

3.2

Independence of the cover. Let ${U_{i}}$ and ${V_{j}}$ be two affine covers of $X$ . By step 1.3 the family of open subschemes $W$ of $X$ that are principal in some $U_{i}$ and in some $V_{j}$ covers $X$ . For such a $W = Spec B$ , with $B = Γ (W, O_{X})$ by [F3], the construction of step 2.1 attaches to $W$ the affine scheme $Spec (B \otimes_{k} K)$ in the glueing over ${U_{i}}$ and, by the same computation, in the glueing over ${V_{j}}$ : in both cases $W$ arises as a principal open of an affine chart, and the attached piece is $Spec$ of the localisation of the chart ring tensored with $K$ , which is $B \otimes_{k} K$ by step 1.1. Both glued schemes are therefore obtained by glueing the same family ${Spec (B \otimes_{k} K)}$ along the same canonical identifications over principal opens of $W$ . By the uniqueness clause of [F1], applied to the two open affine covers of $X_{K}$ and $X_{K}^{'}$ , the canonical chart identifications glue to an isomorphism $X_{K} \to X_{K}^{'}$ over $X$ and $Spec K$ . Any other such isomorphism must preserve each inverse image of $W$ ; on its ring $Γ (W, O_{X}) \otimes_{k} K$ , the induced map fixes both factors because it is over $W$ and over $K$ . It is therefore the identity by [F7]. These opens cover, proving uniqueness with both compatibilities.

F1F3F7step 1.1step 1.3step 2.1

4.1

Fibres. Let $x \in X$ and let $U = Spec A \subseteq X$ be an affine open containing $x$ . By step 3.1 the preimage of $U$ in $X_{K}$ is $Spec (A \otimes_{k} K)$ over $U$ , and the scheme over $κ (x)$ attached to the point $x$ of that affine piece is $Spec ((A \otimes_{k} K) \otimes_{A} κ (x))$ , which by step 1.2 is canonically $Spec (κ (x) \otimes_{k} K)$ over $Spec κ (x)$ . If $V = Spec B$ is a second affine open containing $x$ , choose $W$ principal in $U$ and in $V$ with $x \in W$ , say $W = Spec (A_{f}) = Spec (B_{g})$ , using step 1.3 and [F3]; then $(A \otimes_{k} K) \otimes_{A} A_{f} ≅ A_{f} \otimes_{k} K$ and $(B \otimes_{k} K) \otimes_{B} B_{g} ≅ B_{g} \otimes_{k} K$ are canonically the same ring, so tensoring the identification with $κ (x)$ over the common ring $A_{f} = B_{g} = Γ (W, O_{X})$ identifies $(A \otimes_{k} K) \otimes_{A} κ (x)$ with $(B \otimes_{k} K) \otimes_{B} κ (x)$ canonically. Hence the fibre is independent of the affine neighbourhood and is $Spec (κ (x) \otimes_{k} K)$ as asserted.

F3F4step 1.1step 1.2step 3.1

5.1

Transitivity. Let $K \subseteq L$ be a tower, and write $(X_{K})_{L}$ for the construction applied over the base field $K$ to the extension $L / K$ . The affine pieces $Spec (A_{i} \otimes_{k} K)$ constructed in step 2.1 form an affine cover of $X_{K}$ , so the construction of $(X_{K})_{L}$ glues the schemes $Spec ((A_{i} \otimes_{k} K) \otimes_{K} L)$ . The canonical isomorphisms $(A_{i} \otimes_{k} K) \otimes_{K} L ≅ A_{i} \otimes_{k} L$ of [F7] and [F8] are compatible with the transition identifications of step 2.1, because those are induced by identities of section rings; hence $X_{L}$ , which is glued from the pieces $Spec (A_{i} \otimes_{k} L)$ , and $(X_{K})_{L}$ are glued from corresponding pieces with corresponding identifications. By step 3.2 (applied to the two covers of the same scheme, or directly by the uniqueness clause of [F1]) the displayed chart isomorphisms glue to an isomorphism $(X_{K})_{L} \to X_{L}$ over $X$ and $Spec L$ . It is unique with both compatibilities, by the same two-factor argument as step 3.2, proving clause 4.

F1F7F8step 2.1step 3.1step 3.2∎

DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-27Open item page →

Geometrically regular algebras and geometrically regular fibres

Definition

Let $k$ be a field. A $k$ -algebra $A$ of finite type (Finitely presented modules and finitely presented algebras) is geometrically regular over $k$ when for every finitely generated field extension $K / k$ (Finitely generated field extensions $F (a_{1}, \dots, a_{r})$ ) the $K$ -algebra $A \otimes_{k} K$ (Universal mapping property of the tensor product of commutative algebras) is a regular Noetherian ring (regular noetherian ring). Since a finitely generated extension $K / k$ makes $A \otimes_{k} K$ a finite-type $K$ -algebra, the Noetherian condition in that clause is automatic; the regularity is the content. The quantifier is over the finitely generated extensions only, and the later field-test lemma shows that regularity for those is equivalent to regularity after every field extension of $k$ .

Now let $R \to S$ be a homomorphism of commutative rings with $S$ finitely presented as an $R$ -algebra (Finitely presented modules and finitely presented algebras), let $q \in Spec S$ and let $p = q \cap R$ . The fibre of $R \to S$ over $p$ is $S \otimes_{R} κ (p)$ , a $κ (p)$ -algebra, and it is geometrically regular at $q$ when for every field extension $K / κ (p)$ and every prime $Q$ of $(S \otimes_{R} κ (p)) \otimes_{κ (p)} K$ lying over the image of $q$ in $S \otimes_{R} κ (p)$ , the local ring $((S \otimes_{R} κ (p)) \otimes_{κ (p)} K)_{Q}$ is regular. A fibre is geometrically regular when it is geometrically regular at each of its points, and the condition is imposed on every field extension $K / κ (p)$ , not only on the finitely generated ones; an empty fibre satisfies the pointwise condition vacuously, and the affine line over the residue field is the model case.

Three conventions belong to the definition. First, geometric regularity of a finite-type $k$ -algebra is a statement about all scalar extensions $A \otimes_{k} K$ , so it is strictly stronger than regularity of $A$ and is defined without using smoothness of the structural morphism; the equivalence with local standard smoothness is a theorem, not part of the definition. Second, the fibre condition is pointwise at a prime $q$ and ranges over all field extensions of the residue field $κ (p)$ , so that a rational point over an algebraic closure of $κ (p)$ is covered without appealing to any later theorem. Third, the hypothesis that $S$ is finitely presented over $R$ is part of the definition of the fibre condition, since it is what lets the standard smooth presentations and the local presentation theorem apply to it.

Remarks

The field-case equivalence with local standard smoothness is proved in Locally standard smooth iff flat with geometrically regular fibres, clause 3, under the Axiom of Choice assumed there. Standard smooth presentations and locally standard smooth maps supplies the presentation terminology, rather than the equivalence theorem.

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Standard smooth presentations and locally standard smooth maps

Definition

Let $R$ be a commutative ring. A standard smooth presentation of an $R$ -algebra $S$ consists of integers $n \geq c \geq 0$ , elements $f_{1}, \dots, f_{c}$ of the polynomial ring $P = R [x_{1}, \dots, x_{n}]$ (The polynomial ring $R [x_{i} : i \in I]$ as finitely supported coefficient families on monomials) and an element $g \in P$ such that $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ as $R$ -algebras, and such that the Jacobian matrix of (Differentials of a polynomial quotient and the Jacobian cokernel) $(\frac{\partial f_{j}}{\partial x_{i}})_{\begin{array}{c} 1 \leq j \leq c \\ 1 \leq i \leq n \end{array}}$ has a $c \times c$ minor whose image in $S$ is a unit, that is, an invertible element. The integer $n - c \geq 0$ is the relative dimension of the presentation. The case $c = 0$ is allowed and is exactly a localisation of a polynomial ring, $S ≅ (R [x_{1}, \dots, x_{n}])_{g}$ ; the case $n = c = 0$ presents $S = R []_{g}$ for $g \in R$ , that is a localisation of $R$ itself.

For a homomorphism $R \to S$ of commutative rings that is finitely presented as an $R$ -algebra (Finitely presented modules and finitely presented algebras) and a prime $q \in Spec S$ , the map is standard smooth at $q$ , or has a standard smooth presentation at $q$ , when there is $h \in S ∖ q$ such that $S_{h}$ admits a standard smooth presentation over $R$ . It is locally standard smooth when this holds at every prime of $S$ ; equivalently, when every point of $Spec S$ has an affine open neighbourhood on which $S$ is presented by a single standard smooth presentation.

Two conventions are part of the definition. First, the invertible minor may be assumed to sit in the first $c$ columns: if a minor on columns $i_{1} < \dots < i_{c}$ is a unit, the automorphism of $R [x_{1}, \dots, x_{n}]$ permuting the variables so that these become the first $c$ variables carries the presentation to one whose minor in the first $c$ columns is that unit, and the relative dimension $n - c$ is unchanged. Second, a further principal localisation can be absorbed into the presentation: adjoining a variable $z$ with the single equation $z g - 1$ to a presentation produces a standard smooth presentation of the localisation, the new equation contributing a diagonal entry that keeps a block minor invertible, and it changes $n$ and $c$ by the same amount, so that the relative dimension is again unchanged.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Base change of standard smooth presentations

Statement

Let $R \to R^{'}$ be a homomorphism of commutative rings and let $R \to S$ be an $R$ -algebra carrying a standard smooth presentation (Standard smooth presentations and locally standard smooth maps) of relative dimension $n - c$ , with $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ and with the leading $c \times c$ Jacobian minor $h = \det (\frac{\partial f_{j}}{\partial x_{i}})_{1 \leq i, j \leq c}$ (in the sense of Differentials of a polynomial quotient and the Jacobian cokernel) mapping to a unit of $S$ ; the conventions of the definition allow the invertible minor to be assumed in the first $c$ columns. Let $f_{j}^{'}, g^{'}, h^{'}$ be the images of $f_{j}, g, h$ under the induced map $R [x_{1}, \dots, x_{n}] \to R^{'} [x_{1}, \dots, x_{n}]$ and put $S^{'} = (R^{'} [x_{1}, \dots, x_{n}] / (f_{1}^{'}, \dots, f_{c}^{'}))_{g^{'}} .$ Then:

there is a unique $R^{'}$ -algebra isomorphism $Φ ⁣ : R^{'} \otimes_{R} S \to S^{'}$ with $Φ (a \otimes \overline{F} / g^{N}) = a \overline{F^{'}} / (g^{'})^{N}$ , where $\overline{F}$ is the image of $F \in R [x_{1}, \dots, x_{n}]$ in $S$ and $\overline{F^{'}}$ its image in $S^{'}$ ;
$S^{'}$ is standard smooth over $R^{'}$ with the same $n$ , $c$ and relative dimension $n - c$ ; explicitly, the image $h^{'}$ of $h$ is a unit of $S^{'}$ .

No hypothesis is placed on $R \to R^{'}$ , and the relative dimension is unchanged.

Facts & Assumptions

Given: A ring homomorphism $R \to R^{'}$ and a standard smooth presentation $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ over $R$ whose leading $c \times c$ minor $h$ maps to a unit in $S$ .

[F1]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation consists of integers $n \geq c \geq 0$ , elements $f_{1}, \dots, f_{c} \in R [x_{1}, \dots, x_{n}]$ and $g \in R [x_{1}, \dots, x_{n}]$ with $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ , such that the Jacobian matrix $(\partial f_{j} / \partial x_{i})$ has a $c \times c$ minor whose image in $S$ is a unit; $n - c$ is the relative dimension, and the invertible minor may be assumed to lie in the first $c$ columns.

[F2]

[F3]

Universal property of a polynomial ring on an arbitrary family of indeterminates: for a ring homomorphism $φ ⁣ : R \to T$ and any family $(t_{i})_{i \in I}$ in $T$ there is a unique ring homomorphism $R [x_{i} : i \in I] \to T$ restricting to $φ$ on $R$ with $x_{i} \mapsto t_{i}$ .

[F4]

Universal property of localisation: maps that invert $S$ factor uniquely through $S^{- 1} R$ : if $f ⁣ : R \to A$ is a unital homomorphism of commutative rings carrying a multiplicative set $T$ into the units of $A$ , there is a unique unital ring homomorphism $\tilde{f} ⁣ : T^{- 1} R \to A$ with $\tilde{f} \circ λ_{T} = f$ , namely $\tilde{f} (r / t) = f (r) f (t)^{- 1}$ .

[F5]

A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring: a ring homomorphism whose kernel contains a two-sided ideal $I$ factors uniquely through the quotient $R \to R / I$ .

[F6]

Universal mapping property of the tensor product of commutative algebras: for commutative $R$ -algebras $A, B, C$ and $R$ -algebra homomorphisms $f ⁣ : A \to C$ , $g ⁣ : B \to C$ there is a unique $R$ -algebra homomorphism $h ⁣ : A \otimes_{R} B \to C$ with $h (a \otimes 1) = f (a)$ and $h (1 \otimes b) = g (b)$ , namely $h (a \otimes b) = f (a) g (b)$ .

[F7]

The tensor product of $R$ -algebras has multiplication $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ : $A \otimes_{R} B$ is a commutative $R$ -algebra with $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ and unit $1 \otimes 1$ ; in particular $a \mapsto a \otimes 1$ and $b \mapsto 1 \otimes b$ are ring homomorphisms.

[F8]

Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions: the localisation $T^{- 1} R$ of a commutative ring at a multiplicative subset $T$ is a commutative ring, $λ_{T} ⁣ : R \to T^{- 1} R$ , $r \mapsto r / 1$ , is a ring homomorphism and each $t \in T$ maps to a unit.

[F9]

Differentials of a polynomial quotient and the Jacobian cokernel: $Ω_{P / A}$ is free on $d x_{1}, \dots, d x_{n}$ for $P = A [x_{1}, \dots, x_{n}]$ , the partial derivatives $\partial_{i}$ are computed on the monomial basis by $\partial_{i} (x^{a}) = a_{i} x^{a - e_{i}}$ and extended $A$ -linearly, $d f = \sum_{i} \partial_{i} f d x_{i}$ , and for $I = (f_{1}, \dots, f_{c})$ the module $Ω_{P / I / A}$ is the cokernel of the Jacobian matrix $(\partial_{i} f_{j})$ .

Proof

1.1

Notation. Put $P : = R [x_{1}, \dots, x_{n}]$ , $I : = (f_{1}, \dots, f_{c}) \subseteq P$ , $A : = P / I$ , so that $S = A_{g}$ ; put $P^{'} : = R^{'} [x_{1}, \dots, x_{n}]$ , $I^{'} : = (f_{1}^{'}, \dots, f_{c}^{'}) \subseteq P^{'}$ and $S^{'} : = (P^{'} / I^{'})_{g^{'}}$ . All four are commutative rings by [F2] and [F8], and the coefficient-change map $P \to P^{'}$ , $x_{i} \mapsto x_{i}$ , is a ring homomorphism by [F3] applied to $R \to R^{'}$ .

F1F2F3F8

2.1

The map $ψ ⁣ : S \to S^{'}$ . The composite $P \to P^{'} \to P^{'} / I^{'} \to S^{'}$ kills $I$ and carries $g$ to $g^{'}$ , which is a unit of $S^{'}$ by [F8]; by [F5] it factors uniquely through $P / I = A$ , and by [F4] the resulting map $A \to S^{'}$ factors uniquely through $A_{g} = S$ . This gives a unique ring homomorphism $ψ ⁣ : S \to S^{'}$ with $ψ (\overline{F} / g^{N}) = \overline{F^{'}} / (g^{'})^{N}$ for $F \in P$ , $N \geq 0$ .

F4F5step 1.1F8

2.2

The map $τ ⁣ : S^{'} \to R^{'} \otimes_{R} S$ . By [F7] the assignment $a \mapsto a \otimes 1$ is a ring homomorphism $R^{'} \to R^{'} \otimes_{R} S$ , and $1 \otimes \overline{x_{i}} \in R^{'} \otimes_{R} S$ are elements; by [F3] there is a unique ring homomorphism $P^{'} = R^{'} [x_{1}, \dots, x_{n}] \to R^{'} \otimes_{R} S$ restricting to $a \mapsto a \otimes 1$ and sending $x_{i} \mapsto 1 \otimes \overline{x_{i}}$ . Its kernel contains $I^{'}$ , because $f_{j}^{'} \mapsto 1 \otimes \overline{f_{j}} = 0$ , and it sends $g^{'}$ to $1 \otimes \overline{g}$ , which is a unit with inverse $1 \otimes {\overline{g}}^{- 1}$ in view of [F7] and $g \cdot g^{- 1}$ invertible in $S$ . Applying [F5] and then [F4] gives a unique ring homomorphism $τ ⁣ : S^{'} = (P^{'} / I^{'})_{g^{'}} \to R^{'} \otimes_{R} S$ over $R^{'}$ .

F3F4F5F7step 1.1

3.1

The map $Φ ⁣ : R^{'} \otimes_{R} S \to S^{'}$ . The identity map of $R^{'}$ and the map $ψ ⁣ : S \to S^{'}$ of step 2.1 are $R$ -algebra maps into $S^{'}$ that agree on $R$ ; the latter is induced by the coefficient-change map $R \to R^{'}$ . Hence [F6] provides a unique $R^{'}$ -algebra homomorphism $Φ ⁣ : R^{'} \otimes_{R} S \to S^{'}$ with $Φ (a \otimes 1) = a$ and $Φ (1 \otimes y) = ψ (y)$ , that is, $Φ (a \otimes y) = a ψ (y)$ ; on the elements $a \otimes \overline{F} / g^{N}$ it is $Φ (a \otimes \overline{F} / g^{N}) = a \overline{F^{'}} / (g^{'})^{N}$ .

F6step 2.1step 2.2

4.1

$Φ \circ τ = {i d}_{S^{'}}$ and $τ \circ Φ = {i d}_{R^{'} \otimes_{R} S}$ . Both composites are $R^{'}$ -algebra homomorphisms. The $R^{'}$ -algebra $S^{'}$ is generated by the images of the $x_{i}$ and by $(g^{'})^{- 1}$ : every element of $(P^{'} / I^{'})_{g^{'}}$ is a class $u / (g^{'})^{N}$ with $u \in P^{'}$ , and $u$ is an $R^{'}$ -linear combination of monomials in the $x_{i}$ . A ring homomorphism out of $S^{'}$ is determined by its restriction to $R^{'}$ and the images of the $x_{i}$ , by [F3], [F5] and [F4] applied in that order, so $(Φ \circ τ) (x_{i}^{'}) = x_{i}^{'}$ and $(Φ \circ τ) ((g^{'})^{- 1}) = (g^{'})^{- 1}$ force $Φ \circ τ = {i d}_{S^{'}}$ . Similarly $R^{'} \otimes_{R} S$ is generated as an $R^{'}$ -algebra by $1 \otimes \overline{x_{i}}$ and $1 \otimes {\overline{g}}^{- 1}$ , by [F7], and $τ \circ Φ$ fixes these elements: $τ (Φ (1 \otimes \overline{x_{i}})) = τ (x_{i}^{'}) = 1 \otimes \overline{x_{i}}$ and $τ (Φ (1 \otimes {\overline{g}}^{- 1})) = τ ((g^{'})^{- 1}) = 1 \otimes {\overline{g}}^{- 1}$ , the last because $τ$ is a ring homomorphism sending $g^{'}$ to $1 \otimes \overline{g}$ . Hence $τ \circ Φ = i d$ as well, and $Φ$ is an isomorphism with inverse $τ$ .

F3F4F5F7step 2.1step 2.2step 3.1

5.1

The minor maps to a unit of $S^{'}$ . The element $h \in P$ maps to a unit of $S$ by hypothesis, hence $1 \otimes h \in R^{'} \otimes_{R} S$ is a unit with inverse $1 \otimes h^{- 1}$ by [F7], and $Φ$ carries it to $(1 \otimes h)$ 's image, namely $h^{'} = Φ (1 \otimes h)$ ; as a ring isomorphism $Φ$ carries units to units, so $h^{'}$ is a unit of $S^{'}$ .

F7step 3.1step 4.1

6.1

The Jacobian of the changed polynomials. By the monomial formula of [F9] the partial derivative $\partial_{i}$ is linear over the coefficient ring, so $\partial_{i} (f_{j}^{'})$ is the image of $\partial_{i} (f_{j})$ under $P \to P^{'}$ for all $i, j$ ; hence $h^{'} = \det (\partial_{i} f_{j}^{'})_{1 \leq i, j \leq c}$ is the leading $c \times c$ minor of the Jacobian matrix of $f_{1}^{'}, \dots, f_{c}^{'}$ . By step 5.1 its image in $S^{'}$ is a unit, so $S^{'} ≅ (R^{'} [x_{1}, \dots, x_{n}] / (f_{1}^{'}, \dots, f_{c}^{'}))_{g^{'}}$ is a standard smooth presentation over $R^{'}$ of relative dimension $n - c$ , the same parameters as the given presentation. With step 4.1 this proves both assertions.

F1F9step 4.1step 5.1algebra∎

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Invertible Jacobian minor gives regular parameters in a polynomial fibre

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field, let $n \geq 0$ , let $P = k [x_{1}, \dots, x_{n}]$ , let $q \subseteq P$ be a prime ideal and put $A = P_{q}$ , with maximal ideal $m = q A$ and residue field $κ = A / m$ (Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ , $R_{p}$ is local with unique maximal ideal $p R_{p}$ ). Let $f_{1}, \dots, f_{c} \in q$ and suppose that the leading $c \times c$ minor $h = \det (\frac{\partial f_{j}}{\partial x_{i}})_{1 \leq i, j \leq c}$ of the Jacobian matrix of Differentials of a polynomial quotient and the Jacobian cokernel satisfies $h \notin q$ . Then:

the classes of $f_{1}, \dots, f_{c}$ in $m / m^{2}$ are $κ$ -linearly independent;
$A$ is a regular local ring, $(f_{1}, \dots, f_{c})$ is a regular sequence in $A$ , and $A / (f_{1}, \dots, f_{c})$ is a regular local ring with $\dim A / (f_{1}, \dots, f_{c}) = \dim A - c$ .

This is the fibre computation used when a standard smooth presentation is examined over a field.

Facts & Assumptions

Given: A field $k$ , the polynomial ring $P = k [x_{1}, \dots, x_{n}]$ , a prime $q \subseteq P$ , the localisation $A = P_{q}$ with maximal ideal $m$ and residue field $κ$ , and elements $f_{1}, \dots, f_{c} \in q$ whose leading $c \times c$ Jacobian minor $h$ is not in $q$ ; and the Axiom of Choice.

[F1]

Differentials of a polynomial quotient and the Jacobian cokernel: $Ω_{P / k}$ is free on $d x_{1}, \dots, d x_{n}$ ; the partial derivatives $\partial_{i}$ are defined on the monomial basis by $\partial_{i} (x^{a}) = a_{i} x^{a - e_{i}}$ and extended $k$ -linearly, satisfy the Leibniz rule, and $d f = \sum_{i} \partial_{i} f d x_{i}$ ; for $I = (f_{1}, \dots, f_{c})$ the module $Ω_{P / I / k}$ is the cokernel of the Jacobian matrix $(\partial_{i} f_{j})_{i, j}$ .

[F2]

localisation and polynomial extension of regular rings: under the Axiom of Choice, localizations and finite polynomial extensions of a commutative regular Noetherian ring are regular, and regularity can equivalently be tested at maximal ideals.

[F3]

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 $edim R = \dim R$ .

[F4]

regular noetherian ring: a commutative Noetherian ring is regular when every prime localisation is a regular local ring.

[F5]

regular system of parameters equivalent basis: under the Axiom of Choice, for a nonzero Noetherian local ring $(R, m, k)$ of dimension $d$ and $x = (x_{1}, \dots, x_{d}) \in m^{d}$ , the tuple is a regular system of parameters if and only if its classes form a $k$ -basis of $m / m^{2}$ ; in particular every lift of a cotangent basis generates $m$ and is a system of parameters.

[F6]

regular local rings are domains and cohen macaulay: under the Axiom of Choice, a regular local ring $R$ of dimension $d$ is a domain and Cohen–Macaulay, and for every regular system $(y_{1}, \dots, y_{d})$ of parameters the tuple is $R$ -regular and $R / (y_{1}, \dots, y_{c})$ is regular local of dimension $d - c$ for every $0 \leq c \leq d$ .

[F7]

regular system of parameters: in a regular local ring of dimension $d$ , a regular system of parameters is an ordered minimal generating tuple of the maximal ideal, of length $d$ ; the empty tuple when $d = 0$ .

[F8]

Zorn's lemma gives a basis between any linearly independent set and any spanning set containing it: if $L \subseteq S \subseteq V$ with $L$ independent and $span (S) = V$ , there is a basis $B$ of $V$ with $L \subseteq B \subseteq S$ : under the Axiom of Choice, a linearly independent subset of a vector space extends to a basis.

[F9]

Localisation of modules is exact: localisation at a multiplicative set preserves short exact sequences.

[F10]

$R / P$ is an integral domain if and only if $P$ is a prime ideal: $R / P$ is an integral domain if and only if $P$ is a prime ideal; in particular $(0)$ is prime in a domain.

[F11]

A commutative ring is Noetherian exactly when every ideal is finitely generated, exactly when its ideals satisfy the ascending chain condition, and exactly when every nonempty set of ideals has a maximal member: a commutative ring is Noetherian if and only if every ideal is finitely generated.

[F12]

Field: a field is a commutative ring with $0 \neq 1$ in which every nonzero element has a multiplicative inverse.

[F13]

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

[F14]

A local ring is a nonzero commutative ring with a unique maximal ideal: a local ring is a commutative ring with exactly one maximal ideal; its residue field is the quotient by that ideal.

[F15]

Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ : $P_{q}$ is the localisation at the multiplicative set $P ∖ q$ .

[F16]

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

[F17]

$R_{p} / p R_{p} ≅ Frac (R / p)$ is the residue field at $p$ : the residue field of $P_{q}$ is the fraction field of $P / q$ .

[F18]

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

Proof

1.1

The field $k$ is a regular Noetherian ring. Its ideals are $0$ and $k$ : a nonzero ideal contains a nonzero element, which is a unit by [F12], hence contains $1$ and equals $k$ . Both ideals are finitely generated, so $k$ is Noetherian by [F11]. The only prime ideal of $k$ is $(0)$ : it is prime because $k / (0) ≅ k$ is a domain and every nonzero ideal equals $k$ , which is not prime; so $k$ has exactly one maximal ideal, namely $(0)$ , and is a local ring in the sense of [F14] with residue field $k$ . There is no strict chain of primes, so $\dim k = 0$ by [F13]; and $edim k = \dim_{k} ((0) / (0)^{2}) = \dim_{k} 0 = 0 = \dim k$ , so $k$ is regular local by [F3]. Every prime localisation of $k$ is $k$ itself, hence regular local, so $k$ is a regular Noetherian ring by [F4].

F3F4F11F12F13F14F10F18

2.1

The local ring $A$ is regular. By [F2] applied to the regular Noetherian ring $k$ of step 1.1, the polynomial ring $P = k [x_{1}, \dots, x_{n}]$ is regular, and its localisation $A = P_{q}$ at the prime $q$ is regular as well; by [F4] this says that every prime localisation of the Noetherian ring $A$ is a regular local ring, in particular $A$ itself, whose only maximal ideal is $m = q A$ by [F16]. Hence $A$ is a regular local ring with residue field $κ = A / m$ [F17], and $\dim A = edim A = \dim_{κ} (m / m^{2})$ by [F3].

F2F3F4step 1.1F16F17

3.1

The differential map $δ ⁣ : m / m^{2} \to κ^{n}$ . Localising the exact sequence of $P$ -modules $q^{2} \to q \to q / q^{2} \to 0$ at $P ∖ q$ and using [F9] identifies $m / m^{2} = (q / q^{2}) \otimes_{P} A$ . The assignment $(F m o d q^{2}, a) \mapsto a \cdot (\partial_{i} f m o d q)_{i}$ is $P$ -balanced: for $F \in q$ and $G \in P$ the Leibniz rule of [F1] gives $\partial_{i} (F G) - G \partial_{i} F = F \partial_{i} G \in q$ , and for $F \in q^{2}$ one has $\partial_{i} F \in q$ by the same rule applied to a product of two elements of $q$ . Hence it induces an $A$ -linear map $(q / q^{2}) \otimes_{P} A \to κ^{n}$ , that is, a $κ$ -linear map $δ$ on $m / m^{2}$ , which sends the class of $f_{j}$ to the $j$ -th Jacobian column $(\partial_{i} f_{j} m o d q)_{i}$ .

F1F9F15step 2.1

4.1

The classes of $f_{1}, \dots, f_{c}$ are linearly independent. Let $λ_{1}, \dots, λ_{c} \in A$ satisfy $\sum_{j} λ_{j} \overline{f_{j}} = 0$ in $m / m^{2}$ . Applying $δ$ of step 3.1 and using its $κ$ -linearity gives $\sum_{j} \overline{λ_{j}} (\partial_{i} f_{j} m o d q)_{i} = 0$ in $κ^{n}$ . The first $c$ coordinates are the matrix equation ${\overline{M}}^{T} \overline{λ} = 0$ , where $\overline{M}$ is the image in $κ$ of the $c \times c$ matrix $(\partial_{i} f_{j})_{1 \leq i, j \leq c}$ ; its determinant is the image of $h$ , which is nonzero because $h \notin q$ and $κ = k [x]_{q} / q k [x]_{q}$ has kernel exactly $q$ on $P$ . Hence $\overline{M}$ is invertible over the field $κ$ and $\overline{λ} = 0$ , that is, every $λ_{j} \in m$ . Therefore no nontrivial $κ$ -linear relation exists among the classes of $f_{1}, \dots, f_{c}$ .

F1F17step 2.1step 3.1algebra

5.1

A regular system of parameters. Put $d : = \dim A = \dim_{κ} (m / m^{2})$ by step 2.1. By step 4.1 the family of classes $(\overline{f_{1}}, \dots, \overline{f_{c}})$ in the $κ$ -vector space $m / m^{2}$ is linearly independent, so by [F8] it extends to a $κ$ -basis $(\overline{f_{1}}, \dots, \overline{f_{c}}, \overline{z_{c + 1}}, \dots, \overline{z_{d}})$ with $z_{i} \in m$ ; here $c \leq d$ because the independent family has at most $\dim_{κ} (m / m^{2}) = d$ members. Define $y_{i} : = f_{i}$ for $1 \leq i \leq c$ and $y_{i} : = z_{i}$ for $c < i \leq d$ . The classes of $y_{1}, \dots, y_{d}$ form a $κ$ -basis of $m / m^{2}$ , so $(y_{1}, \dots, y_{d})$ is a regular system of parameters of $A$ by [F5], of length $d = \dim A$ as required by [F7].

F5F7F8step 2.1step 4.1

6.1

Conclusion. By [F6] applied to the regular local ring $A$ of step 2.1 and its regular system of parameters $(y_{1}, \dots, y_{d})$ of step 5.1, the tuple $(y_{1}, \dots, y_{d})$ is an $A$ -regular sequence and $A / (y_{1}, \dots, y_{c})$ is a regular local ring of dimension $d - c = \dim A - c$ . Since $y_{i} = f_{i}$ for $i \leq c$ , the quotient $A / (y_{1}, \dots, y_{c})$ is $A / (f_{1}, \dots, f_{c})$ and the initial segment $(f_{1}, \dots, f_{c})$ is a regular sequence. This proves both assertions.

F6step 2.1step 5.1∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Local flatness criterion by regular parameters

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(R, m) \to (S, n)$ be a local homomorphism of Noetherian local rings and let $M$ be a finite $S$ -module. If ${Tor}_{1}^{R} (R / m, M) = 0$ , then $M$ is flat over $R$ . The module $M$ is not assumed finite over $R$ .

Consequently, if $R$ and $S$ are regular local rings and the images in $S$ of a regular system of parameters of $R$ extend to a regular system of parameters of $S$ , then $S$ is flat over $R$ .

Facts & Assumptions

Given: A local homomorphism $(R, m) \to (S, n)$ of Noetherian local rings and a finite $S$ -module $M$ with ${Tor}_{1}^{R} (R / m, M) = 0$ ; for the second assertion regular local $R, S$ and a regular system of parameters of $R$ whose images extend to one of $S$ ; and the Axiom of Choice.

[F1]

The long exact Tor sequence in the right-module variable: under Dependent Choice, a short exact sequence $0 \to N^{'} \to N \to N^{''} \to 0$ of right $R$ -modules and a left module $M$ with a supplied projective resolution give the natural long exact sequence $\dots \to {Tor}_{i}^{R} (N^{'}, M) \to {Tor}_{i}^{R} (N, M) \to {Tor}_{i}^{R} (N^{''}, M) \to {Tor}_{i - 1}^{R} (N^{'}, M) \to \dots$ , with the usual tensor-product tail.

[F2]

Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests: $M$ is flat over $R$ if and only if $I \otimes_{R} M \to M$ is injective for every finitely generated ideal $I \subseteq R$ .

[F3]

Artin-Rees controls intersections of submodules with high ideal powers: for a Noetherian ring $S$ , an ideal $J$ , a finite $S$ -module $N$ and a submodule $K$ there is $c \geq 0$ with $J^{n} N \cap K = J^{n - c} (J^{c} N \cap K)$ for every $n \geq c$ .

[F4]

The Krull intersection is the $(1 - a)$ -torsion submodule, and it vanishes in the Jacobson-radical case: for a Noetherian ring $S$ , an ideal $J \subseteq J (S)$ and a finite $S$ -module $N$ , the intersection $⋂_{n \geq 0} J^{n} N$ is zero.

[F5]

Composition series and length of a module: a composition series of a module is a finite chain with simple factors, the length is the number of factors, and the zero module has length $0$ .

[F6]

Module length is additive in short exact sequences: for $0 \to N^{'} \to N \to N^{''} \to 0$ , the module $N$ has finite length if and only if $N^{'}$ and $N^{''}$ do, and then $ℓ (N) = ℓ (N^{'}) + ℓ (N^{''})$ .

[F7]

Simple module: a nonzero module with no proper nonzero submodule: a module is simple when it is nonzero and has no proper nonzero submodule.

[F8]

A local ring is a nonzero commutative ring with a unique maximal ideal: a local ring has exactly one maximal ideal.

[F9]

Left and right Noetherian rings: in a Noetherian ring every ideal is finitely generated.

[F10]

Tor from a projective resolution of the right module: for a right module $N$ with a specified projective resolution $Q_{∙}$ one sets ${Tor}_{n}^{R, Q} (N, M) = H_{n} (Q_{∙} \otimes_{R} M)$ .

[F11]

The balanced Tor bifunctor: under Dependent Choice, ${Tor}_{i}^{R} (N, M)$ is the balanced bifunctor obtained from either a resolution of $N$ or one of $M$ , identified by the left-right comparison theorem.

[F12]

The left and right projective constructions of Tor are naturally isomorphic: under Dependent Choice there is a natural isomorphism $H_{i} (N \otimes_{R} P_{∙}) ≅ H_{i} (Q_{∙} \otimes_{R} M)$ for supplied projective resolutions.

[F13]

The recursion theorem: for a set $X$ , an element $x \in X$ and a function $f ⁣ : X \to X$ there is a unique $g ⁣ : N \to X$ with $g (0) = x$ and $g (n + 1) = f (g (n))$ .

[F14]

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain: for every nonempty set $X$ , every entire relation $R$ on $X$ and every $a \in X$ there is $x ⁣ : N \to X$ with $x_{0} = a$ and $x_{n} R x_{n + 1}$ for all $n$ .

[F15]

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

[F16]

regular local residue field koszul resolution: under the Axiom of Choice, for a regular local ring $(R, m, κ)$ of dimension $d$ the Koszul complex on any regular system of parameters is a minimal free resolution of $κ$ of length $d$ .

[F17]

Koszul Complex Of A Sequence With Coefficients: $K (x; M)$ has degree- $p$ term $⋀^{p} R^{d} \otimes_{R} M$ and differential $d (e_{i_{1}} \land \dots \land e_{i_{p}} \otimes m) = \sum_{j} (- 1)^{j - 1} e_{i_{1}} \land \dots \hat{e_{i_{j}}} \dots \land e_{i_{p}} \otimes x_{i_{j}} m$ .

[F18]

Regular Sequences Give Acyclic Koszul Complexes: every finite $M$ -regular sequence is $M$ -Koszul-regular, that is $H_{i} (K (x; M)) = 0$ for $i > 0$ .

[F19]

regular local rings are domains and cohen macaulay: under the Axiom of Choice, for a regular local ring of dimension $d$ every regular system of parameters $(y_{1}, \dots, y_{d})$ is a regular sequence and $R / (y_{1}, \dots, y_{c})$ is regular local of dimension $d - c$ for $0 \leq c \leq d$ ; in particular an initial segment of a regular system of parameters is a regular sequence.

[F20]

Tensoring is right exact: tensoring an exact sequence $A^{'} \to B^{'} \to C^{'} \to 0$ with a module preserves exactness at the right.

Proof

1.1

AC gives DC, so the Dependent-Choice suppliers [F1], [F11] and [F12] are available. Given a nonempty set $X$ , an entire relation $R$ on $X$ and $a \in X$ , apply [F15] to the family of nonempty subsets of $X$ to obtain $g$ with $g (T) \in T$ for every nonempty $T \subseteq X$ , and put $f (x) : = g ({y \in X : x R y})$ , a function $X \to X$ because $R$ is entire. By [F13] there is $x ⁣ : N \to X$ with $x_{0} = a$ and $x_{n + 1} = f (x_{n})$ ; then $x_{n} R x_{n + 1}$ for all $n$ because $f (x_{n}) \in {y : x_{n} R y}$ . This is exactly the statement of [F14]. Supply a free resolution of $M$ by mapping the free module on its underlying set onto $M$ , then repeating this construction on each successive kernel; recursion gives the required resolution for [F1].

F13F14F15

1.2

Ideals of finite colength. Let $J \subseteq R$ be an ideal containing $m^{n}$ for some $n \geq 0$ . For $n = 0$ , $J = R$ and $R / J = 0$ . For $n \geq 1$ , $R / J$ has finite length over $R$ : the ring $R / m^{n}$ carries the finite chain $0 \subseteq m^{n - 1} / m^{n} \subseteq \dots \subseteq R / m^{n}$ whose successive quotients are $m^{i} / m^{i + 1} = m^{i} / (m \cdot m^{i})$ ; each $m^{i}$ is finitely generated over the Noetherian ring $R$ by [F9], so each $m^{i} / m^{i + 1}$ is a finitely generated module over the field $κ = R / m$ , hence a finite-dimensional $κ$ -vector space, which has a finite composition series with simple factors $κ$ and therefore finite length by [F5]; a finite extension of modules of finite length has finite length with additive length by [F6], so $ℓ_{R} (R / m^{n}) < \infty$ , and $R / J$ , being a quotient of $R / m^{n}$ , has finite length as well.

F5F6F9

2.1

Vanishing on finite length. Suppose ${Tor}_{1}^{R} (κ, M) = 0$ with $κ = R / m$ . Then ${Tor}_{1}^{R} (N, M) = 0$ for every $R$ -module $N$ of finite length $ℓ_{R} (N)$ : prove this by induction on $ℓ_{R} (N)$ . For $ℓ_{R} (N) = 0$ we have $N = 0$ . If $ℓ_{R} (N) \geq 1$ , choose a proper submodule $N^{'} \subset N$ that is maximal for inclusion, which exists because $N$ has finite length; then $N / N^{'}$ is simple by [F7], so choosing $0 \neq s \in N / N^{'}$ presents it as $(R / Ann (s)) \cdot s$ , and $Ann (s)$ is a maximal ideal of $R$ , because for a proper ideal $J ⊋ Ann (s)$ the submodule $J s$ is nonzero and hence all of $N / N^{'}$ , forcing $J s = N / N^{'}$ and $1 \in J$ . By [F8] the only maximal ideal is $m$ , so $N / N^{'} ≅ κ$ . By [F6] $ℓ_{R} (N^{'}) = ℓ_{R} (N) - 1$ , so the induction hypothesis applies to $N^{'}$ , and the exact sequence ${Tor}_{1}^{R} (N^{'}, M) \to {Tor}_{1}^{R} (N, M) \to {Tor}_{1}^{R} (κ, M) = 0$ from [F1] has vanishing outer terms, whence ${Tor}_{1}^{R} (N, M) = 0$ .

step 1.1F1F5F6F7F8

3.1

Injectivity for finite colength ideals. Let $J \subseteq R$ be any ideal. Since the free module $R$ with the resolution concentrated in degree $0$ satisfies ${Tor}_{1}^{R} (R, M) = 0$ by [F10], the long exact sequence of [F1] for $0 \to J \to R \to R / J \to 0$ exhibits $\ker (J \otimes_{R} M \to M)$ as the image of ${Tor}_{1}^{R} (R / J, M) \to J \otimes_{R} M$ . Hence if $J \supseteq m^{n}$ and ${Tor}_{1}^{R} (κ, M) = 0$ , then $R / J$ has finite length by step 1.2 and ${Tor}_{1}^{R} (R / J, M) = 0$ by step 2.1, so $J \otimes_{R} M \to M$ is injective.

step 1.1step 1.2step 2.1F1F10

4.1

The diagram chase. Let $I \subseteq R$ be a finitely generated ideal and let $K : = \ker (I \otimes_{R} M \to M)$ . For every $n$ the sequence $0 \to I \cap m^{n} \to I \oplus m^{n} \to I + m^{n} \to 0$ , with maps $x \mapsto (x, - x)$ and $(a, b) \mapsto a + b$ , is exact, so after tensoring with $M$ and using [F20] the sequence $(I \cap m^{n}) \otimes_{R} M \to I \otimes_{R} M \oplus m^{n} \otimes_{R} M \to (I + m^{n}) \otimes_{R} M \to 0$ is exact; the vertical maps to $M$ are the multiplication maps, which are injective on $m^{n} \otimes_{R} M$ and on $(I + m^{n}) \otimes_{R} M$ because both ideals contain $m^{n}$ and step 3.1 applies. Given $k \in K$ , its image $(k, 0)$ in the middle maps to zero in $(I + m^{n}) \otimes_{R} M$ and hence, by exactness at the middle, equals $(x, - x)$ for some $x \in (I \cap m^{n}) \otimes_{R} M$ ; then $x$ maps to $k$ in $I \otimes_{R} M$ and to $0$ in $m^{n} \otimes_{R} M$ , so $x \in \ker ((I \cap m^{n}) \otimes_{R} M \to M)$ and $K$ lies in the image of $(I \cap m^{n}) \otimes_{R} M \to I \otimes_{R} M$ .

step 3.1F20

5.1

Concluding $K = 0$ . Apply Artin–Rees [F3] over the Noetherian ring $R$ to the finite module $R$ , its submodule $I$ and the ideal $m$ . It gives $c \geq 0$ such that $I \cap m^{n} = m^{n - c} (I \cap m^{c}) \subseteq m^{n - c} I$ for every $n \geq c$ . Put $N : = I \otimes_{R} M$ , a finite $S$ -module because $I$ is finite over $R$ and $M$ is finite over $S$ . By step 4.1 and this inclusion, $K \subseteq (m S)^{n - c} N$ for every $n \geq c$ : an elementary tensor $r a \otimes m$ with $r \in m^{n - c}$ equals $r (a \otimes m)$ . The local homomorphism gives $m S \subseteq n = J (S)$ , so Krull intersection [F4] on the finite $S$ -module $N$ gives $⋂_{j \geq 0} (m S)^{j} N = 0$ . Hence $K = 0$ .

step 4.1F3F4F8

6.1

Since $I \subseteq R$ was an arbitrary finitely generated ideal and $I \otimes_{R} M \to M$ is injective, [F2] shows that $M$ is flat over $R$ . This proves the first assertion.

step 5.1F2

7.1

The regular-parameter case. Let $x_{1}, \dots, x_{d}$ be a regular system of parameters of $R$ and let ${\overline{x}}_{1}, \dots, {\overline{x}}_{d} \in S$ be their images, extending to a regular system of parameters $y_{1}, \dots, y_{e}$ of $S$ . By [F19] the tuple $(y_{1}, \dots, y_{e})$ is $S$ -regular, hence so is its initial segment ${\overline{x}}_{1}, \dots, {\overline{x}}_{d}$ . By [F16] the Koszul complex $K_{R} (x_{1}, \dots, x_{d}; R)$ is a free resolution of $κ = R / m$ , and tensoring its defining formulas, [F17], with $- \otimes_{R} S$ replaces each $x_{i}$ by ${\overline{x}}_{i}$ and reproduces the Koszul complex $K_{S} ({\overline{x}}_{1}, \dots, {\overline{x}}_{d}; S)$ of [F17] term by term, so $K_{R} (x) \otimes_{R} S ≅ K_{S} (\overline{x}; S)$ as complexes. Therefore, using [F10] for the right-resolution construction and [F12] (available by step 1.1) to identify it with the balanced Tor of [F11], ${Tor}_{1}^{R} (κ, S) = H_{1} (K_{R} (x) \otimes_{R} S) = H_{1} (K_{S} (\overline{x}; S)) = 0$ by [F18], as $\overline{x}$ is an $S$ -regular sequence. The module $S$ is a finite $S$ -module, so the first assertion of this lemma, applied to $M = S$ , gives that $S$ is flat over $R$ .

step 1.1step 6.1F10F11F12F16F17F18F19∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Standard smooth algebras are finitely presented and flat

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $R$ be a commutative ring and let $S$ be a standard smooth $R$ -algebra (Standard smooth presentations and locally standard smooth maps), so that $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ for some $n \geq c \geq 0$ , some $f_{1}, \dots, f_{c}, g \in R [x_{1}, \dots, x_{n}]$ , and with the leading $c \times c$ Jacobian minor $h = \det (\partial f_{j} / \partial x_{i})_{1 \leq i, j \leq c}$ mapping to a unit of $S$ . Then:

$S$ is a finitely presented $R$ -algebra (Finitely presented modules and finitely presented algebras);
$S$ is flat over $R$ .

No hypothesis is placed on $R$ : it may be non-Noetherian, and it may have zero divisors.

Facts & Assumptions

Given: A commutative ring $R$ , a standard smooth presentation $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ whose leading $c \times c$ minor $h$ maps to a unit of $S$ , and the Axiom of Choice. Write $P : = R [x_{1}, \dots, x_{n}]$ and $I : = (f_{1}, \dots, f_{c})$ .

[F1]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation consists of integers $n \geq c \geq 0$ , elements $f_{1}, \dots, f_{c} \in P$ and $g \in P$ with $S ≅ (P / (f_{1}, \dots, f_{c}))_{g}$ , such that the Jacobian matrix has a $c \times c$ minor whose image in $S$ is a unit; $n - c$ is the relative dimension, and the invertible minor may be assumed to lie in the first $c$ columns.

[F2]

Base change of standard smooth presentations: for any ring map $R \to R^{'}$ and a standard smooth presentation as above, there is a unique $R^{'}$ -algebra isomorphism $R^{'} \otimes_{R} S \to (R^{'} [x_{1}, \dots, x_{n}] / (f_{1}^{'}, \dots, f_{c}^{'}))_{g^{'}}$ sending $a \otimes \overline{F} / g^{N}$ to $a \overline{F^{'}} / (g^{'})^{N}$ , and the target is standard smooth over $R^{'}$ with the same $n, c$ and relative dimension, its $c \times c$ minor $h^{'}$ still a unit.

[F3]

Invertible Jacobian minor gives regular parameters in a polynomial fibre: under the Axiom of Choice, if $k$ is a field, $q \subseteq k [x_{1}, \dots, x_{n}]$ is prime and $f_{1}, \dots, f_{c} \in q$ have leading $c \times c$ Jacobian minor $h \notin q$ , then the classes of $f_{1}, \dots, f_{c}$ in $q k [x]_{q} / (q k [x]_{q})^{2}$ are linearly independent, $k [x]_{q}$ is regular local, $(f_{1}, \dots, f_{c})$ is a regular sequence in it and the quotient is regular local of dimension $\dim k [x]_{q} - c$ .

[F4]

[F5]

Finitely presented modules and finitely presented algebras: a commutative $R$ -algebra $A$ is finitely presented when $A ≅ R [x_{1}, \dots, x_{m}] / a$ for some $m \in N$ and a finitely generated ideal $a$ ; the boundary values $m = 0$ and $a = 0$ are admitted.

[F6]

Universal property of a polynomial ring on an arbitrary family of indeterminates: for a ring homomorphism $φ ⁣ : R \to T$ and a family $(t_{i})$ in $T$ there is a unique ring homomorphism $R [x_{i}] \to T$ restricting to $φ$ with $x_{i} \mapsto t_{i}$ .

[F7]

A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring: a ring homomorphism whose kernel contains an ideal $I$ factors uniquely through $R / I$ .

[F8]

Universal property of localisation: maps that invert $S$ factor uniquely through $S^{- 1} R$ : a unital homomorphism $f ⁣ : R \to A$ carrying a multiplicative set $T$ into the units of $A$ factors uniquely through $λ_{T} ⁣ : R \to T^{- 1} R$ .

[F9]

Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions: the localisation $T^{- 1} A$ of a commutative ring at a multiplicative set $T$ consists of the classes $a / t$ , and $λ_{T} (a) = a / 1$ with every $t \in T$ a unit.

[F10]

A module is flat if and only if all prime localizations are flat, equivalently all maximal localizations are flat: for an $R$ -module $M$ , $M$ is flat over $R$ if and only if $M_{p}$ is flat over $R_{p}$ for every prime $p \subseteq R$ , equivalently for every maximal ideal.

[F11]

[F12]

Every localization is flat, and localizing a flat module preserves flatness: $S^{- 1} R$ is a flat $R$ -algebra, and if $N$ is flat over $R$ then $S^{- 1} N$ is flat over $S^{- 1} R$ .

[F13]

Under the stated choice boundary, free modules are projective and hence flat: for every commutative ring $R$ and free $R$ -module $F$ , $F$ is flat over $R$ regardless of choice.

[F14]

Localisation of modules is extension of scalars: for a commutative ring $A$ , a multiplicative set $T$ and an $A$ -module $M$ there is an isomorphism $T^{- 1} M ≅ (T^{- 1} A) \otimes_{A} M$ , $m / t \mapsto (1 / t) \otimes m$ .

[F15]

Tensoring is right exact: tensoring an exact sequence $A^{'} \to B^{'} \to C^{'} \to 0$ with a module preserves exactness; tensoring preserves cokernels and surjections.

[F16]

Symmetry and associativity isomorphisms for tensor products over a commutative ring: tensor products of modules over a commutative ring are commutative and associative, so $(M \otimes_{A} N) \otimes_{A} P ≅ M \otimes_{A} (N \otimes_{A} P)$ .

[F17]

For a finite module, support is the set of primes containing the annihilator: for a finitely generated module $M$ over a commutative ring $R$ , ${Supp}_{R} (M) = {p : {Ann}_{R} (M) \subseteq p}$ ; in particular a finitely generated module whose localisations at all maximal ideals vanish is zero, because a proper ideal lies in a maximal ideal.

[F18]

In a nonzero commutative ring, every proper ideal is contained in a maximal ideal: every proper ideal of a nonzero commutative ring is contained in a maximal ideal.

[F19]

Finitely generated modules over a left Noetherian ring are Noetherian: submodules of finitely generated modules over a Noetherian ring are again finitely generated.

[F20]

Every quotient and every localisation of a Noetherian ring is Noetherian: quotients and localisations of a Noetherian commutative ring are Noetherian.

[F21]

If $R$ is Noetherian then $R [x_{1}, \dots, x_{n}]$ is Noetherian for every $n \in N$ : for a Noetherian commutative ring $R$ and $n \in N$ the polynomial ring $R [x_{1}, \dots, x_{n}]$ is Noetherian.

[F22]

[F23]

Every subgroup of $(Z, +)$ is $⟨ n ⟩ = n Z$ for exactly one natural number $n$ : every subgroup of $(Z, +)$ is generated by one element.

[F24]

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.

[F26]

The long exact Tor sequence in the left-module variable: under the Axiom of Dependent Choice, a short exact sequence $0 \to M^{'} \to M \to M^{''} \to 0$ of modules and a module $N$ give a natural long exact sequence $\dots \to {Tor}_{1}^{R} (N, M^{'}) \to {Tor}_{1}^{R} (N, M) \to {Tor}_{1}^{R} (N, M^{''}) \to N \otimes_{R} M^{'} \to N \otimes_{R} M \to \dots$ .

[F27]

Localisation commutes with kernels images and cokernels: localisation commutes with kernels, images and cokernels of module homomorphisms.

[F28]

Extension of scalars carries flat modules to flat modules: if $M$ is a flat $R$ -module and $R \to R^{'}$ is a ring homomorphism, then $R^{'} \otimes_{R} M$ is a flat $R^{'}$ -module.

[F29]

Equality, vanishing, and the kernel of the localisation map: in a localisation, $r / s = 0$ if and only if $u r = 0$ for some $u \in T$ , and $r / s = r^{'} / s^{'}$ if and only if $u (r s^{'} - r^{'} s) = 0$ for some $u \in T$ .

[F30]

Localisation commutes with quotient rings: $S^{- 1} R / S^{- 1} I ≅ {\overset{ˉ}{S}}^{- 1} (R / I)$ : $(T^{- 1} A) / (T^{- 1} I) ≅ {\overset{ˉ}{T}}^{- 1} (A / I)$ for an ideal $I$ of $A$ and multiplicative $T$ , where $\overset{ˉ}{T}$ is the image of $T$ in $A / I$ .

[F31]

A local ring is a nonzero commutative ring with a unique maximal ideal: a local ring is a commutative ring with exactly one maximal ideal; a local homomorphism $R \to S$ of local rings is one carrying the maximal ideal of $R$ into that of $S$ .

[F32]

[F33]

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

[F34]

The polynomial ring $R [x_{i} : i \in I]$ as finitely supported coefficient families on monomials: $R [x_{1}, \dots, x_{n}]$ is the commutative $R$ -algebra of polynomials in the indeterminates; a ring map $R \to R^{'}$ induces a ring map $R [x_{1}, \dots, x_{n}] \to R^{'} [x_{1}, \dots, x_{n}]$ sending each coefficient, and this map is injective when $R \to R^{'}$ is injective.

[F35]

The Krull intersection is the $(1 - a)$ -torsion submodule, and it vanishes in the Jacobson-radical case: if $T$ is a Noetherian commutative ring, $J \subseteq J (T)$ is an ideal and $M$ is a finite $T$ -module, then $⋂_{r \geq 0} J^{r} M = 0$ .

Proof

1.1

The encoded presentation. Let $Q : = R [x_{1}, \dots, x_{n}, z] / (f_{1}, \dots, f_{c}, z g - 1)$ . The composite $P \to S$ sends $g$ to a unit of $S$ , so by [F6] and [F7], and then [F8], there is a unique $R$ -algebra homomorphism $Q \to S$ with $x_{i} \mapsto \overline{x_{i}}$ and $z \mapsto {\overline{g}}^{- 1}$ ; here $z$ is the new variable and the relations $f_{j} \mapsto 0$ , $z g - 1 \mapsto {\overline{g}}^{- 1} \overline{g} - 1 = 0$ hold. Conversely, the substitution $x_{i} \mapsto x_{i}$ , $z \mapsto z$ gives a ring homomorphism $P \to Q$ whose kernel contains $I$ because each $f_{j}$ maps to $0$ , and which sends $g$ to a unit of $Q$ with inverse $z$ ; hence [F7] and [F8] produce a unique $R$ -algebra homomorphism $S = (P / I)_{g} \to Q$ with $\overline{F} / g^{N} \mapsto F z^{N}$ . The two composites fix the generators $x_{1}, \dots, x_{n}, z$ of $Q$ over $R$ and the generators $\overline{x_{1}}, \dots, \overline{x_{n}}, {\overline{g}}^{- 1}$ of $S$ , so by the uniqueness clauses of [F6], [F7] and [F8] they are the respective identities; thus $S ≅ Q$ .

F6F7F8F9

1.2

Reduction of flatness to the Noetherian local case. Assume first that $R$ is Noetherian. By [F10] it suffices to prove that $S \otimes_{R} R_{p}$ is flat over $R_{p}$ for every prime $p \subseteq R$ ; by [F2] the algebra $S \otimes_{R} R_{p}$ is standard smooth over $R_{p}$ with the same $n, c$ and a minor that is still a unit. Hence it suffices to prove: if $(R, m)$ is a Noetherian local ring and $S$ is standard smooth over $R$ , then $S$ is flat over $R$ .

F2F10F1

1.3

Dependent choice is available. Given the Axiom of Choice [F33], let $X$ be a nonempty set with an entire relation $R$ and let $a \in X$ ; choosing an element of each nonempty subset of $X$ and setting $f (x)$ to be the chosen element of ${y : x R y}$ gives a function $X \to X$ , so recursion produces $x ⁣ : N \to X$ with $x_{0} = a$ and $x_{n} R x_{n + 1}$ . Thus the Dependent Choice supplier [F26] is available throughout this proof.

F15F26F32F33

1.4

The local criterion, prepared. Let now $R$ be Noetherian local with maximal ideal $m$ and residue field $κ = R / m$ , and let $S = (P / I)_{g}$ be standard smooth over $R$ . Then $S$ is Noetherian: $P$ is Noetherian by [F21], $P / I$ is Noetherian by [F20] and so is its localisation $S$ by [F20]. Fix a finitely generated ideal $J \subseteq R$ and put $K_{J} : = \ker (J \otimes_{R} S \to S)$ ; this is a finitely generated $S$ -module by [F19], since $J \otimes_{R} S$ is a finitely generated $S$ -module. For every maximal ideal $n \subseteq S$ the localisation $(K_{J})_{n}$ equals the kernel of $J \otimes_{R} S_{n} \to S_{n}$ , by [F27] and [F14] applied to the localisation $S \to S_{n}$ . Hence if $S_{n}$ is flat over $R$ , then $(K_{J})_{n} = 0$ ; and if that holds for every maximal $n$ , then $K_{J} = 0$ by [F17] and [F18]. Therefore, by [F11], it is enough to prove that $S_{n}$ is flat over $R$ for every maximal ideal $n \subseteq S$ .

F11F14F17F18F19F20F21F27

1.5

Set-up at the contracted prime. Fix a maximal ideal $n \subseteq S$ and put $r = n \cap R$ . Before the local computation, replace the base and presentation by $R_{r}$ and $S \otimes_{R} R_{r}$ , using [F2]. The ideal $n$ induces a maximal ideal there with the same local ring $S_{n}$ . In steps 1.6, 2.2, 3.1 and 4.1 only, write $R, m, κ, P$ for this localized base, its maximal ideal and residue field, and its polynomial ring. Let $q^{'} \subseteq P$ be the prime over $n$ , so $q^{'} \cap R = m$ , $I \subseteq q^{'}$ and $g \notin q^{'}$ . Put $T = P_{q^{'}}$ and $N_{i} = T / (f_{1}, \dots, f_{i}) T$ ; then $N_{c} = S_{n}$ by [F30]. The module $T$ is flat over this base: tensoring an injection with the free $R$ -module $P$ preserves injectivity by [F13], and localizing the result preserves it by [F27], with the tensor identifications of [F14, F16]. Each $N_{i}$ is Noetherian local, and $m T \subseteq q^{'} T$ makes $R \to N_{i}$ local. Once the computation proves $N_{c}$ flat over $R_{r}$ , it is flat over the original base by clause 2 of Every localization is flat, and localizing a flat module preserves flatness.

F2F12F13F14F16F20F21F27F30F31

1.6

The fibre at $κ$ . The quotient map $P \to P / (f_{1}, \dots, f_{i})$ tensored with $κ$ has cokernel $κ [x_{1}, \dots, x_{n}] / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{i})$ by [F15]; localising this at the prime ${\overset{ˉ}{q}}^{'}$ induced by $q^{'}$ and applying [F14] twice together with [F16] gives $N_{i} \otimes_{R} κ ≅ κ [x_{1}, \dots, x_{n}]_{{\overset{ˉ}{q}}^{'}} / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{i})$ , where ${\overset{ˉ}{f}}_{j}$ denotes the image of $f_{j}$ in $κ [x_{1}, \dots, x_{n}]$ .

F14F15F16

1.7

$Z$ is Noetherian. Every ideal of $Z$ is an additive subgroup, hence generated by one element by [F23] and therefore finitely generated; so $Z$ is Noetherian by [F22].

F22F23

1.8

The unit witness for the minor. Since $h$ maps to a unit of $S = (P / I)_{g}$ , there are $u \in P$ and $N \geq 0$ with $\overline{h} \cdot \overline{u} / {\overline{g}}^{N} = 1$ in $S$ , that is, $\overline{h u - g^{N}} = 0$ in $(P / I)_{g}$ ; by [F29] applied to the localisation $(P / I) \to S$ there is $M \geq 0$ with $g^{M} (h u - g^{N}) \in I$ . Putting $w : = g^{M} u$ and $N^{'} : = M + N$ gives $h w - g^{N^{'}} \in I$ , so there are $m_{1}, \dots, m_{c} \in P$ with $h w - g^{N^{'}} = \sum_{j = 1}^{c} m_{j} f_{j}$ in $P$ .

F29algebra

2.1

Finite presentation. By step 1.1 the $R$ -algebra $S$ is isomorphic to the quotient of the polynomial ring $R [x_{1}, \dots, x_{n}, z]$ by the ideal generated by the finitely many elements $f_{1}, \dots, f_{c}, z g - 1$ ; by [F5] this exhibits $S$ as a finitely presented $R$ -algebra.

F5step 1.1

2.2

The minor survives in the fibre. The element $h$ maps to a unit of $S$ , hence its image in the localisation $S_{n} = N_{c}$ is a unit, hence its image in $N_{c} \otimes_{R} κ$ is a unit. By step 1.6 with $i = c$ this ring is $κ [x_{1}, \dots, x_{n}]_{{\overset{ˉ}{q}}^{'}} / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c})$ , and the image of $h$ there is the image of $\overset{ˉ}{h}$ ; since a unit of a local ring does not lie in the maximal ideal, $\overset{ˉ}{h} \notin {\overset{ˉ}{q}}^{'}$ . So [F3] applies over the field $κ$ : the images ${\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c}$ form a regular sequence in the regular local ring $κ [x_{1}, \dots, x_{n}]_{{\overset{ˉ}{q}}^{'}}$ , and consequently ${\overset{ˉ}{f}}_{i + 1}$ is a nonzerodivisor on $κ [x_{1}, \dots, x_{n}]_{{\overset{ˉ}{q}}^{'}} / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{i})$ for every $i < c$ . By step 1.6 this says that $f_{i + 1}$ is a nonzerodivisor on $N_{i} \otimes_{R} κ$ for every $0 \leq i < c$ .

F3step 1.5step 1.6algebra

2.3

Descent to a finitely generated subring. Let $R_{0} \subseteq R$ be the $Z$ -subalgebra generated by the finitely many coefficients occurring in the polynomials $f_{1}, \dots, f_{c}$ , $g$ , $h$ , $w$ , $m_{1}, \dots, m_{c}$ . Then $R_{0}$ is a finitely generated $Z$ -algebra, hence Noetherian by steps 1.7 and [F24]. The polynomial identity of step 1.8 has all its coefficients in $R_{0}$ , and $R_{0} [x_{1}, \dots, x_{n}] \to R [x_{1}, \dots, x_{n}]$ is injective by [F34], so $h w - g^{N^{'}} = \sum_{j} m_{j} f_{j}$ holds already in $R_{0} [x_{1}, \dots, x_{n}]$ ; therefore $S_{0} : = (R_{0} [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ is a standard smooth $R_{0}$ -algebra with the same $n, c, f_{j}, g$ and the minor $h$ a unit.

F24F34step 1.7step 1.8

3.1

Induction: a fibre nonzerodivisor lifts across a flat map. Suppose $N_{i}$ is flat over $R$ for some $i < c$ , put $J : = m N_{i}$ , and let $f : = f_{i + 1}$ . By step 2.2, multiplication by the image of $f$ on $N_{i} / J$ is injective. For every $r \geq 0$ , flatness of $N_{i}$ over $R$ applied to $0 \to m^{r + 1} \to m^{r} \to m^{r} / m^{r + 1} \to 0$ gives the natural isomorphism $J^{r} / J^{r + 1} ≅ (m^{r} / m^{r + 1}) \otimes_{κ} (N_{i} / J), κ = R / m .$ After choosing a $κ$ -basis of $m^{r} / m^{r + 1}$ , this is a direct sum of copies of $N_{i} / J$ , so multiplication by $f$ is injective on each graded piece $J^{r} / J^{r + 1}$ . If $f x = 0$ in $N_{i}$ , induction on $r$ now gives $x \in J^{r}$ for every $r$ : injectivity modulo $J$ starts the induction, and injectivity on $J^{r} / J^{r + 1}$ advances it. The ring $N_{i}$ is Noetherian local and $J$ lies in its maximal ideal by step 1.5, so [F35] gives $⋂_{r \geq 0} J^{r} = 0$ and therefore $x = 0$ . Thus $f_{i + 1}$ is a nonzerodivisor on $N_{i}$ , and $0 \to N_{i} \overset{f_{i + 1}}{\to} N_{i} \to N_{i + 1} \to 0$ is a short exact sequence.

F35step 1.5step 2.2algebra

4.1

Induction: the Tor vanishing and flatness pass to the next quotient. In the situation of step 3.1 the long exact Tor sequence of [F26] for $0 \to N_{i} \overset{f_{i + 1}}{\to} N_{i} \to N_{i + 1} \to 0$ , together with ${Tor}_{1}^{R} (κ, N_{i}) = 0$ from the flatness of $N_{i}$ , exhibits ${Tor}_{1}^{R} (κ, N_{i + 1})$ as the kernel of $N_{i} \otimes_{R} κ \overset{f_{i + 1}}{\to} N_{i} \otimes_{R} κ$ , which is zero by step 2.2. The ring $N_{i + 1}$ is a Noetherian local ring, $R \to N_{i + 1}$ is a local homomorphism by step 1.5 and $N_{i + 1}$ is a finite $N_{i + 1}$ -module, so [F4] gives that $N_{i + 1}$ is flat over $R$ .

F4F26step 1.5step 2.2step 3.1

5.1

Conclusion of the induction and of the local case. Steps 3.1 and 4.1, starting with $N_{0}$ of step 1.5, prove that every $N_{i}$ is flat over the localized base $R_{r}$ . In particular the original local ring $S_{n}$ is flat over $R_{r}$ , hence over the original $R$ by step 1.5. As $n$ was arbitrary, step 1.4 gives that $S$ is flat over the original base. This proves flatness over every Noetherian local base, and step 1.2 proves it over every Noetherian base.

F12step 1.2step 1.4step 1.5step 3.1step 4.1

6.1

Flatness of $S_{0}$ and base change back to $R$ . By step 2.3 the algebra $S_{0}$ is standard smooth over the Noetherian ring $R_{0}$ , so $S_{0}$ is flat over $R_{0}$ by step 5.1; and by [F2] applied to the ring map $R_{0} \to R$ there is an $R$ -algebra isomorphism $R \otimes_{R_{0}} S_{0} ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g} = S$ . Hence $S ≅ R \otimes_{R_{0}} S_{0}$ is flat over $R$ by [F28]. This proves the second assertion for arbitrary $R$ , and step 2.1 proves the first.

F2F28step 2.1step 5.1step 2.3∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Fibres of standard smooth algebras are regular of relative dimension

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $R$ be a commutative ring and let $S$ be a standard smooth $R$ -algebra (Standard smooth presentations and locally standard smooth maps), presented as $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with leading $c \times c$ Jacobian minor $h$ mapping to a unit of $S$ . Let $p \in Spec R$ , put $κ = κ (p)$ , and let $K / κ$ be a field extension. Write $F = S \otimes_{R} κ, F_{K} = F \otimes_{κ} K,$ so that, by base change of the presentation, $F_{K} ≅ (K [x_{1}, \dots, x_{n}] / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c}))_{\overset{ˉ}{g}}$ , where ${\overset{ˉ}{f}}_{j}, \overset{ˉ}{g}$ are the images of $f_{j}, g$ in $K [x_{1}, \dots, x_{n}]$ . Then:

every local ring $(F_{K})_{Q}$ of $F_{K}$ at a prime $Q$ is a regular local ring; if $Q^{'} \subseteq K [x_{1}, \dots, x_{n}]$ is the prime corresponding to $Q$ , then $\dim (F_{K})_{Q} = ht (Q^{'}) - c$ (The height of a prime ideal);
every irreducible component of $Spec F_{K}$ has dimension $n - c$ ; equivalently, for every minimal prime $P$ of $F_{K}$ one has $\dim (F_{K} / P) = n - c$ .

Both clauses are vacuous when $F_{K}$ is the zero ring, and no hypothesis is placed on $R$ or on the field extension $K / κ$ . The relative dimension $n - c$ is the dimension of the components, not the dimension of every local ring: a local ring at the generic point of a component has dimension $0$ .

Facts & Assumptions

Given: A commutative ring $R$ , a standard smooth presentation $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with leading $c \times c$ minor $h$ a unit of $S$ , a prime $p \in Spec R$ , a field extension $K / κ (p)$ , and the Axiom of Choice.

[F1]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation consists of integers $n \geq c \geq 0$ , elements $f_{1}, \dots, f_{c}, g \in R [x_{1}, \dots, x_{n}]$ with $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ , such that the Jacobian matrix $(\partial f_{j} / \partial x_{i})$ has a $c \times c$ minor whose image in $S$ is a unit; $n - c$ is the relative dimension, and the invertible minor may be assumed to be the leading one, in the first $c$ columns.

[F2]

Base change of standard smooth presentations: for a ring map $R \to R^{'}$ and a standard smooth presentation as above, $R^{'} \otimes_{R} S ≅ (R^{'} [x_{1}, \dots, x_{n}] / (f_{1}^{'}, \dots, f_{c}^{'}))_{g^{'}}$ with the same $n, c$ , and the image of $h$ is again a unit.

[F3]

Invertible Jacobian minor gives regular parameters in a polynomial fibre: under the Axiom of Choice, if $k$ is a field, $q \subseteq k [x_{1}, \dots, x_{n}]$ is prime and $f_{1}, \dots, f_{c} \in q$ have leading $c \times c$ Jacobian minor $h \notin q$ , then $k [x_{1}, \dots, x_{n}]_{q}$ is regular local, $(f_{1}, \dots, f_{c})$ is a regular sequence in it, and the quotient is regular local of dimension $\dim k [x_{1}, \dots, x_{n}]_{q} - c$ .

[F5]

Minimal primes are exactly the primes of height zero: a minimal prime ideal of a commutative ring has height $0$ .

[F7]

Equality, vanishing, and the kernel of the localisation map: for a multiplicative set $S \subseteq R$ and $r \in R$ , the class $r / 1$ is zero in $S^{- 1} R$ if and only if $u r = 0$ for some $u \in S$ ; a fraction $r / s$ equals $r^{'} / s^{'}$ if and only if $u (r s^{'} - r^{'} s) = 0$ for some $u \in S$ .

[F8]

Prime ideals of a localization are exactly the primes disjoint from the denominator set: for a multiplicative set $S \subseteq R$ , contraction along $R \to S^{- 1} R$ is an inclusion-preserving bijection from $Spec (S^{- 1} R)$ onto the primes of $R$ disjoint from $S$ , with inverse $p \mapsto S^{- 1} p$ .

[F9]

Localising twice is localising once at the multiplicative set generated by both denominator sets: for multiplicative sets $S, T \subseteq R$ with images $\overset{ˉ}{T}$ in $S^{- 1} R$ and $U$ the multiplicative set generated by $S \cup T$ , there is a unique $R$ -algebra isomorphism ${\overset{ˉ}{T}}^{- 1} (S^{- 1} R) ≅ U^{- 1} R$ .

[F10]

Localisation commutes with quotient rings: $S^{- 1} R / S^{- 1} I ≅ {\overset{ˉ}{S}}^{- 1} (R / I)$ : for an ideal $I ⊴ R$ and a multiplicative set $S$ , the image $\overset{ˉ}{S}$ of $S$ in $R / I$ gives a canonical isomorphism $(S^{- 1} R) / (S^{- 1} I) ≅ {\overset{ˉ}{S}}^{- 1} (R / I)$ , with both sides zero when $S \cap I \neq \emptyset$ .

[F11]

Height plus quotient dimension equals ambient dimension in an affine domain: under the Axiom of Choice, for a field $k$ , a finite-type $k$ -domain $A$ and $p \in Spec A$ one has $ht (p) + \dim (A / p) = \dim A$ .

[F12]

A polynomial ring in n variables over a field has dimension n: for a field $k$ and $n \geq 0$ , $\dim k [x_{1}, \dots, x_{n}] = n$ .

[F13]

Irreducible components of the spectrum correspond to minimal prime ideals: under the Axiom of Choice, the irreducible components of $Spec R$ are exactly the closed sets $V (p)$ for minimal primes $p$ of $R$ , each minimal prime giving a unique component.

[F14]

The spectrum of a quotient is a closed subspace: for an ideal $I ⊴ R$ , contraction along $R \to R / I$ is a homeomorphism from $Spec (R / I)$ onto $V (I)$ .

[F15]

The height of a prime ideal: the height of a prime ideal $p$ is $ht (p) = \dim (R_{p})$ .

[F16]

Affine-domain dimension equals transcendence degree: a finite-type domain $A$ over a field $K$ has dimension ${trdeg}_{K} Frac (A)$ .

Proof

1.1

Set $B : = K [x_{1}, \dots, x_{n}] / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c})$ and $\overset{ˉ}{h}$ for the image of $h$ in $B$ . By [F2] applied to $R \to κ$ the fibre is $F ≅ (κ [x_{1}, \dots, x_{n}] / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c}))_{\overset{ˉ}{g}}$ , and applying [F2] again to the ring map $κ \to K$ shows that $F_{K} ≅ B_{\overset{ˉ}{g}}$ , with the image of $\overset{ˉ}{h}$ in $B_{\overset{ˉ}{g}}$ a unit; the relative dimension $n - c$ is unchanged throughout.

F1F2

2.1

Primes of $B_{\overset{ˉ}{g}}$ correspond under [F8] to the primes $P \subseteq B$ with $\overset{ˉ}{g} \notin P$ , and these in turn correspond to the primes $Q^{'} \subseteq K [x_{1}, \dots, x_{n}]$ with $({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c}) \subseteq Q^{'}$ and $\overset{ˉ}{g} \notin Q^{'}$ . For such a $Q^{'}$ one has $\overset{ˉ}{h} \notin Q^{'}$ : since the image of $\overset{ˉ}{h}$ in $B_{\overset{ˉ}{g}}$ is a unit, there are $v \in B$ and $N \geq 0$ with $\overset{ˉ}{h} v = {\overset{ˉ}{g}}^{N}$ in $B_{\overset{ˉ}{g}}$ , so by [F7] there is $M \geq 0$ with ${\overset{ˉ}{g}}^{M} (\overset{ˉ}{h} v - {\overset{ˉ}{g}}^{N}) = 0$ in $B$ ; if $\overset{ˉ}{h}$ lay in $Q^{'}$ , hence in $P = Q^{'} / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c})$ , the contradiction ${\overset{ˉ}{g}}^{M + N} \in P$ with $\overset{ˉ}{g} \notin P$ would follow.

F7F8step 1.1

3.1

Let $Q$ be a prime of $B_{\overset{ˉ}{g}}$ , let $P \subseteq B$ be the prime it contracts to and let $Q^{'} \subseteq K [x_{1}, \dots, x_{n}]$ be the corresponding prime; then $(F_{K})_{Q} = B_{P} = (K [x_{1}, \dots, x_{n}]_{Q^{'}}) / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c})$ . Indeed $B_{\overset{ˉ}{g}} = (K [x_{1}, \dots, x_{n}] / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c}))_{\overset{ˉ}{g}}$ , localising further at $Q$ gives $B_{P}$ by [F9], and [F10] identifies $B_{P}$ with the quotient of $K [x_{1}, \dots, x_{n}]_{Q^{'}}$ by the ideal generated by the ${\overset{ˉ}{f}}_{j}$ .

F9F10step 2.1

4.1

Regularity of local rings. In the situation of step 3.1 the elements ${\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c}$ lie in the prime $Q^{'}$ and, by step 2.1, have leading minor $\overset{ˉ}{h} \notin Q^{'}$ ; so [F3] applies over the field $K$ and shows that $(F_{K})_{Q} = (K [x_{1}, \dots, x_{n}]_{Q^{'}}) / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c})$ is a regular local ring with $\dim (F_{K})_{Q} = \dim K [x_{1}, \dots, x_{n}]_{Q^{'}} - c = ht (Q^{'}) - c$ , the last equality by [F15]. This proves clause 1, including for $c = 0$ , where the empty leading minor is $1$ and the quotient is $K [x_{1}, \dots, x_{n}]_{Q^{'}}$ .

F3F15step 2.1step 3.1

5.1

Height of the ambient prime at a component. Let $P$ be a minimal prime of $F_{K} = B_{\overset{ˉ}{g}}$ , let $P$ be its contraction to $B$ , and let $Q^{'}$ be the corresponding prime of $K [x_{1}, \dots, x_{n}]$ . The prime $P$ is minimal in $B$ : a prime strictly below it avoids $\overset{ˉ}{g}$ and would localize to a prime strictly below $P$ by [F8]. The local ring $(F_{K})_{P}$ has dimension zero by [F5, F15], so clause 1, already proved in step 4.1, gives $0 = ht (Q^{'}) - c$ . Thus $ht (Q^{'}) = c$ , also when $c = 0$ .

F5F8F15step 4.1

6.1

Component dimension. Put $A : = K [x_{1}, \dots, x_{n}] / Q^{'}$ , a finite-type $K$ -domain, and let $g_{A}$ be the image of $\overset{ˉ}{g}$ . Then $g_{A} \neq 0$ , and [F10] gives $F_{K} / P ≅ A_{g_{A}}$ . The localization is again a finite-type $K$ -domain (adjoin $z$ with $z g_{A} = 1$ ), and has the same fraction field as $A$ . Applying [F16] to both rings gives $\dim A_{g_{A}} = \dim A$ . Now [F11, F12] and step 5.1 give $\dim A = n - ht (Q^{'}) = n - c$ . By [F13, F14], the component $V (P)$ is homeomorphic to $Spec (F_{K} / P)$ , so has dimension $n - c$ .

F10F11F12F13F14F16step 5.1

7.1

Both clauses hold: every local ring of $F_{K}$ is regular local of dimension $ht (Q^{'}) - c$ by step 4.1, and every irreducible component has dimension $n - c$ by step 6.1; if $F_{K}$ is the zero ring there are no primes and no components, so both clauses are vacuous. ∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Regularity ascends and descends along a flat local homomorphism

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $(R, m) \to (S, n)$ be a flat local homomorphism (Flat and faithfully flat modules and ring homomorphisms) of Noetherian local rings, so that $S / m S$ is again a Noetherian local ring. Then:

ascent. if $R$ is a regular local ring and the closed fibre $S / m S$ is regular, then $S$ is regular;
descent. if $S$ is regular, then $R$ is regular.

No regularity of the closed fibre is assumed in the descent statement, and no finiteness of the field extension $S / n$ over $R / m$ is assumed anywhere.

Facts & Assumptions

Given: A flat local homomorphism $(R, m) \to (S, n)$ of Noetherian local rings and the Axiom of Choice.

[F1]

[F2]

regular system of parameters: a regular system of parameters of a regular local ring of dimension $d$ is an ordered minimal generating tuple of its maximal ideal, of length $d$ ; the empty tuple when $d = 0$ .

[F3]

quotient and lifting regularity across a regular element: under the Axiom of Choice, if $(R, m)$ is nonzero Noetherian local, $x \in m$ is a nonzerodivisor and $R / (x)$ is regular, then $R$ is regular.

[F4]

A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra: under the Axiom of Choice, a flat ring homomorphism $f ⁣ : R \to S$ is faithfully flat if and only if for every proper ideal $I ⊊ R$ the extended ideal $I S$ is proper.

[F5]

Localisation And Faithfully Flat Base Change Of Regular Sequences: a regular sequence remains regular after faithfully flat base change.

[F6]

regular local residue field projective dimension dimension: under the Axiom of Choice, for a regular local ring $(R, m, k)$ of dimension $d$ one has ${pd}_{R} k = d$ .

[F7]

finite local modules admit minimal free resolutions: under the Axiom of Choice, every finite module over a nonzero Noetherian local ring has a resolution by finite-rank free modules.

[F8]

Flat and faithfully flat modules and ring homomorphisms: an $R$ -module $M$ is flat when $- \otimes_{R} M$ preserves exact sequences, and a ring map is flat when the target is flat as a module over the source.

[F9]

Tensoring is right exact: $- \otimes_{R} M$ is right exact, so applying it to $R \to R / m \to 0$ gives $(R / m) \otimes_{R} S ≅ S / m S$ .

[F10]

Projective dimension at most n iff the nth syzygy is projective: for $n \geq 1$ and a projective resolution $P_{∙} \to M$ , one has $pd (M) \leq n$ if and only if the $n$ -th syzygy is projective.

[F11]

Every projective module over a commutative ring is flat: every projective module over a commutative ring is flat, with no use of the Axiom of Choice.

[F12]

Finitely generated modules over a left Noetherian ring are Noetherian: every finitely generated module over a Noetherian ring is Noetherian, so its submodules are finitely generated.

[F13]

Flatness descends along faithfully flat base change: for a faithfully flat ring map $R \to S$ and an $R$ -module $N$ , the module $N$ is flat over $R$ if and only if $N \otimes_{R} S$ is flat over $S$ .

[F14]

A finite flat module over a Noetherian ring is finite projective: a finite flat module over a Noetherian commutative ring is finite projective.

[F15]

auslander buchsbaum serre regularity criterion: under the Axiom of Choice, a nonzero Noetherian local ring is regular if and only if its global dimension is finite, and then the global dimension equals the projective dimension of the residue field and equals the dimension.

[F16]

Every faithfully flat ring map is injective: a faithfully flat ring homomorphism is injective.

[F17]

regular local rings are domains and cohen macaulay: under the Axiom of Choice, a regular local ring is a domain, and every regular system of parameters is a regular sequence.

[F18]

A Noetherian local domain has dimension zero exactly when it is a field: a Noetherian local domain of dimension zero is a field.

[F19]

In a nonzero commutative ring, every proper ideal is contained in a maximal ideal: under the Axiom of Choice every proper ideal of a commutative ring is contained in a maximal ideal.

[F20]

A local ring is a nonzero commutative ring with a unique maximal ideal: a local ring has a unique maximal ideal, which contains every proper ideal.

[F21]

Field: a field is a nonzero commutative ring in which every nonzero element is a unit.

Proof

1.1

The map is faithfully flat. Since the homomorphism is local, $m S \subseteq n ⊊ S$ ; for every proper ideal $I ⊊ R$ the ideal $I$ is contained in the maximal ideal $m$ by [F20], so $I S \subseteq m S$ is proper. Hence $R \to S$ is faithfully flat by [F4].

F4F20

1.2

Descent, the case $d \geq 1$ : setting up the resolution. Assume $S$ regular of dimension $d \geq 1$ and put $k : = R / m$ . By [F7] choose a resolution $F_{∙} \to k \to 0$ by finite free $R$ -modules and let $Ω$ be its $d$ -th syzygy, a finitely generated $R$ -module by [F12]. Tensoring with the flat $R$ -module $S$ preserves exactness by [F8], and [F9] identifies the tensor of the augmentation with $S / m S$ , so $F_{∙} \otimes_{R} S \to S / m S \to 0$ is a free resolution of the $S$ -module $S / m S$ whose $d$ -th syzygy is $Ω \otimes_{R} S$ .

F7F8F9F12F1F6

2.1

Ascent, set-up. Assume $R$ regular of dimension $d$ with regular system of parameters $y_{1}, \dots, y_{d}$ , so that $(y_{1}, \dots, y_{d}) = m$ by [F2]; if $d = 0$ then $m = 0$ and $S = S / m S$ is regular by hypothesis. For $d \geq 1$ , the parameters are an $R$ -regular sequence by [F17]; hence [F5] and the faithful flatness of step 1.1 make $y_{1}, \dots, y_{d}$ an $S$ -regular sequence, and $S / (y_{1}, \dots, y_{d}) S = S / m S$ .

F2F5F17step 1.1

2.2

Descent, the case $d = 0$ . Assume now that $S$ is regular of dimension $d = 0$ . Then $S$ is a domain by [F17] and hence a field by [F18]. Because $R \to S$ is faithfully flat by step 1.1, the extension $m S$ is proper by [F4], and $m S$ is an ideal of the field $S$ , so $m S = 0$ ; moreover $R \to S$ is injective by [F16], so $m = 0$ . A nonzero element $x$ of the local ring $R$ is then a unit: otherwise $(x)$ would be a proper ideal, hence would lie in a maximal ideal by [F19], necessarily the unique maximal ideal $m = 0$ , forcing $x = 0$ . Thus $R$ is a field by [F21], in particular regular.

F4F16F17F18F19F21step 1.1

2.3

Descent, the case $d \geq 1$ : the syzygy is projective. Since $S$ is regular local of dimension $d$ , its global dimension is $d$ by [F15], so every $S$ -module, in particular $S / m S$ , has projective dimension at most $d$ ; by [F10] applied to the resolution of step 1.2, the $S$ -module $Ω \otimes_{R} S$ is projective, hence flat over $S$ by [F11].

F10F11F15step 1.2

3.1

Ascent, induction. For $0 \leq i \leq d$ put $S_{i} : = S / (y_{1}, \dots, y_{i}) S$ , a nonzero Noetherian local ring, so that $S_{0} = S$ , $S_{d} = S / m S$ and $S_{i} / (y_{i + 1} S_{i}) ≅ S_{i + 1}$ for $i < d$ with the image of $y_{i + 1}$ a nonzerodivisor on $S_{i}$ . If $S_{i + 1}$ is regular for some $0 \leq i < d$ , then [F3] applied to the nonzero Noetherian local ring $S_{i}$ and the nonzerodivisor $y_{i + 1}$ makes $S_{i}$ regular. Since $S_{d} = S / m S$ is regular by hypothesis when $d \geq 1$ , downward induction gives that $S = S_{0}$ is regular; together with the case $d = 0$ of step 2.1 this proves the ascent claim.

F3step 2.1

3.2

Descent, conclusion. By [F13] and the faithful flatness of step 1.1, the finite $R$ -module $Ω$ is flat over $R$ , hence finite projective by [F14]. Therefore the resolution $F_{∙} \to k$ of step 1.2 has projective $d$ -th syzygy, so ${pd}_{R} k \leq d$ by [F10], and [F15] makes the Noetherian local ring $R$ regular. Combined with step 2.2 this proves the descent claim for every $d$ .

F10F13F14F15step 1.1step 1.2step 2.3

4.1

Both claims are proved: the ascent in step 3.1 and the descent in steps 2.2 and 3.2; the case $d = 0$ was separated out in steps 2.2 and 3.2 because [F10] requires $n \geq 1$ . ∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Separable generation after finite purely inseparable extensions

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $K / k$ be a finitely generated field extension (Finitely generated field extensions $F (a_{1}, \dots, a_{r})$ ) whose characteristic is $p > 0$ . There are finite purely inseparable extensions $k^{'} / k$ and $K^{'} / K$ fitting into a commutative square of field embeddings $\begin{array}{ccc} k & ⟶ & K \\ ↓ & ↓ \\ k^{'} & ⟶ & K^{'} \end{array}$ such that $K^{'} / k^{'}$ is separably generated (Separating transcendence basis and separably generated extensions for the separability terminology). No perfectness of $k$ is assumed.

In characteristic $0$ the same conclusion holds with $k^{'} = k$ and $K^{'} = K$ , since a finitely generated extension of a perfect field is separably generated; the statement above is the positive-characteristic case, where $K^{'} / K$ and $k^{'} / k$ may both be nontrivial.

Facts & Assumptions

Given: A finitely generated field extension $K / k$ of characteristic $p > 0$ and the Axiom of Choice.

[F1]

Separating transcendence basis and separably generated extensions: for a finitely generated extension $K / k$ , a finite tuple $t_{1}, \dots, t_{r}$ is a separating transcendence basis when it is a transcendence basis and $K / k (t_{1}, \dots, t_{r})$ is finite separable; $K / k$ is separably generated when it admits such a tuple.

[F2]

Algebraic and transcendental elements and algebraic extensions: an element is algebraic over a subfield when it satisfies a nonzero polynomial over it, transcendental otherwise, and a set is algebraically independent when it satisfies no nonzero polynomial relation.

[F3]

A maximal algebraically independent set is a transcendence basis: an algebraically independent subset maximal for inclusion is a transcendence basis.

[F4]

An extension generated by finitely many algebraic elements is finite: a finitely generated algebraic field extension is finite.

[F5]

Separable algebraic elements and separable extensions: $α$ is separable over $F$ when it is algebraic with separable minimal polynomial, and $K / F$ is separable when every element of $K$ is separable over $F$ .

[F6]

The separable closure of the base inside an algebraic extension: for algebraic $K / F$ , the separable closure $K_{s} = {a \in K : a separable over F}$ is the largest intermediate field separable over $F$ .

[F7]

An algebraic extension is purely inseparable over its separable closure: $K / K_{s}$ is purely inseparable.

[F8]

Pure inseparability and its conjugate, embedding, and separable-degree criteria: for algebraic $K / F$ of characteristic $p$ , $K / F$ is purely inseparable if and only if every $α \in K$ has $α^{p^{e}} \in F$ for some $e \geq 0$ ; a finite extension is purely inseparable exactly when $[K : F]_{s} = 1$ .

[F9]

$[K : F] = [K : F]_{s} [K : F]_{i}$ , and in positive characteristic the inseparable degree is a power of $p$ : $[K : F] = [K : F]_{s} [K : F]_{i}$ for finite $K / F$ , and $[K : F]_{i}$ is a power of $p$ in characteristic $p$ .

[F10]

For a finite extension, $[K : F]_{s} = [K_{s} : F]$ : $[K : F]_{s} = [K_{s} : F]$ for finite $K / F$ .

[F11]

If $a$ is not a $p$ th power in a characteristic- $p$ field, then $x^{p^{n}} - a$ is irreducible for every $n \geq 1$ : if $F$ has characteristic $p > 0$ and $a \in F$ is not a $p$ -th power, then $x^{p^{n}} - a$ is irreducible in $F [x]$ for every $n \geq 1$ .

[F12]

Repeated roots in extension fields and separable polynomials: a polynomial is separable over $F$ when it has no repeated root in any extension field.

[F13]

A nonzero polynomial over a field is separable exactly when its gcd with its derivative is $1$ : $0 \neq f \in F [x]$ is separable if and only if $\gcd (f, f^{'}) = 1$ .

[F14]

The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element: $f (a) = 0$ if and only if the minimal polynomial $m_{a}$ of $a$ over $F$ divides $f$ .

[F15]

The binomial theorem over an arbitrary commutative ring: $(u + v)^{m} = \sum_{k} (\binom{m}{k}) u^{k} v^{m - k}$ in every commutative ring.

[F16]

A prime $p$ divides $(\binom{p}{k})$ for $0 < k < p$ : $p ∣ (\binom{p}{k})$ for $0 < k < p$ , so in characteristic $p$ the binomial theorem gives $(u + v)^{p} = u^{p} + v^{p}$ and, more generally, $(\sum_{i} u_{i})^{p} = \sum_{i} u_{i}^{p}$ .

[F17]

Pure inseparability is transitive in towers and stable under composita: purely inseparable extensions compose, and the compositum of purely inseparable subextensions of a common algebraic extension is purely inseparable over the base.

[F18]

Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ : for $F \subseteq K \subseteq L$ with $K / F$ and $L / K$ finite, $[L : F] = [L : K] [K : F]$ .

[F19]

Perfect fields: every irreducible polynomial is separable, A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective and Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: a field of characteristic $0$ is perfect, and in characteristic $p > 0$ perfectness is equivalent to every element being a $p$ -th power.

[F20]

Finitely generated extensions of a perfect field are separably generated: every finitely generated extension of a perfect field is separably generated.

[F21]

Finitely generated field extensions $F (a_{1}, \dots, a_{r})$ : $K / k$ is finitely generated when $K = k (α_{1}, \dots, α_{n})$ for finitely many elements.

Proof

1.1

Reduction and set-up. If the characteristic is $0$ , then $k$ is perfect by [F19], so [F20] makes $K / k$ separably generated and $k^{'} = k$ , $K^{'} = K$ are finite purely inseparable over their bases by [F8] (in characteristic $0$ the only purely inseparable extension is the trivial one). Assume from now on that the characteristic is $p > 0$ . Fix a finite generating set $α_{1}, \dots, α_{n}$ of $K$ over $k$ by [F21] and choose a maximal algebraically independent subset ${x_{1}, \dots, x_{r}}$ of it, which is a transcendence basis of $K / k$ by [F3, F2]. Put $F_{0} : = k (x_{1}, \dots, x_{r})$ ; every $α_{i}$ is algebraic over $F_{0}$ by maximality, so $K / F_{0}$ is finitely generated algebraic, hence finite by [F4]. Let $K_{s}$ be the separable closure of $F_{0}$ in $K$ ; then $K_{s} / F_{0}$ is finite separable by [F6, F4] and $K / K_{s}$ is purely inseparable by [F7], finite by [F18], so that $d : = \log_{p} [K : K_{s}]$ is a nonnegative integer by [F9, F8].

F2F3F4F6F7F8F9F18F19F20F21F10

1.2

An element of $K$ with a $p$ -th root of the right shape. Assume $d \geq 1$ , so $K \neq K_{s}$ . By [F8] and [F7] applied to an element of $K ∖ K_{s}$ there is $β_{0} \in K$ and a minimal $e \geq 1$ with $β_{0}^{p^{e}} \in K_{s}$ ; then $β : = β_{0}^{p^{e - 1}}$ satisfies $β \notin K_{s}$ by minimality of $e$ and $α : = β^{p} \in K_{s}$ . Since $α \in K_{s}$ is separable over $F_{0}$ by [F5, F6], its minimal polynomial $P \in F_{0} [T]$ over $F_{0}$ is separable by [F12, F5], hence $\gcd (P, P^{'}) = 1$ by [F13] and $P$ has pairwise distinct roots; write $P = \sum_{i} a_{i} T^{i}$ with $a_{i} \in F_{0} = k (x_{1}, \dots, x_{r})$ .

F5F6F7F8F12F13

1.3

A finite purely inseparable base change making the coefficients $p$ -th powers. Write each $a_{i} = f_{i} / g_{i}$ with $f_{i}, g_{i} \in k [x_{1}, \dots, x_{r}]$ and $g_{i} \neq 0$ , and let $C \subseteq k$ be the finite set of all coefficients occurring in the finitely many polynomials $f_{i}, g_{i}$ . Choose an algebraic closure $Ω$ of $K$ and inside it put $k^{'} : = k (c^{1 / p} : c \in C)$ , so that $k^{'} / k$ is finite purely inseparable by [F17, F8]; put $F : = k^{'} (x_{1}^{1 / p}, \dots, x_{r}^{1 / p}) \subseteq Ω$ , the compositum of $k^{'}$ and $F_{0}$ . Each monomial $x^{m}$ with $m \in N^{r}$ is a $p$ -th power in $F$ : writing $m = p m^{'} + m^{''}$ with $m^{''} \in {0, \dots, p - 1}^{r}$ one has $x^{m} = (x^{m^{'}} (x^{1 / p})^{m^{''}})^{p}$ . Each $c \in C$ equals $(c^{1 / p})^{p}$ , so every $f_{i}$ and $g_{i}$ is a finite sum of $p$ -th powers, hence a $p$ -th power by [F16], and therefore each $a_{i} = f_{i} / g_{i}$ is a $p$ -th power in $F$ . Write $a_{i} = c_{i}^{p}$ with $c_{i} \in F$ and put $R : = \sum_{i} c_{i} T^{i} \in F [T]$ .

F8F16F17

1.4

The $p$ -th root of $α$ is separable. By [F15] and [F16], $P (T^{p}) = \sum_{i} a_{i} T^{p i} = \sum_{i} c_{i}^{p} T^{p i} = (\sum_{i} c_{i} T^{i})^{p} = R (T)^{p}$ in $F [T]$ . Let $α_{1}, \dots, α_{n} \in Ω$ be the distinct roots of $P$ (so $n = \deg P = \deg R$ , and $n \geq 1$ ), and for each $j$ choose $γ_{j} \in Ω$ with $γ_{j}^{p} = α_{j}$ , which exists because $Ω$ is algebraically closed. Then $R (γ_{j})^{p} = P (γ_{j}^{p}) = P (α_{j}) = 0$ , so $R (γ_{j}) = 0$ , and the $γ_{j}$ are distinct because $γ_{j}^{p} = α_{j}$ are. Hence $R$ has $\deg R$ distinct roots in $Ω$ , so $R$ is separable over $F$ by [F12] and [F13] applied to its distinct-root factorisation. Since $α = β^{p}$ is a root of $P$ , the element $β$ satisfies $R (β)^{p} = P (β^{p}) = P (α) = 0$ , hence $R (β) = 0$ ; and $β \in L : = K \cdot F \subseteq Ω$ , a compositum which is finitely generated over $k^{'}$ . As $R$ is separable with the root $β$ , the minimal polynomial of $β$ over $F$ divides $R$ by [F14] and is separable, so $β$ is separable over $F$ by [F5] and lies in the separable closure $L_{s}$ of $F$ in $L$ by [F6].

F5F6F12F13F14F15F16

2.1

The case $d = 0$ . If $d = 0$ then $[K : K_{s}] = 1$ , so $K = K_{s}$ and $K / F_{0}$ is finite separable by [F6]; then $x_{1}, \dots, x_{r}$ is a separating transcendence basis of $K / k$ by [F1], and with $k^{'} = k$ , $K^{'} = K$ the conclusion holds trivially.

F1F6step 1.1

2.2

The separable closure of $F$ in $L$ and the degree drop. $L / K$ is finite purely inseparable and $F / F_{0}$ is purely inseparable. First, $K_{s} F / F$ is separable: $K_{s} / F_{0}$ is finite separable by step 1.1, and a compositum of a separable algebraic extension with a further extension is separable because the minimal polynomial over the larger field divides the separable minimal polynomial over the smaller one by [F14]. Second, $L / K_{s} F$ is purely inseparable by [F17]. Hence $L_{s} = K_{s} F$ : one inclusion holds because $K_{s} F / F$ is separable and $L_{s}$ is the largest separable intermediate field by [F6], and for the other, an element $γ \in L_{s}$ is separable over $F$ and has $γ^{p^{e}} \in K_{s} F$ for some $e \geq 0$ by [F8], so it is simultaneously separable and purely inseparable over $K_{s} F$ and therefore already lies in $K_{s} F$ . Since $β \in L_{s}$ and $β \notin K_{s}$ , the field $K_{s} (β)$ satisfies $[K_{s} (β) : K_{s}] = p$ : the polynomial $T^{p} - α \in K_{s} [T]$ vanishes at $β$ while $α$ is not a $p$ -th power in $K_{s}$ (a relation $α = γ^{p}$ would give $(β / γ)^{p} = 1$ , hence $β = γ$ by [F16]), so $T^{p} - α$ is irreducible by [F11] and is the minimal polynomial of $β$ over $K_{s}$ by [F14]. Put $E : = K \cap L_{s}$ , an intermediate field of $K / K_{s}$ containing $K_{s} (β)$ , and choose $θ_{1}, \dots, θ_{m} \in K$ with $K = E (θ_{1}, \dots, θ_{m})$ , possible since $K / E$ is finite by [F18]. Then $L = L_{s} \cdot K$ equals $L_{s} (θ_{1}, \dots, θ_{m})$ , and for each $i$ the minimal polynomial of $θ_{i}$ over $L_{s} (θ_{1}, \dots, θ_{i - 1})$ divides its minimal polynomial over $E (θ_{1}, \dots, θ_{i - 1})$ by [F14], so those two extensions satisfy the degree inequality $[L_{s} (θ_{1}, \dots, θ_{i}) : L_{s} (θ_{1}, \dots, θ_{i - 1})] \leq [E (θ_{1}, \dots, θ_{i}) : E (θ_{1}, \dots, θ_{i - 1})]$ ; multiplying over $i$ and applying [F18] twice yields $[L : L_{s}] \leq [K : E] \leq [K : K_{s} (β)] = [K : K_{s}] / p$ , the last equality being [F18] for the tower $K_{s} \subseteq K_{s} (β) \subseteq K$ .

F6F8F11F14F16F17F18step 1.4

3.1

Induction. The extension $L / k^{'}$ is finitely generated by [F21] and has characteristic $p$ , and by step 2.2 its invariant $\log_{p} [L : L_{s}]$ is strictly smaller than $d = \log_{p} [K : K_{s}]$ . Applying the induction hypothesis (on the nonnegative integer $d$ , with the same statement for the pair $L / k^{'}$ ) produces finite purely inseparable extensions $k^{''} / k^{'}$ and $L^{'} / L$ with $L^{'} / k^{''}$ separably generated. Then $k^{''} / k$ is finite purely inseparable by [F17, F9] and $L^{'} / K$ is finite purely inseparable by [F17] since $L / K$ is, so $k^{''}$ and $L^{'}$ satisfy the conclusion for $K / k$ . The base case $d = 0$ of the induction is step 2.1, so the assertion holds for every $d \geq 0$ .

F8F9F17F21step 2.1step 2.2

4.1

Conclusion. In characteristic $p > 0$ steps 1.2–1.4, 2.1, 2.2 and 3.1 produce the required finite purely inseparable $k^{'} / k$ and $K^{'} / K$ with $K^{'} / k^{'}$ separably generated, the induction being on the integer $d = \log_{p} [K : K_{s}]$ of step 1.1; the characteristic $0$ case is step 1.1. ∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Field tests for geometric regularity

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and let $A$ be a finite-type $k$ -algebra (Geometrically regular algebras and geometrically regular fibres). Then:

$A$ is geometrically regular over $k$ if and only if $A$ is a regular ring (regular noetherian ring) and $A \otimes_{k} k^{'}$ is regular for every finite purely inseparable field extension $k^{'} / k$ ;
if $A$ is geometrically regular over $k$ , then $A \otimes_{k} K$ is a regular ring for every field extension $K / k$ , not only for the finitely generated ones appearing in the definition;
conversely, if $K / k$ is a field extension such that $A \otimes_{k} K$ is geometrically regular over $K$ , then $A$ is geometrically regular over $k$ .

Clause 1 is the finite purely inseparable test, clause 2 removes the finite generation from the scalar extension, and clause 3 is descent of geometric regularity along the faithfully flat field extension $k \to K$ . The zero algebra is regular vacuously and all clauses hold for it.

Facts & Assumptions

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

[F1]

Geometrically regular algebras and geometrically regular fibres: $A$ is geometrically regular over $k$ when $A \otimes_{k} K$ is a regular Noetherian ring for every finitely generated field extension $K / k$ ; the condition on a single scalar extension is tested at primes, and geometric regularity of $A$ over $k$ does not presuppose that $A$ is regular, which is why clause 1 of the statement carries regularity of $A$ as a separate hypothesis.

[F2]

regular noetherian ring: a commutative Noetherian ring is regular when every prime localisation is a regular local ring, the zero ring being regular vacuously; the maximal-ideal test is proved with the localisation and polynomial-extension theorem.

[F3]

Separable generation after finite purely inseparable extensions: under the Axiom of Choice, for a finitely generated field extension $K / k$ of characteristic $p > 0$ there are finite purely inseparable extensions $k^{'} / k$ and $K^{'} / K$ with $K^{'} / k^{'}$ separably generated, and in characteristic $0$ one may take $k^{'} = k$ , $K^{'} = K$ .

[F4]

localisation and polynomial extension of regular rings: under the Axiom of Choice, localisations and finite polynomial extensions of a commutative regular Noetherian ring are regular, and regularity may be tested at maximal ideals.

[F5]

Separating transcendence basis and separably generated extensions: $K / k$ is separably generated when it has a finite transcendence basis $t_{1}, \dots, t_{s}$ with $K / k (t_{1}, \dots, t_{s})$ finite separable.

[F6]

A finite extension generated by elements all but possibly one of which are separable is simple: a finite extension generated by elements all but at most one of which are separable is simple; in particular a finite separable extension is simple.

[F7]

Regularity ascends and descends along a flat local homomorphism: under the Axiom of Choice, for a flat local homomorphism $(R, m) \to (S, n)$ of Noetherian local rings: if $R$ and $S / m S$ are regular then $S$ is regular, and if $S$ is regular then $R$ is regular.

[F8]

A nonzero polynomial over a field is separable exactly when its gcd with its derivative is $1$ : $0 \neq f \in F [x]$ is separable if and only if $\gcd (f, f^{'}) = 1$ .

[F9]

Every nonzero nonunit polynomial over a field factors into irreducible polynomials and For a nonconstant $p$ in $F [x]$ , the ideal $(p)$ is maximal and $F [x] / (p)$ is a field exactly when $p$ is irreducible: a nonzero nonunit polynomial over a field is a product of irreducibles, and the quotient by a nonconstant polynomial is a field exactly when that polynomial is irreducible.

[F10]

Tensoring is right exact: $- \otimes_{R} M$ is right exact, so $F [T] / (f) \otimes_{F} κ ≅ κ [T] / (\overset{ˉ}{f})$ for a field extension $κ / F$ .

[F11]

auslander buchsbaum serre regularity criterion: under the Axiom of Choice, a nonzero Noetherian local ring is regular exactly when every finite module over it has finite projective dimension, and then its global dimension equals its dimension.

[F12]

Noether normalisation yields module finiteness over a polynomial subring: a nonzero finite-type algebra over a field $k$ is module-finite over a polynomial ring $k [z_{1}, \dots, z_{d}]$ .

[F13]

Injective integral extensions preserve Krull dimension and A polynomial ring in n variables over a field has dimension n: an injective integral extension of nonzero commutative rings preserves dimension, and $\dim k [z_{1}, \dots, z_{d}] = d$ .

[F14]

Localisation does not increase Krull dimension: localising a commutative ring does not increase its dimension.

[F15]

Projective dimension of an object and Left and right global dimension of a ring: $pd (M)$ is the infimum of the lengths of projective resolutions of $M$ , and the global dimension of a ring is the supremum of the projective dimensions of its modules.

[F16]

Extension of scalars carries flat modules to flat modules and Every localization is flat, and localizing a flat module preserves flatness: extension of scalars along a flat ring map preserves flatness, and localisations are flat.

Proof

1.1

The forward implication of clause 1 is immediate: taking $K = k$ in [F1] exhibits $A$ itself as regular, and every finite purely inseparable $k^{'} / k$ is a finitely generated field extension, so [F1] makes $A \otimes_{k} k^{'}$ regular.

F1F2

1.2

A base-change fact used below: if $B$ is a regular Noetherian $k$ -algebra and $K / k$ is a separably generated field extension, then $B \otimes_{k} K$ is regular. Indeed, choose a separating transcendence basis $t_{1}, \dots, t_{s}$ and put $F : = k (t_{1}, \dots, t_{s})$ , so that $K / F$ is finite separable by [F5] and $K = F (γ)$ by [F6] with minimal polynomial $P \in F [T]$ satisfying $\gcd (P, P^{'}) = 1$ by [F8]. The ring $B_{1} : = B \otimes_{k} F$ is a localisation of the polynomial extension $B [t_{1}, \dots, t_{s}]$ , hence regular by [F4]; let $q$ be a prime of $B_{2} : = B \otimes_{k} K = B_{1} \otimes_{F} K$ and $p : = q \cap B_{1}$ , so that $(B_{1})_{p} \to (B_{2})_{q}$ is a flat local homomorphism of Noetherian local rings whose closed fibre is a localisation of $κ (p) \otimes_{F} K ≅ κ (p) [T] / (\overset{ˉ}{P})$ by [F10]. Reducing the Bezout identity for $\gcd (P, P^{'}) = 1$ modulo $p$ shows $\gcd (\overset{ˉ}{P}, {\overset{ˉ}{P}}^{'}) = 1$ , so $\overset{ˉ}{P}$ is separable by [F8] and factors into pairwise distinct irreducibles by [F9], whence $κ (p) [T] / (\overset{ˉ}{P})$ is a product of fields and its further localisation is a field, in particular regular; [F7] then makes $(B_{2})_{q}$ regular. As $q$ was arbitrary, $B_{2}$ is regular by [F2].

F2F4F5F6F7F8F9F10

2.1

The finite purely inseparable test implies the finitely generated case of clause 1: assume $A$ is regular and $A \otimes_{k} k^{'}$ is regular for every finite purely inseparable $k^{'} / k$ , and let $K / k$ be finitely generated. By [F3] there are finite purely inseparable extensions $k^{'} / k$ and $K^{'} / K$ with $K^{'} / k^{'}$ separably generated (in characteristic $0$ take both extensions trivial). The ring $A \otimes_{k} k^{'}$ is regular by hypothesis, so step 1.2 applied to $B : = A \otimes_{k} k^{'}$ and the separably generated extension $K^{'} / k^{'}$ makes $A \otimes_{k} K^{'} = (A \otimes_{k} k^{'}) \otimes_{k^{'}} K^{'}$ regular. Since $K^{'} / K$ is finite purely inseparable, the field extension $K \to K^{'}$ is faithfully flat and local, and for every prime $q$ of $A \otimes_{k} K$ and every prime $q^{'}$ of $A \otimes_{k} K^{'}$ over it the induced map $(A \otimes_{k} K)_{q} \to (A \otimes_{k} K^{'})_{q^{'}}$ is a flat local homomorphism of Noetherian local rings with regular target, so [F7] descends regularity; as $q$ was arbitrary, $A \otimes_{k} K$ is regular by [F2]. With step 1.1 this proves clause 1, since [F1] defines geometric regularity by the regular rings $A \otimes_{k} K$ over finitely generated $K / k$ .

F1F2F3F7step 1.1step 1.2

3.1

Arbitrary field extensions. Assume $A$ is geometrically regular over $k$ and let $K / k$ be any field extension, written as the filtered union of its finitely generated subextensions $K_{i}$ . Then $A \otimes_{k} K = ⋃_{i} A \otimes_{k} K_{i}$ is a filtered union, and each $A \otimes_{k} K_{i}$ is regular by clause 1 as proved in step 2.1. Let $q$ be a prime of $A \otimes_{k} K$ and put $R : = (A \otimes_{k} K)_{q}$ and $R_{i} : = (A \otimes_{k} K_{i})_{q_{i}}$ with $q_{i} : = q \cap (A \otimes_{k} K_{i})$ , so that $R = ⋃_{i} R_{i}$ is a filtered union of regular local rings. There is a uniform bound on dimensions: for $A \neq 0$ , if $A$ is module-finite over $k [z_{1}, \dots, z_{d}]$ by [F12], then base change along $k \to K_{i}$ presents $A \otimes_{k} K_{i}$ as a quotient of a finite free $K_{i} [z_{1}, \dots, z_{d}]$ -module by [F10], so it is integral over $K_{i} [z_{1}, \dots, z_{d}]$ and has dimension $d$ by [F13]; hence $\dim R_{i} \leq d$ by [F14]. For $A = 0$ the ring is the zero ring, regular by [F2].

F2F10F12F13F14step 2.1

4.1

The filtered union is regular. In the notation of step 3.1 let $M$ be a finitely generated $R$ -module; presenting $M$ by finitely many generators and relations, all structure constants lie in some stage, so $M ≅ M_{i} \otimes_{R_{i}} R$ for a finitely generated $R_{i}$ -module $M_{i}$ and some $i$ . By [F11] the regular local ring $R_{i}$ of dimension at most $d$ has $gldim R_{i} = \dim R_{i} \leq d$ , so $M_{i}$ has a projective resolution of length at most $d$ by [F15]; tensoring it with $R$ , which is flat over $R_{i}$ because it is a localisation of the flat base change $R_{i} \to R_{i} \otimes_{K_{i}} K$ by [F16], gives an exact sequence of projective $R$ -modules of length at most $d$ (a direct summand of a free module stays such after tensoring), hence ${pd}_{R} M \leq d$ by [F15]. As every finitely generated $R$ -module has finite projective dimension, [F11] makes $R$ regular; therefore $A \otimes_{k} K$ is regular by [F2], which is clause 2.

F2F11F15F16step 3.1

5.1

Descent, clause 3. Let $K / k$ be a field extension with $A \otimes_{k} K$ geometrically regular over $K$ , and let $E / k$ be a finitely generated field extension; we show $A \otimes_{k} E$ is regular. The $k$ -algebra $E \otimes_{k} K$ is nonzero, so choose a prime of it and let $L$ be its residue field, a field receiving both $E$ and $K$ for which $L / K$ is a finitely generated field extension. Then $A \otimes_{k} L = (A \otimes_{k} K) \otimes_{K} L = (A \otimes_{k} E) \otimes_{E} L$ is regular by clause 2, applied over $K$ to the geometrically regular $K$ -algebra $A \otimes_{k} K$ ; and $A \otimes_{k} E \to A \otimes_{k} L$ is faithfully flat and local at corresponding primes because $E \to L$ is a field extension, so [F7] descends regularity to every local ring of $A \otimes_{k} E$ , making it regular by [F2]. As $E$ was arbitrary, $A$ is geometrically regular over $k$ by [F1].

F1F2F7step 4.1

6.1

Clause 1 is step 2.1, clause 2 is step 4.1 and clause 3 is step 5.1; the zero algebra is covered by the vacuous regularity of [F2]. ∎

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Regular algebras over a perfect field are geometrically regular

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 $A$ be a finite-type $k$ -algebra that is regular (regular noetherian ring). Then $A \otimes_{k} K$ is a regular ring for every field extension $K / k$ , not only for the finitely generated ones. In particular a regular finite-type algebra over a perfect field is geometrically regular (Field tests for geometric regularity).

No assumption is made on the extension field $K$ : it need not be perfect. In characteristic $0$ the hypothesis that $k$ is perfect is automatic, and the statement says that regular finite-type algebras over such fields stay regular after arbitrary scalar extension.

Facts & Assumptions

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

[F1]

A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective: $F$ is perfect if and only if either $char F = 0$ , or $char F = p > 0$ and the Frobenius map $a \mapsto a^{p}$ is surjective; iterating, every element of a perfect field of characteristic $p$ is a $p^{e}$ -th power for every $e \geq 0$ .

[F2]

Perfect fields: every irreducible polynomial is separable: a field is perfect when every algebraic extension of it is separable, equivalently (in characteristic $p$ ) when its Frobenius endomorphism is surjective.

[F3]

Pure inseparability and its conjugate, embedding, and separable-degree criteria: for algebraic $K / F$ of characteristic $p$ , $K / F$ is purely inseparable if and only if every $α \in K$ has $α^{p^{e}} \in F$ for some $e \geq 0$ ; in characteristic $0$ a purely inseparable extension is trivial.

[F4]

The binomial theorem over an arbitrary commutative ring and A prime $p$ divides $(\binom{p}{k})$ for $0 < k < p$ : in a commutative ring of characteristic $p$ one has $(u + v)^{p} = \sum_{k} (\binom{p}{k}) u^{k} v^{p - k} = u^{p} + v^{p}$ and hence $(u - v)^{p^{e}} = u^{p^{e}} - v^{p^{e}}$ for every $e \geq 0$ ; in a field $t^{p^{e}} = 0$ forces $t = 0$ .

[F5]

Field tests for geometric regularity: under the Axiom of Choice, a finite-type $k$ -algebra $A$ is geometrically regular over $k$ if and only if $A$ is regular and $A \otimes_{k} k^{'}$ is regular for every finite purely inseparable $k^{'} / k$ ; and then $A \otimes_{k} K$ is regular for every field extension $K / k$ .

[F6]

regular noetherian ring: a commutative Noetherian ring is regular when all its prime localisations are regular local rings.

Proof

1.1

Every finite purely inseparable extension of $k$ is trivial. Let $k^{'} / k$ be finite purely inseparable. If the characteristic is $0$ , then $k^{'} = k$ by [F3]. If the characteristic is $p > 0$ , fix $a \in k^{'}$ ; by [F3] there is $e \geq 0$ with $a^{p^{e}} \in k$ , and since the Frobenius map of $k$ is surjective by [F1] we may write $a^{p^{e}} = b^{p^{e}}$ with $b \in k$ . Then $(a - b)^{p^{e}} = a^{p^{e}} - b^{p^{e}} = 0$ by [F4], and $k^{'}$ is a field, so $a = b \in k$ . Hence $k^{'} = k$ .

F1F3F4

2.1

Applying the field tests. The algebra $A$ is regular by hypothesis, and by step 1.1 the only finite purely inseparable extension of $k$ is $k$ itself, for which $A \otimes_{k} k = A$ is regular. Hence [F5] makes $A$ geometrically regular over $k$ , and its second clause makes $A \otimes_{k} K$ regular for every field extension $K / k$ .

F5step 1.1F6

3.1

Thus a regular finite-type algebra over a perfect field is geometrically regular, and every scalar extension $A \otimes_{k} K$ is regular, whether or not $K$ is perfect or finitely generated; in characteristic $0$ the perfectness hypothesis is automatic by

F1F2∎

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-sol)audited 2026-09-27Open item page →

Jacobian criterion and 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), let $P = k [x_{1}, \dots, x_{n}]$ with $n \geq 0$ , let $I \subseteq P$ be an ideal and put $A : = P / I$ . Fix elements $f_{1}, \dots, f_{r} \in P$ generating $I$ and let $J = (\partial f_{j} / \partial x_{i})$ be the Jacobian matrix (Differentials of a polynomial quotient and the Jacobian cokernel), of size $r \times n$ , with entries viewed in $A$ .

Closed-point criterion. Let $m \subseteq A$ be a maximal ideal whose residue field $κ = A / m$ is a finite separable extension of $k$ , and let $J (m)$ be the matrix over $κ$ obtained by evaluating the entries of $J$ . Then $A_{m}$ is a regular local ring if and only if ${rank}_{κ} J (m) = n - \dim A_{m} .$ In particular the rank of $J (m)$ does not depend on the chosen generating set of $I$ .
Local charts. If $q \in Spec A$ and $A_{q}$ is regular, then there are $g_{1}, \dots, g_{c} \in I$ and $t \in A ∖ q$ such that $I$ is generated by $g_{1}, \dots, g_{c}$ after inverting $t$ , some $c \times c$ minor of the Jacobian matrix $(\partial g_{j} / \partial x_{i})$ is a unit of $A_{t}$ , and therefore $A_{t}$ is a standard smooth $k$ -algebra (Standard smooth presentations and locally standard smooth maps). Here $c = ht (q \cap P) - \dim A_{q}$ (The height of a prime ideal).
Openness and density. The regular locus ${q \in Spec A : A_{q} regular}$ 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.

The remaining relations of $I$ beyond the $c$ chosen ones are killed by a Nakayama argument, so no appeal to a later local-presentation theorem is needed.

Facts & Assumptions

Given: A perfect field $k$ , the polynomial ring $P = k [x_{1}, \dots, x_{n}]$ , an ideal $I \subseteq P$ generated by $f_{1}, \dots, f_{r}$ , the algebra $A = P / I$ , and the Axiom of Choice.

[F1]

Differentials of a polynomial quotient and the Jacobian cokernel: $Ω_{P / k}$ is free on $d x_{1}, \dots, d x_{n}$ , and for $I = (f_{1}, \dots, f_{r})$ the module $Ω_{P / I / k}$ is the cokernel of the Jacobian matrix $(\partial f_{j} / \partial x_{i})$ acting from $A^{r}$ to $A^{n}$ .

[F2]

Separable residue and the cotangent sequence of a local algebra: for a Noetherian local $k$ -algebra $R$ with maximal ideal $m$ and residue field $κ$ finitely generated and separably generated over $k$ , the sequence $0 \to m / m^{2} \to Ω_{R / k} \otimes_{R} κ \to Ω_{κ / k} \to 0$ is exact; if $κ / k$ is finite separable, then $Ω_{κ / k} = 0$ and the first map is an isomorphism.

[F3]

Finitely generated extensions of a perfect field are separably generated: every finitely generated field extension of a perfect field is separably generated.

[F4]

Tensoring is right exact: tensoring is right exact, so the cokernel of the Jacobian matrix base changes to the cokernel of the base-changed matrix.

[F5]

regular system of parameters equivalent basis: under the Axiom of Choice, in a Noetherian local ring the classes of a regular system of parameters form a basis of $m / m^{2}$ , and conversely a lift of any basis generates the maximal ideal as a system of parameters.

[F6]

regular local regular quotient ideal is parameter generated: under the Axiom of Choice, for a regular local ring $(R, m, k)$ of dimension $d$ and an ideal $I \subseteq m$ , the quotient $R / I$ is regular if and only if $I$ is generated by an initial part of a regular system of parameters, if and only if $\dim_{k} ((I + m^{2}) / m^{2}) = d - \dim (R / I)$ .

[F7]

localisation and polynomial extension of regular rings: under the Axiom of Choice, localisations and finite polynomial extensions of a regular Noetherian ring are regular; in particular every prime localisation $P_{p}$ of the polynomial ring $P$ is a regular local ring of dimension $ht (p)$ .

[F8]

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

[F9]

Fibres of standard smooth algebras are regular of relative dimension: under the Axiom of Choice, for a standard smooth presentation with $n$ variables and $c$ equations over a commutative ring $R$ , every local ring of the base change to any field extension of the residue field $κ (p)$ of any $p \in Spec R$ is regular; over the field $R = k$ and the extension $k / k$ this says that every local ring of a standard smooth $k$ -algebra is regular.

[F10]

Prime ideals of a localization are exactly the primes disjoint from the denominator set: contraction is a bijection from the primes of a localisation onto the primes of the base ring avoiding the multiplicative set, so an element outside a prime stays outside every prime of the localisation at that prime and is a unit after further inverting it.

[F11]

Minimal primes are exactly the primes of height zero: a minimal prime ideal has height zero, that is, $\dim A_{p} = 0$ for a minimal prime $p$ of $A$ .

[F12]

A Noetherian ring is Artinian exactly when every prime ideal is maximal and An Artinian local ring has nilpotent maximal ideal, and its finite modules have finite length: under the Axiom of Choice, a Noetherian ring is Artinian exactly when all its primes are maximal, and the maximal ideal of an Artinian local ring is nilpotent.

[F13]

Irreducible topological spaces and irreducible subsets in the subspace topology: a nonempty open subset of an irreducible topological space is dense.

[F14]

Perfect fields: every irreducible polynomial is separable: a field is perfect when every algebraic extension of it is separable, equivalently when its Frobenius endomorphism is surjective in characteristic $p$ .

Proof

1.1

The closed-point criterion. Let $m \subseteq A$ be maximal with $κ = A / m$ finite separable over $k$ . The local ring $A_{m}$ is a Noetherian local $k$ -algebra with residue field $κ$ , which is finitely generated and separably generated over the perfect field $k$ by [F3]; since $κ / k$ is finite separable, [F2] gives $m / m^{2} ≅ Ω_{A_{m} / k} \otimes_{A_{m}} κ$ , and $Ω_{A_{m} / k} \otimes κ ≅ Ω_{A / k} \otimes_{A} κ$ . By [F1] and [F4] the latter is the cokernel of the matrix $J (m)$ acting from $κ^{r}$ to $κ^{n}$ , so its dimension is $n - {rank}_{κ} J (m)$ . Hence $edim A_{m} = n - {rank}_{κ} J (m)$ by [F8], and by [F8] again $A_{m}$ is regular if and only if $edim A_{m} = \dim A_{m}$ , that is, if and only if $rank J (m) = n - \dim A_{m}$ ; both $edim$ and $\dim$ are intrinsic, so the rank is independent of the generating set of $I$ .

F1F2F3F4F8F14

1.2

Local charts at regular points. Let $q \in Spec A$ with $A_{q}$ regular, let $p = q \cap P$ correspond to $q$ , and put $R : = P_{p}$ , a regular local ring of dimension $d : = ht (p)$ by [F7], with ideal $I^{'} : = I R$ and quotient $R / I^{'} = A_{q}$ of dimension $\dim A_{q}$ . By [F6] there are $y_{1}, \dots, y_{c} \in I^{'}$ forming an initial part of a regular system of parameters of $R$ , with $c = d - \dim A_{q}$ , and generating $I^{'}$ . Lifting the fractions to elements $g_{1}, \dots, g_{c} \in I$ that still generate $I^{'}$ , the classes of the $y_{j}$ span $p R / p^{2}$ modulo $p^{2}$ with $\dim_{κ (p)} ((I^{'} + p^{2}) / p^{2}) = c$ by [F6], and they are linearly independent by [F5]. The map $p R / p^{2} \to Ω_{R / k} \otimes_{R} κ (p) ≅ κ (p)^{n}$ is injective by [F2], the residue field $κ (p)$ being finitely generated and separably generated over the perfect field $k$ by [F3], and the images of the classes of the $g_{j}$ are the vectors $(\partial g_{j} / \partial x_{i} m o d p)_{i}$ by [F1]; these vectors are therefore linearly independent over $κ (p)$ , so some $c \times c$ minor $h$ of the Jacobian matrix of $g_{1}, \dots, g_{c}$ satisfies $h \notin p$ . Since $I$ is finitely generated and $I^{'} = (g_{1}, \dots, g_{c}) R$ , there is $s \notin p$ with $s I \subseteq (g_{1}, \dots, g_{c})$ ; put $t : = s h \in A$ . By [F10] the element $t$ is not in $q$ and becomes a unit in $A_{t}$ , and in $A_{t}$ the ideal $I$ is generated by $g_{1}, \dots, g_{c}$ while $h$ is a unit, so $A_{t} ≅ (P / (g_{1}, \dots, g_{c}))_{t}$ is a standard smooth $k$ -algebra with $n$ variables and $c$ equations.

F1F2F3F5F6F7F10

2.1

Openness of the regular locus. In the situation of step 1.2, [F9] applied over the field $k$ shows that every local ring of the standard smooth $k$ -algebra $A_{t}$ is regular; hence the whole distinguished open $D (t)$ consists of regular points and is a neighbourhood of $q$ . As $q$ was an arbitrary regular point, the regular locus is open in $Spec A$ .

F9step 1.2

3.1

Density along a generically reduced component. Let $p$ be a minimal prime of $A$ with $A_{p}$ reduced. By [F11] the local ring $A_{p}$ is Noetherian local of dimension $0$ , so all its primes are maximal and it is Artinian by [F12]; its maximal ideal is therefore nilpotent by [F12], and reducedness forces it to be zero, so $A_{p}$ is a field, in particular regular. Step 2.1 then provides $t \notin p$ with $D (t)$ contained in the regular locus, and $D (t) \cap V (p)$ is a nonempty open subset of the irreducible space $V (p)$ , hence dense in $V (p)$ by [F13]. When $A$ is reduced, every localisation $A_{p}$ at a minimal prime is reduced, so this applies to every irreducible component.

F11F12F13step 2.1

4.1

The three assertions are steps 1.1, 1.2 and 2.1 with the density statement of step 3.1; all of them use only the perfectness of $k$ through [F3] and [F2], and no later local-presentation theorem. ∎

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Flat maps with geometrically regular fibres have standard smooth local presentations

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $R \to S$ be a ring map of finite presentation. Choose a presentation $S ≅ P / I$ with $P = R [x_{1}, \dots, x_{n}]$ and $I = (f_{1}, \dots, f_{m})$ , and let $q^{'} \subseteq P$ be the preimage of $q \in Spec S$ ; put $p = q \cap R$ , $κ = κ (p)$ and $F : = S \otimes_{R} κ, I_{κ} : = I κ [x_{1}, \dots, x_{n}] .$ Assume

the local ring homomorphism $R_{p} \to S_{q}$ is flat, and
the fibre $F$ is geometrically regular at $q$ (Geometrically regular algebras and geometrically regular fibres): for every field extension $K / κ$ and every prime of $F \otimes_{κ} K$ lying over the prime of $F$ corresponding to $q$ , the local ring there is regular.

Then there are an integer $0 \leq c \leq n$ , elements $f_{1}, \dots, f_{c}$ selected from the chosen generating list for $I$ , a polynomial $u \in P ∖ q^{'}$ , and a $c \times c$ Jacobian minor of these $f_{j}$ whose image is a unit in $(P / (f_{1}, \dots, f_{c}))_{u}$ , such that, writing $\overset{ˉ}{u}$ for the image of $u$ in $S$ , there is an $R$ -algebra isomorphism $S_{\overset{ˉ}{u}} ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{u} .$ Thus $S_{\overset{ˉ}{u}}$ is standard smooth over $R$ and witnesses that $R \to S$ is standard smooth at $q$ (Standard smooth presentations and locally standard smooth maps). This is the converse direction of the equivalence between local standard smoothness and flatness with geometrically regular fibres.

Facts & Assumptions

Given: A ring map $R \to S$ of finite presentation, a presentation $S ≅ R [x_{1}, \dots, x_{n}] / I$ with $I = (f_{1}, \dots, f_{m})$ , a prime $q \subseteq S$ with $p = q \cap R$ , the primes $q^{'} \subseteq R [x_{1}, \dots, x_{n}]$ and $\overset{ˉ}{q} \subseteq F$ lying over $q$ , the fibre $F = S \otimes_{R} κ (p)$ , flatness of $R_{p} \to S_{q}$ , geometric regularity of $F$ at $q$ , and the Axiom of Choice.

[F1]

Geometrically regular algebras and geometrically regular fibres: for a finitely presented $R$ -algebra $S$ and $q \in Spec S$ over $p \in Spec R$ , the fibre $S \otimes_{R} κ (p)$ is geometrically regular at $q$ when for every field extension $K / κ (p)$ every local ring of $(S \otimes_{R} κ (p)) \otimes_{κ (p)} K$ at a prime lying over the prime corresponding to $q$ is a regular local ring; the fibre is $S \otimes_{R} κ (p)$ , and $κ (p) = R_{p} / p R_{p}$ .

[F2]

Standard smooth presentations and locally standard smooth maps: a standard smooth $R$ -presentation is a presentation $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with $n \geq c \geq 0$ in which some $c \times c$ minor of the Jacobian matrix $(\partial f_{j} / \partial x_{i})$ has image a unit of $S$ ; the invertible minor may be taken to be the leading one, in the first $c$ columns, and $n - c$ is the relative dimension.

[F3]

Differentials of a polynomial quotient and the Jacobian cokernel: for $P = A [x_{1}, \dots, x_{n}]$ the module $Ω_{P / A}$ is free on $d x_{1}, \dots, d x_{n}$ , the partial derivatives are computed on the monomial basis and extended $A$ -linearly, $d f = \sum_{i} \partial_{i} f d x_{i}$ , and the Jacobian matrix $(\partial_{i} f_{j})$ governs $Ω_{P / I / A}$ .

[F4]

Jacobian criterion and openness of the regular locus over a perfect field: under the Axiom of Choice, for a perfect field $k$ , $P = k [x_{1}, \dots, x_{n}]$ , $A = P / I$ and a maximal ideal $m \subseteq A$ whose residue field is a finite separable extension of $k$ , the local ring $A_{m}$ is regular if and only if ${rank}_{κ} J (m) = n - \dim A_{m}$ , where $J (m)$ is the Jacobian matrix of a generating set of $I$ evaluated at $m$ .

[F5]

regular local regular quotient ideal is parameter generated: under the Axiom of Choice, for a regular local ring $(R, m, k)$ of dimension $d$ and an ideal $I \subseteq m$ , the following are equivalent: $R / I$ is regular; $I$ is generated by an initial part of a regular system of parameters; and $\dim_{k} ((I + m^{2}) / m^{2}) = d - \dim (R / I)$ .

[F6]

Assuming the Axiom of Choice, Nakayama's lemma: under the Axiom of Choice, if $R$ is a commutative ring, $I \subseteq J (R)$ and $M$ is a finitely generated $R$ -module with $I M = M$ , then $M = 0$ .

[F7]

Assuming Choice, every field has an algebraic closure, An algebraic closure of a field and Fields of characteristic zero, finite fields, and algebraically closed fields are perfect: under the Axiom of Choice every field has an algebraic closure; an algebraic closure of a field is an algebraically closed algebraic extension; and every algebraically closed field is perfect.

[F8]

Height plus quotient dimension equals ambient dimension in an affine domain and A polynomial ring in n variables over a field has dimension n: under the Axiom of Choice, for a field $k$ , a finite-type $k$ -domain $A$ and $p \in Spec A$ one has $ht (p) + \dim (A / p) = \dim A$ , and $\dim k [x_{1}, \dots, x_{n}] = n$ .

[F9]

Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests and The long exact Tor sequence in the left-module variable: $M$ is flat over $R$ exactly when $I \otimes_{R} M \to M$ is injective for every ideal $I$ ; and under the Axiom of Dependent Choice, a short exact sequence $0 \to M^{'} \to M \to M^{''} \to 0$ of left $R$ -modules and a right module $N$ give the long exact sequence $\dots \to {Tor}_{1} (N, M) \to {Tor}_{1} (N, M^{''}) \to N \otimes_{R} M^{'} \to N \otimes_{R} M \to \dots$ ; in particular ${Tor}_{1}^{R} (R / I, M) = \ker (I \otimes_{R} M \to M)$ for flat $M$ .

[F10]

Localisation of modules is extension of scalars and Tensoring is right exact: localisation of modules is given by tensoring with the localised ring, so localising commutes with base change of scalars, and $- \otimes_{R} N$ is right exact, so a surjection stays surjective and $(M / L) \otimes_{R} N = (M \otimes_{R} N) / (L \otimes_{R} N)$ .

[F11]

Equality, vanishing, and the kernel of the localisation map and Universal property of localisation: maps that invert $S$ factor uniquely through $S^{- 1} R$ : in $S^{- 1} R$ a fraction $r / s$ is zero exactly when $u r = 0$ for some $u \in S$ . If a module $M$ is finitely generated, then $M_{q} = 0$ exactly when some $g \notin q$ annihilates $M$ : choose an annihilator outside $q$ for each of its finitely many generators and take their product. For a ring map carrying a multiplicative set into the units there is a unique extension to the localisation, so elements of the multiplicative set become units.

[F12]

The height of a prime ideal and $R_{p}$ is local with unique maximal ideal $p R_{p}$ : $ht (p) = \dim (R_{p})$ , and for a prime $p$ the localisation $R_{p}$ is a local ring with maximal ideal $p R_{p}$ .

[F13]

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain: the Axiom of Dependent Choice, used only through the Tor long exact sequence of [F9]; it is a consequence of the Axiom of Choice assumed in the statement.

[F14]

localisation and polynomial extension of regular rings: under the Axiom of Choice, finite polynomial extensions and localizations of a commutative regular Noetherian ring are regular. In particular, a finite polynomial ring over a field and its localization at any prime are regular local at that prime.

Proof

1.1

Set-up. Write $P : = R [x_{1}, \dots, x_{n}]$ , $S = P / I$ , $P_{κ} : = κ [x_{1}, \dots, x_{n}]$ and $I_{κ} = I P_{κ}$ , so that $F = P_{κ} / I_{κ}$ and the primes $q^{'} \subseteq P$ , $\overset{ˉ}{q} \subseteq F$ correspond to one another and contract to $q$ ; in the fibre, $\overset{ˉ}{q}$ corresponds to the prime ${\overset{ˉ}{q}}^{'} : = q^{'} P_{κ} + I_{κ} \subseteq P_{κ}$ and $F / \overset{ˉ}{q} = (S / q) \otimes_{R} κ$ has fraction field $κ (q)$ because $S / q$ is a domain.

F1F10given

2.1

The fibre is regular at its own residue field, so the fibre ideal has $c$ generators. Taking for $K$ the identity extension $κ / κ$ in [F1], the local ring $A^{'} : = P_{κ, {\overset{ˉ}{q}}^{'}}$ is regular local by [F14], since the field $κ$ is regular Noetherian, and has dimension $ht ({\overset{ˉ}{q}}^{'})$ by [F12], and its quotient $F_{\overset{ˉ}{q}} = A^{'} / I_{κ} A^{'}$ is a regular local ring by the hypothesis. Since $I_{κ} = (f_{1}, \dots, f_{m})$ and every $f_{j}$ lies in ${\overset{ˉ}{q}}^{'}$ , [F5] applies and shows that $I_{κ} A^{'}$ is generated by an initial part of a regular system of parameters of $A^{'}$ ; put $c : = ht ({\overset{ˉ}{q}}^{'}) - \dim F_{\overset{ˉ}{q}},$ so that $I_{κ} A^{'}$ is generated by $c$ elements whose classes in $I_{κ} A^{'} / {\overset{ˉ}{q}}^{'} (I_{κ} A^{'})$ are $κ (q)$ -independent. The classes of the images of $f_{1}, \dots, f_{m}$ span that $κ (q)$ -vector space of dimension $c$ , so after renumbering, the images of $f_{1}, \dots, f_{c}$ form a basis and generate $I_{κ} A^{'}$ by [F6] applied to the finitely generated $A^{'}$ -module $I_{κ} A^{'}$ .

F3F5F6F12F14step 1.1given

2.2

The $K$ -rational point of the fibre and the rank computation. By [F7] choose an algebraic closure $K / κ (q) \supseteq κ$ ; it is perfect by [F7]. The composite $F \otimes_{κ} K \to κ (q) \otimes_{κ} K \to K$ , $a \otimes b \mapsto a b$ , obtained from the algebraic closure $κ (q) \subseteq K$ , is a surjective $K$ -algebra homomorphism (it is $K$ -linear and hits $K$ ), and restricting it to $F$ recovers the quotient map $F \to F / \overset{ˉ}{q} ↪ κ (q)$ ; hence its kernel $Q$ is a maximal ideal of $F_{K} : = F \otimes_{κ} K$ with $F_{K} / Q ≅ K$ , so $Q$ lies over $\overset{ˉ}{q}$ and its preimage $\overset{ˉ}{n} \subseteq K [x_{1}, \dots, x_{n}]$ is maximal with $K [x] / \overset{ˉ}{n} ≅ K$ . By [F1] applied to $K / κ$ the local ring $(F_{K})_{Q}$ is regular local; put $d : = \dim (F_{K})_{Q}$ .

F1F7step 1.1given

3.1

The chosen equations also generate the ideal of the $K$ -fibre. Put $K [x] : = K [x_{1}, \dots, x_{n}]$ and $I_{K} : = I K [x] = I_{κ} K [x]$ , so $F_{K} = K [x] / I_{K}$ . Since the images of $f_{1}, \dots, f_{c}$ generate $I_{κ} A^{'}$ by step 2.1, the cokernel $I_{κ} A^{'} / (f_{1}, \dots, f_{c}) A^{'}$ is zero, so $I_{K} K [x]_{\overset{ˉ}{n}} = (f_{1}, \dots, f_{c}) K [x]_{\overset{ˉ}{n}}$ by [F10] (right exactness of base change of scalars, and localisation commuting with it). Hence $T_{K} : = K [x] / (f_{1}, \dots, f_{c})$ satisfies $(T_{K})_{\overset{ˉ}{n}} = K [x]_{\overset{ˉ}{n}} / I_{K} K [x]_{\overset{ˉ}{n}} = (F_{K})_{Q}$ , which is regular local of dimension $d$ . Now $\overset{ˉ}{n}$ is a maximal ideal of the finite-type $K$ -domain $K [x]$ , so $ht (\overset{ˉ}{n}) + \dim (K [x] / \overset{ˉ}{n}) = \dim K [x] = n$ by [F8], that is, $ht (\overset{ˉ}{n}) = n$ .

F8F10step 2.1step 2.2

3.2

The remaining relations die after inverting. Put $T : = R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c})$ and $J : = \ker (T \to S) = I / (f_{1}, \dots, f_{c})$ ; the ideal $J$ is finitely generated because $I$ is. Since $f_{1}, \dots, f_{c}$ generate $I_{κ} P_{κ, {\overset{ˉ}{q}}^{'}}$ by step 2.1, the map $T_{q^{'}} \otimes_{R_{p}} κ ⟶ S_{q} \otimes_{R_{p}} κ, κ [x]_{{\overset{ˉ}{q}}^{'}} / ({\overset{ˉ}{f}}_{1}, \dots, {\overset{ˉ}{f}}_{c}) ⟶ κ [x]_{{\overset{ˉ}{q}}^{'}} / I_{κ},$ is an isomorphism. The ring $S_{q}$ is flat over $R_{p}$ by hypothesis, so [F9] gives ${Tor}_{1}^{R_{p}} (κ, S_{q}) = 0$ ; the Tor long exact sequence of [F9] applied to $0 \to J_{q^{'}} \to T_{q^{'}} \to S_{q} \to 0$ therefore makes $J_{q^{'}} \otimes_{R_{p}} κ \to T_{q^{'}} \otimes_{R_{p}} κ$ injective, while its composite with the isomorphism above is zero; hence $J_{q^{'}} \otimes_{R_{p}} κ = 0$ , that is, $J_{q^{'}} = p J_{q^{'}}$ . The Axiom of Dependent Choice assumed through [F9] is a consequence of the Axiom of Choice assumed in the statement, and is used only here.

F9F13step 2.1given

4.1

The rank and the minor. Apply [F4] with $k : = K$ (perfect), $P : = K [x]$ , $I : = (f_{1}, \dots, f_{c})$ — a generating set of that ideal, with the $f_{j}$ now viewed in $K [x]$ — $A : = T_{K}$ and $m : = \overset{ˉ}{n} / (f_{1}, \dots, f_{c}) \subseteq A$ : this maximal ideal has residue field $A / m = K$ , a finite separable extension of $K$ , and $A_{m} = (T_{K})_{\overset{ˉ}{n}} = (F_{K})_{Q}$ is regular local of dimension $d$ by step 3.1. The criterion gives ${rank}_{K} J (m) = n - \dim A_{m} = n - d = : μ_{K},$ where $J$ is the $c \times n$ Jacobian matrix of $f_{1}, \dots, f_{c}$ . Moreover $μ_{K} = \dim_{K} (I_{K} K [x]_{\overset{ˉ}{n}} / \overset{ˉ}{n} I_{K} K [x]_{\overset{ˉ}{n}}) = \dim_{K} ((I_{κ} A^{'} / {\overset{ˉ}{q}}^{'} I_{κ} A^{'}) \otimes_{κ (q)} K) \geq \dim_{κ (q)} (I_{κ} A^{'} / {\overset{ˉ}{q}}^{'} I_{κ} A^{'}) = c,$ where the middle identification is base change of scalars by [F10] applied to the quotient $I_{κ} A^{'} / (f_{1}, \dots, f_{c}) A^{'} = 0$ of step 2.1 and the last inequality is that the dimension of a vector space cannot drop under a field extension. Therefore ${rank}_{K} J (m) = c$ , so some $c \times c$ minor $h$ of the Jacobian matrix $(\partial f_{j} / \partial x_{i})$ has nonzero image in $K [x] / \overset{ˉ}{n} = K$ .

F4F10step 2.2step 3.1algebra

5.1

The minor descends to $q$ . By the monomial formula of [F3], partial differentiation is linear over the coefficient ring, so the image in $K [x]$ of the minor $h \in R [x_{1}, \dots, x_{n}]$ is the corresponding minor of the images of $f_{1}, \dots, f_{c}$ ; since its image in $K [x] / \overset{ˉ}{n}$ is nonzero we get $h \notin \overset{ˉ}{n}$ , hence $h \notin \overset{ˉ}{n} \cap R [x] = q^{'}$ because $\overset{ˉ}{n}$ lies over $q^{'}$ , and hence $h \notin q$ .

F3step 4.1given

6.1

Conclusion. The ring $T_{q^{'}}$ is local with maximal ideal $q^{'} T_{q^{'}}$ , which contains $p T_{q^{'}}$ because $p \subseteq q^{'}$ , and $J_{q^{'}}$ is a finitely generated $T_{q^{'}}$ -module; so $J_{q^{'}} = p J_{q^{'}}$ forces $J_{q^{'}} = 0$ by [F6]. By [F11] there is $g \in P ∖ q^{'}$ , with $J_{g} = 0$ , that is $T_{g} ≅ S_{\overset{ˉ}{g}}$ as $R$ -algebras, where $\overset{ˉ}{g}$ is the image of $g$ in $S$ . Since $h \notin q^{'}$ by step 5.1, put $u : = g h \in P ∖ q^{'}$ and write $\overset{ˉ}{u}$ for its image in $S$ . The image of $h$ is a unit in $S_{\overset{ˉ}{u}}$ because $u = g h$ is inverted, and $J_{u} = 0$ because $J_{g} = 0$ ; hence $S_{\overset{ˉ}{u}} ≅ T_{u} = (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{u}$ with the $c \times c$ Jacobian minor $h$ a unit, a standard smooth presentation of relative dimension $n - c$ by [F2]. Since $u \notin q^{'}$ , its image $\overset{ˉ}{u} \notin q$ , so this chart witnesses that $R \to S$ is standard smooth at $q$ .

F2F6F11step 5.1step 3.2∎

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Locally standard smooth iff flat with geometrically regular fibres

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $R \to S$ be a ring map of finite presentation (Finitely presented modules and finitely presented algebras).

Pointwise criterion. Let $q \in Spec S$ , put $p = q \cap R$ and $κ = κ (p)$ . Then $R \to S$ is standard smooth at $q$ (Standard smooth presentations and locally standard smooth maps) if and only if the local ring homomorphism $R_{p} \to S_{q}$ is flat and the fibre $S \otimes_{R} κ (p)$ is geometrically regular at $q$ (Geometrically regular algebras and geometrically regular fibres).
Global form. The map $R \to S$ is locally standard smooth if and only if $R \to S$ is flat and every fibre $S \otimes_{R} κ (p)$ , $p \in Spec R$ , is geometrically regular.
Field case. Let $k$ be a field and $A$ a finite-type $k$ -algebra. Then $A$ is geometrically regular over $k$ if and only if the structure map $k \to A$ is locally standard smooth; equivalently, if and only if $A$ admits a standard smooth presentation over $k$ at every prime. In that case the relative dimension of a standard smooth chart at a prime $q$ is the dimension of the regular local ring $A_{q}$ when $q$ is a $k$ -rational point.

Clause 1 is the pointwise form of the classical equivalence between smoothness and flatness with geometrically regular fibres; clause 2 is its global form, and finite presentation is needed in both directions (locally standard smooth maps are finitely presented by definition, and the fibre condition is only defined for a finitely presented $R$ -algebra). No hypothesis is placed on $R$ .

Facts & Assumptions

Given: A ring map $R \to S$ of finite presentation, a prime $q \in Spec S$ with $p = q \cap R$ and $κ = κ (p)$ , the fibre $F = S \otimes_{R} κ (p)$ , a finite-type $k$ -algebra $A$ in clause 3, and the Axiom of Choice.

[F1]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation of an $R$ -algebra $S$ consists of integers $n \geq c \geq 0$ , elements $f_{1}, \dots, f_{c}$ and $g$ of $R [x_{1}, \dots, x_{n}]$ with $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ such that some $c \times c$ minor of the Jacobian matrix $(\partial f_{j} / \partial x_{i})$ has image a unit of $S$ ; $n - c$ is the relative dimension, the invertible minor may be assumed leading, and a further principal localisation may be absorbed. The map $R \to S$ is standard smooth at $q$ when $S_{h}$ has a standard smooth presentation over $R$ for some $h \notin q$ , and locally standard smooth when this holds at every prime; finite presentation of $S$ over $R$ is part of the definition of standard smoothness at a prime, as well as of the fibre condition.

[F2]

Standard smooth algebras are finitely presented and flat: under the Axiom of Choice, a standard smooth $R$ -algebra $S$ is a finitely presented $R$ -algebra and is flat over $R$ , for every commutative ring $R$ .

[F3]

Fibres of standard smooth algebras are regular of relative dimension: under the Axiom of Choice, for a standard smooth $R$ -algebra $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with leading minor a unit, a prime $p \in Spec R$ and a field extension $K / κ (p)$ , every local ring $(F_{K})_{Q}$ of the fibre $F_{K} = (S \otimes_{R} κ (p)) \otimes_{κ (p)} K$ is a regular local ring with $\dim (F_{K})_{Q} = ht (Q^{'}) - c$ , where $Q^{'} \subseteq K [x_{1}, \dots, x_{n}]$ is the prime corresponding to $Q$ , and every irreducible component of $Spec F_{K}$ has dimension $n - c$ .

[F4]

Flat maps with geometrically regular fibres have standard smooth local presentations: under the Axiom of Choice, if $R \to S$ is of finite presentation, $q \in Spec S$ , $p = q \cap R$ , the local homomorphism $R_{p} \to S_{q}$ is flat and the fibre $S \otimes_{R} κ (p)$ is geometrically regular at $q$ , then there is $g \notin q$ such that $S_{g}$ admits a standard smooth presentation over $R$ ; that is, $R \to S$ is standard smooth at $q$ .

[F5]

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 $K / k$ ; for a finitely presented $R$ -algebra $S$ , a prime $q$ with $p = q \cap R$ , the fibre is $S \otimes_{R} κ (p)$ and it is geometrically regular at $q$ when for every field extension $K / κ (p)$ and every prime of $(S \otimes_{R} κ (p)) \otimes_{κ (p)} K$ lying over the image of $q$ the local ring there is regular; a fibre is geometrically regular when it is geometrically regular at each of its points.

[F6]

Field tests for geometric regularity: under the Axiom of Choice, for a finite-type $k$ -algebra $A$ : $A$ is geometrically regular over $k$ if and only if $A$ is regular and $A \otimes_{k} k^{'}$ is regular for every finite purely inseparable $k^{'} / k$ ; if $A$ is geometrically regular over $k$ then $A \otimes_{k} K$ is regular for every field extension $K / k$ ; and if $A \otimes_{k} K$ is geometrically regular over $K$ for one field extension $K / k$ then $A$ is geometrically regular over $k$ .

[F7]

Modules over a field are projective, flat, and injective: under the Axiom of Choice every module over a field $k$ is free, hence projective and flat.

[F8]

Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests, Assuming the Axiom of Choice, local criteria for zero modules and for injective, surjective, and bijective maps: an $R$ -module $M$ is flat if and only if $I \otimes_{R} M \to M$ is injective for every ideal $I \subseteq R$ ; and under the Axiom of Choice an $R$ -module $M$ is zero if and only if $M_{m} = 0$ for every maximal ideal $m \subseteq R$ , equivalently for every prime.

[F9]

Every localization is flat, and localizing a flat module preserves flatness: for a commutative ring $R$ and multiplicative set $T \subseteq R$ , the localisation $T^{- 1} R$ is a flat $R$ -algebra, and a $T^{- 1} R$ -module is flat over $R$ if and only if it is flat over $T^{- 1} R$ .

[F10]

Localisation of modules is extension of scalars, Localisation commutes with kernels images and cokernels, Injective module maps remain injective after localisation, Localising twice is localising once at the multiplicative set generated by both denominator sets: localisation of modules is given by tensoring with the localised ring and commutes with kernels, images and cokernels, so localising preserves injectivity and commutes with base change of scalars; and for multiplicative sets $T, U$ the iterated localisation $(T^{- 1} R)_{U}$ is the localisation at the multiplicative set generated by $T$ and $U$ .

[F11]

regular noetherian ring: a commutative Noetherian ring is regular when its localisation at every prime is a regular local ring; this holds vacuously for the zero ring.

[F12]

Finitely presented modules and finitely presented algebras: a commutative $R$ -algebra $A$ is finitely presented when $A ≅ R [x_{1}, \dots, x_{m}] / a$ for some $m$ and a finitely generated ideal $a$ .

[F13]

The Axiom of Choice: the Axiom of Choice, assumed in the statement and used through [F2], [F3], [F4], [F6], [F7] and [F8].

[F14]

Every affine scheme is quasi-compact: every affine scheme is quasi-compact, so $Spec S$ and $Spec A$ are quasi-compact, and a family of principal opens covering either of them has a finite subcover whose elements generate the unit ideal.

[F15]

A polynomial ring in n variables over a field has dimension n, Maximal ideals of an affine domain have full height: for a field $k$ one has $\dim k [x_{1}, \dots, x_{n}] = n$ , and a maximal ideal of a finite-type $k$ -domain has height equal to the dimension of that domain; in particular a maximal ideal $Q^{'}$ of $k [x_{1}, \dots, x_{n}]$ satisfies $ht (Q^{'}) = n$ .

[F16]

Every algebra of finite type over a Noetherian ring is finitely presented: a finite-type algebra over a Noetherian ring is finitely presented; in particular this holds over a field.

Proof

1.1

Set-up and conventions. Write $F : = S \otimes_{R} κ (p)$ for the fibre over $p$ ; by [F5] the fibre is defined because $R \to S$ is finitely presented, and "geometrically regular at $q$ " means that for every field extension $K / κ (p)$ and every prime $Q$ of $F_{K} = F \otimes_{κ (p)} K$ lying over the image of $q$ in $F$ , the local ring $(F_{K})_{Q}$ is regular. Since a standard smooth chart at $q$ is by definition a standard smooth presentation of some $S_{h}$ , $h \notin q$ [F1], and since $S_{h}$ is finitely presented over $R$ when $S$ is [F2, F12], both sides of clause 1 only concern finitely presented $R$ -algebras.

F1F2F5F12givenF13

1.2

Flatness from a principal cover. Suppose $g_{1}, \dots, g_{k} \in S$ generate the unit ideal of $S$ and each $S_{g_{i}}$ is flat over $R$ . Then $S$ is flat over $R$ . Indeed, let $I \subseteq R$ be an ideal and let $K : = \ker (I \otimes_{R} S \to S)$ ; localising the map at $g_{i}$ gives the map $I \otimes_{R} S_{g_{i}} \to S_{g_{i}}$ by [F10], which is injective because $S_{g_{i}}$ is flat over $R$ , so $K_{g_{i}} = 0$ for every $i$ by [F10]. If $K \neq 0$ , then $K_{m} \neq 0$ for some maximal ideal $m \subseteq S$ by [F8], and since the $g_{i}$ generate the unit ideal some $g_{i} \notin m$ ; then $K_{m} = (K_{g_{i}})_{m} = 0$ by [F10], a contradiction. Hence $K = 0$ , so every such multiplication map is injective and $S$ is flat over $R$ by [F8].

F8F10given

1.3

Clause 1, only-if: flatness at $q$ . Assume $R \to S$ is standard smooth at $q$ , and choose $g \notin q$ such that $A : = S_{g}$ has a standard smooth presentation over $R$ [F1]. Then $A$ is flat over $R$ by [F2], so for every ideal $I \subseteq R$ the map $I \otimes_{R} A \to A$ is injective by [F8]; the localisation $S_{q} = T^{- 1} A$ at $T = S ∖ q$ is a localisation of the $R$ -module $A$ , so $I \otimes_{R} S_{q} \to S_{q}$ is the localisation of that injective map and is injective by [F10]; hence $S_{q}$ is flat over $R$ by [F8]. Because $R ∖ p$ maps into $T$ , the ring $S_{q}$ is an $R_{p}$ -algebra, so [F9] upgrades flatness over $R$ to flatness over $R_{p}$ : the local homomorphism $R_{p} \to S_{q}$ is flat.

F1F2F8F9F10given

1.4

Clause 3, if direction. Conversely let $k \to A$ be locally standard smooth, and let $q \in Spec A$ ; choose $g \notin q$ with $A_{g}$ standard smooth over $k$ [F1]. For every field extension $K / k$ , [F3] applied to the standard smooth $k$ -algebra $A_{g}$ with $p = (0)$ and fibre $(A_{g} \otimes_{k} k) \otimes_{k} K = A_{g} \otimes_{k} K$ shows that every local ring of $A_{g} \otimes_{k} K$ is regular; a prime $Q \subseteq A \otimes_{k} K$ lying over $q$ does not contain $\overset{ˉ}{g}$ , and [F10] identifies $(A \otimes_{k} K)_{Q}$ with the local ring of $A_{g} \otimes_{k} K$ at the corresponding prime, so it is regular. As $K$ was arbitrary, $A$ is geometrically regular at $q$ in the sense of [F5]; in particular, taking $K$ finitely generated over $k$ , the finite-type $K$ -algebra $A \otimes_{k} K$ has all its prime localisations regular, so it is a regular Noetherian ring by [F11] and $A$ is geometrically regular over $k$ .

F1F3F5F10F11given

2.1

Clause 1, only-if: geometric regularity of the fibre at $q$ . Keep the chart $A = S_{g}$ of step 1.3. Since $R \to S$ is finitely presented, so is the coefficient extension $R \to A$ , and $A \otimes_{R} κ (p) ≅ (S \otimes_{R} κ (p))_{g} = F_{g}$ by [F10]; write $\overset{ˉ}{g}$ for the image of $g$ in $F$ , so $\overset{ˉ}{g} \notin q_{F}$ , where $q_{F} \subseteq F$ is the image of $q$ . Let $K / κ (p)$ be a field extension and let $Q \subseteq F_{K}$ be a prime lying over $q_{F}$ ; then $\overset{ˉ}{g} \notin Q$ , and [F10] identifies $(F_{K})_{Q}$ with the local ring of $(F_{K})_{\overset{ˉ}{g}} = (A \otimes_{R} κ (p)) \otimes_{κ (p)} K$ at the corresponding prime. That local ring is regular by [F3] applied to the standard smooth $R$ -algebra $A$ and the extension $K / κ (p)$ . Since $K$ and $Q$ were arbitrary, $F$ is geometrically regular at $q$ by [F5].

F3F5F10step 1.3given

3.1

Clause 1, if direction. If $R_{p} \to S_{q}$ is flat and the fibre $F$ is geometrically regular at $q$ , then [F4] produces $g \notin q$ with $S_{g}$ standard smooth over $R$ , that is, $R \to S$ is standard smooth at $q$ . Steps 1.3 and 2.1 give the converse, so clause 1 holds.

F4step 1.3step 2.1

3.2

Clause 2, only-if. Assume $R \to S$ is locally standard smooth. For each $q \in Spec S$ choose $g_{q} \notin q$ with $S_{g_{q}}$ standard smooth over $R$ [F1]; the open sets $D (g_{q})$ cover the affine, hence quasi-compact, scheme $Spec S$ [F14], so finitely many of them, say for $q_{1}, \dots, q_{k}$ , already cover, and their elements $g_{1}, \dots, g_{k}$ generate the unit ideal of $S$ . Each $S_{g_{i}}$ is flat over $R$ by [F2], so $S$ is flat over $R$ by step 1.2. For the fibres, let $p \in Spec R$ and let $Q \subseteq Spec F$ be a point of the fibre, with image $q \in Spec S$ ; choosing the chart at that $q$ and applying step 2.1 shows that the local ring of the fibre at the prime corresponding to $Q$ — after any field extension of $κ (p)$ — is regular, so $F$ is geometrically regular at $q$ and hence the whole fibre over $p$ is geometrically regular by [F5]. As $p$ was arbitrary, $R \to S$ is flat with geometrically regular fibres.

F1F2F5F14step 1.2step 2.1

4.1

Clause 2, if direction. Assume $R \to S$ is flat and every fibre $S \otimes_{R} κ (p)$ is geometrically regular. Fix $q \in Spec S$ , $p = q \cap R$ . Flatness of $S$ over $R$ localises: $S_{q}$ is flat over $R$ by [F9, F10] applied to the localisation of the flat $R$ -module $S$ , hence flat over $R_{p}$ by [F9] since $S_{q}$ is an $R_{p}$ -module. The fibre condition is exactly hypothesis 2 of [F4] at $q$ , because a fibre that is geometrically regular at each of its points is geometrically regular at $q$ [F5]. So [F4] gives $g \notin q$ with $S_{g}$ standard smooth over $R$ . As $q$ was arbitrary, $R \to S$ is locally standard smooth.

F4F5F9F10step 3.1

4.2

Clause 3, only-if. Let $k$ be a field and $A$ a finite-type, hence finitely presented [F16], $k$ -algebra that is geometrically regular over $k$ ; then the local homomorphism $k \to A_{q}$ is flat for every prime $q \subseteq A$ because every $k$ -module is flat [F7, F9], and the only prime of $k$ is $(0)$ with residue field $k$ , so the fibre is $A \otimes_{k} k ≅ A$ . By [F6] the geometric regularity of $A$ over $k$ makes $A \otimes_{k} K$ a regular ring for every field extension $K / k$ , not only the finitely generated ones; by [F11] this says precisely that every local ring of $(A \otimes_{k} k) \otimes_{k} K$ at a prime lying over a given prime of $A$ is regular, so $A$ is geometrically regular at every prime in the sense of [F5]. Clause 1 (step 3.1) then gives a standard smooth chart of $A$ over $k$ at every prime, that is, $k \to A$ is locally standard smooth.

F5F6F7F9F11F16step 3.1

5.1

The relative-dimension clause of clause 3. Let $q \in Spec A$ be a $k$ -rational point of the finite-type $k$ -algebra $A$ , that is $A / q = k$ as $k$ -algebras, and let $A_{h} ≅ (k [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ , $h \notin q$ , be a standard smooth chart of $A$ over $k$ at $q$ with leading $c \times c$ minor a unit of the localisation [F1]. Write $Q^{'} \subseteq k [x_{1}, \dots, x_{n}]$ for the prime corresponding to $q$ . The composite $k [x_{1}, \dots, x_{n}] \to A_{h} \to A_{q} \to A_{q} / q A_{q} = k$ is a $k$ -algebra map whose kernel is $Q^{'}$ , so $k [x] / Q^{'}$ is a $k$ -subalgebra of the field $k$ containing the image of $k$ , hence equal to $k$ ; thus $Q^{'}$ is maximal and $ht (Q^{'}) = n$ by [F15]. Applying [F3] to the standard smooth $k$ -algebra $A_{h}$ with $p = (0)$ , $K = k$ and the local ring $(A_{h})_{q} = A_{q}$ gives $\dim A_{q} = ht (Q^{'}) - c = n - c$ , which is the relative dimension of the chart, in the situation of step 1.4.

F1F3F15step 1.4given∎

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Base change and composition of standard smooth presentations

Statement

Let $R \to S$ be a homomorphism of commutative rings and let $R \to R^{'}$ be an arbitrary ring homomorphism. Write standard smooth presentations (Standard smooth presentations and locally standard smooth maps) as $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}, T ≅ (S [y_{1}, \dots, y_{m}] / (f_{1}^{''}, \dots, f_{d}^{''}))_{g^{''}},$ of relative dimensions $n - c$ and $m - d$ , with leading Jacobian minors $h$ and $h^{''}$ mapping to units.

Base change. $R^{'} \otimes_{R} S$ is a standard smooth $R^{'}$ -algebra with the same parameters $n, c$ , relative dimension $n - c$ , and with the image of $h$ a unit. If moreover $R \to S$ is standard smooth at a prime $q \in Spec S$ , then $R^{'} \to R^{'} \otimes_{R} S$ is standard smooth at every prime of $R^{'} \otimes_{R} S$ lying over $q$ ; consequently locally standard smooth maps are stable under arbitrary base change of the base ring.
Composition. $T$ carries a standard smooth $R$ -presentation with $n + m$ variables, $c + d$ equations and relative dimension $(n - c) + (m - d)$ ; thus the relative dimensions of these displayed presentations add. If $R \to S$ is standard smooth at $q$ and $S \to T$ is standard smooth at $n \in Spec T$ with $n \cap S = q$ , then $R \to T$ is standard smooth at $n$ ; consequently a composite of locally standard smooth maps is locally standard smooth.

No hypothesis is placed on $R \to R^{'}$ or on $R \to S$ , no regularity theorem is used, and no form of the Axiom of Choice is used: all statements are formal consequences of the displayed polynomial presentations. The relative dimension of a presentation is the integer $n - c$ ; its identification with the dimension of a nonempty fibre is a separate matter, proved under the Axiom of Choice elsewhere on this page and used nowhere below.

Facts & Assumptions

Given: A homomorphism $R \to S$ with a standard smooth presentation of relative dimension $n - c$ and leading minor $h$ a unit, an arbitrary ring homomorphism $R \to R^{'}$ , and an $S$ -algebra $T$ with a standard smooth $S$ -presentation of relative dimension $m - d$ and leading minor $h^{''}$ a unit (with localisation denominator $g^{''}$ ).

[F1]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation of an $R$ -algebra $S$ consists of $n \geq c \geq 0$ , $f_{1}, \dots, f_{c} \in R [x_{1}, \dots, x_{n}]$ and $g \in R [x_{1}, \dots, x_{n}]$ with $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ such that some $c \times c$ minor of the Jacobian matrix has image a unit of $S$ ; $n - c$ is the relative dimension and the invertible minor may be assumed to be the leading one in the first $c$ columns. For a finitely presented $R$ -algebra $S$ , the map $R \to S$ is standard smooth at $q$ when $S_{u}$ has a standard smooth presentation over $R$ for some $u \notin q$ , and locally standard smooth when this holds at every prime.

[F2]

Base change of standard smooth presentations: for any ring map $R \to R^{'}$ and a standard smooth presentation $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with minor $h$ a unit, there is a unique $R^{'}$ -algebra isomorphism $R^{'} \otimes_{R} S \to (R^{'} [x_{1}, \dots, x_{n}] / (f_{1}^{'}, \dots, f_{c}^{'}))_{g^{'}}$ sending $a \otimes \overline{F} / g^{N}$ to $a \overline{F^{'}} / (g^{'})^{N}$ , and the target is standard smooth over $R^{'}$ with the same $n, c$ and relative dimension, the image $h^{'}$ of $h$ again a unit.

[F3]

Differentials of a polynomial quotient and the Jacobian cokernel: for $P = A [x_{1}, \dots, x_{n}]$ the partial derivatives $\partial_{i}$ are computed on the monomial basis by $\partial_{i} (x^{a}) = a_{i} x^{a - e_{i}}$ and extended $A$ -linearly, so that $\partial_{i}$ is $A$ -linear and is zero on polynomials not involving $x_{i}$ ; $d f = \sum_{i} \partial_{i} f d x_{i}$ , and the Jacobian matrix $(\partial_{i} f_{j})$ governs the cokernel presentation of $Ω_{P / I / A}$ .

[F4]

Universal mapping property of the tensor product of commutative algebras, Localisation of modules is extension of scalars, A polynomial ring on a finite ordered family agrees canonically with the iterated polynomial-ring construction: for a ring homomorphism $R \to S$ there is an $S$ -algebra isomorphism $S \otimes_{R} R [x] ≅ S [x]$ ; for a multiplicative set $Σ$ of an $R$ -algebra $A$ the localisation $Σ^{- 1} M$ of an $A$ -module is $(Σ^{- 1} A) \otimes_{A} M$ , so $A_{g} \otimes_{A} A [y] ≅ (A [y])_{g}$ ; and the iterated polynomial ring $R [x_{1}] \dots [x_{n}]$ is canonically $R [x_{1}, \dots, x_{n}]$ .

[F5]

Tensoring is right exact: tensoring an exact sequence $A^{'} \to B^{'} \to C^{'} \to 0$ with a module preserves exactness; in particular for an ideal $I \subseteq B$ one has $(B / I) \otimes_{B} C ≅ C / I C$ , giving $(R [x] / (f_{1}, \dots, f_{c})) [y] ≅ R [x, y] / (f_{1}, \dots, f_{c}) R [x, y]$ .

[F6]

Universal property of localisation: maps that invert $S$ factor uniquely through $S^{- 1} R$ , Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions: a unital homomorphism carrying a multiplicative set into the units factors uniquely through the localisation, and in $A_{u}$ the element $u$ is a unit; localisation is functorial for ring maps.

[F7]

Localising twice is localising once at the multiplicative set generated by both denominator sets: for multiplicative sets $Σ, Υ$ of a commutative ring $A$ , the iterated localisation $(Σ^{- 1} A)_{Υ}$ is the localisation of $A$ at the multiplicative set generated by $Σ \cup Υ$ ; in particular localising successively at $g$ and at $H$ is localising at $g H$ , and an element which is a unit remains a unit.

Proof

1.1

Notation. Fix a standard smooth presentation $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with leading $c \times c$ minor $h$ a unit of $S$ [F1], put $I : = (f_{1}, \dots, f_{c}) \subseteq R [x_{1}, \dots, x_{n}]$ and $A : = R [x_{1}, \dots, x_{n}] / I$ , so that $S = A_{g}$ . Fix also a standard smooth $S$ -presentation $T ≅ (S [y_{1}, \dots, y_{m}] / (f_{1}^{''}, \dots, f_{d}^{''}))_{g^{''}}$ with leading $d \times d$ minor $h^{''}$ a unit of $T$ [F1]. Finally fix a ring map $R \to R^{'}$ .

F1given

2.1

Base change of presentations. By [F2] applied to the presentation of step 1.1 and the ring map $R \to R^{'}$ there is an $R^{'}$ -algebra isomorphism $R^{'} \otimes_{R} S ≅ (R^{'} [x_{1}, \dots, x_{n}] / (f_{1}^{'}, \dots, f_{c}^{'}))_{g^{'}}$ , where $f_{j}^{'}, g^{'}$ are the images of $f_{j}, g$ ; the target is a standard smooth $R^{'}$ -presentation with the same $n, c$ and relative dimension $n - c$ , and the image $h^{'}$ of $h$ is a unit. This is the first assertion of clause 1.

F2step 1.1

2.2

The polynomial presentation of $S [y_{1}, \dots, y_{m}]$ . The coefficient extension $A [y_{1}, \dots, y_{m}] ≅ R [x_{1}, \dots, x_{n}, y_{1}, \dots, y_{m}] / I R [x_{1}, \dots, x_{n}, y_{1}, \dots, y_{m}]$ holds by [F5], since $A = R [x] / I$ and $R [x_{1}, \dots, x_{n}] [y_{1}, \dots, y_{m}] = R [x_{1}, \dots, x_{n}, y_{1}, \dots, y_{m}]$ by [F4]; combining it with $A_{g} [y_{1}, \dots, y_{m}] ≅ (A [y_{1}, \dots, y_{m}])_{g}$ from [F4] and with $S = A_{g}$ gives $S [y_{1}, \dots, y_{m}] ≅ (R [x_{1}, \dots, x_{n}, y_{1}, \dots, y_{m}] / (f_{1}, \dots, f_{c}))_{g} .$ We use this isomorphism to read the presentation of $T$ in the polynomial ring over $R$ .

F4F5step 1.1

3.1

Base change at a prime. Finite presentation is preserved by base change: tensoring $R [z_{1}, \dots, z_{a}] / (r_{1}, \dots, r_{b})$ with $R^{'}$ gives $R^{'} [z_{1}, \dots, z_{a}] / (r_{1}^{'}, \dots, r_{b}^{'})$ by [F4, F5], where the primes denote coefficient images. Suppose $R \to S$ is standard smooth at $q \in Spec S$ , witnessed by an element $u \notin q$ with $S_{u}$ standard smooth over $R$ [F1]; by [F2] the base change $R^{'} \otimes_{R} S_{u}$ is standard smooth over $R^{'}$ . Let $Q \subseteq R^{'} \otimes_{R} S$ be a prime with $Q \cap S = q$ , i.e. lying over $q$ ; then $u \notin Q$ , since $u \in Q$ would give $u \in Q \cap S = q$ . Hence $Q$ lies in the principal open $D (u)$ of $Spec (R^{'} \otimes_{R} S)$ , and the localisation $(R^{'} \otimes_{R} S)_{u}$ — which is $R^{'} \otimes_{R} S_{u}$ by [F4] — is standard smooth over $R^{'}$ [F6]. Therefore $R^{'} \to R^{'} \otimes_{R} S$ is standard smooth at $Q$ ; as $Q$ was an arbitrary prime over $q$ , this gives the pointwise form of clause 1, and taking the witnessing chart at every prime of $S$ gives stability of local standard smoothness under base change.

F1F2F4F6step 2.1

3.2

Clearing denominators and the composite presentation. By step 2.2 the $S$ -presentation of $T$ is a presentation in the ring $(R [x, y] / (f_{1}, \dots, f_{c}))_{g}$ , with $y = (y_{1}, \dots, y_{m})$ ; write $f_{k}^{''} = \sum_{α} \overline{a_{k α}} y^{α}$ and $g^{''} = \sum_{β} \overline{c_{β}} y^{β}$ with coefficients in $S = A_{g}$ , and choose representatives $a_{k α} = g^{- N_{k α}} b_{k α}$ and $c_{β} = g^{- M_{β}} d_{β}$ with $b_{k α}, d_{β} \in R [x_{1}, \dots, x_{n}]$ and $N_{k α}, M_{β} \geq 0$ . Put $N : = \max ({0} \cup {N_{k α}})$ , $M : = \max ({0} \cup {M_{β}})$ (so empty families give $0$ ) and define $F_{k} : = \sum_{α} b_{k α} g^{N - N_{k α}} y^{α} \in R [x, y], H : = \sum_{β} d_{β} g^{M - M_{β}} y^{β} \in R [x, y] .$ Multiplying the displayed identities by $g^{N}$ and $g^{M}$ shows $F_{k} = g^{N} f_{k}^{''}$ and $H = g^{M} g^{''}$ in $S [y]$ . Since $g$ is a unit of $S [y]_{g H}$ , the ideals $(F_{1}, \dots, F_{d})$ and $(f_{1}^{''}, \dots, f_{d}^{''})$ coincide there, and $H$ is a unit multiple of $g^{''}$ ; by [F7] localising at $g$ and then at $g^{''}$ is localising at $g H$ . Hence $T = (S [y] / (f_{1}^{''}, \dots, f_{d}^{''}))_{g^{''}} ≅ (R [x_{1}, \dots, x_{n}, y_{1}, \dots, y_{m}] / (f_{1}, \dots, f_{c}, F_{1}, \dots, F_{d}))_{g H},$ the composite presentation of $T$ over $R$ .

F1F4F6F7step 2.2

4.1

The Jacobian minor of the composite. In the ring $S [y]$ , the sum formula and coefficient linearity of [F3] give $\partial F_{k} / \partial y_{l} = \sum_{α} b_{k α} g^{N - N_{k α}} \partial (y^{α}) / \partial y_{l} = g^{N} \partial f_{k}^{''} / \partial y_{l}$ , because the coefficients $g^{N - N_{k α}} b_{k α}$ represent $g^{N} g^{- N_{k α}} b_{k α} = g^{N} \overline{a_{k α}}$ and $\partial / \partial y_{l}$ is $S$ -linear on $S [y]$ . Hence the leading $d \times d$ block $(\partial F_{k} / \partial y_{l})_{1 \leq k, l \leq d}$ has determinant $g^{N d} h^{''}$ , a unit of $T$ because $g$ and $h^{''}$ are units. Moreover $\partial f_{j} / \partial y_{l} = 0$ for all $j, l$ , since $f_{j} \in R [x_{1}, \dots, x_{n}]$ does not involve the $y$ 's [F3]. Therefore the $(c + d) \times (c + d)$ minor of the Jacobian matrix of $(f_{1}, \dots, f_{c}, F_{1}, \dots, F_{d})$ on the columns $x_{1}, \dots, x_{c}$ and $y_{1}, \dots, y_{d}$ is block triangular with diagonal blocks $(\partial f_{j} / \partial x_{i})_{1 \leq i, j \leq c}$ and $(\partial F_{k} / \partial y_{l})_{1 \leq k, l \leq d}$ , so its determinant is $h \cdot g^{N d} h^{''},$ which is a unit of $T$ because $h$ maps to a unit of $S$ and hence of $T$ , and $g$ and $h^{''}$ are units of $T$ .

F1F3step 3.2algebra

5.1

The composite is standard smooth. By step 3.2 the algebra $T$ is presented over $R$ as $(R [x_{1}, \dots, x_{n}, y_{1}, \dots, y_{m}] / (f_{1}, \dots, f_{c}, F_{1}, \dots, F_{d}))_{g H}$ with $n + m$ variables and $c + d$ equations, and by step 4.1 the displayed $(c + d) \times (c + d)$ minor of the Jacobian matrix is a unit of $T$ ; moreover the invertible minor may be assumed leading after permuting variables, so this is a standard smooth $R$ -presentation [F1]. Its relative dimension is $(n + m) - (c + d) = (n - c) + (m - d)$ , the sum of the relative dimensions of the two given presentations. This proves the first assertion of clause 2.

F1step 3.2step 4.1

6.1

Composition at a point. Finite presentation is preserved by composition: from $S = R [z_{1}, \dots, z_{a}] / (r_{1}, \dots, r_{b})$ and $T = S [w_{1}, \dots, w_{e}] / (s_{1}, \dots, s_{l})$ , lift the finitely many coefficients of the $s_{i}$ to $R [z]$ ; then $T = R [z, w] / (r_{1}, \dots, r_{b}, {\tilde{s}}_{1}, \dots, {\tilde{s}}_{l})$ by [F4, F5]. Thus the finite-presentation prerequisite in [F1] holds for the composite. Suppose $R \to S$ is standard smooth at $q$ and $S \to T$ is standard smooth at $n$ with $n \cap S = q$ . Choose $u \notin q$ with $S_{u}$ standard smooth over $R$ and $v \notin n$ with $T_{v}$ standard smooth over $S$ [F1]. Since $u \notin q = n \cap S$ , both $u$ and $v$ lie outside $n$ , so $u v \notin n$ . Base change of the standard smooth $S$ -presentation of $T_{v}$ along $S \to S_{u}$ gives the standard smooth $S_{u}$ -algebra $S_{u} \otimes_{S} T_{v} ≅ (T_{v})_{u} = T_{u v}$ by step 2.1 and [F4, F7]. Applying step 5.1 to $S_{u}$ over $R$ and $T_{u v}$ over $S_{u}$ exhibits $T_{u v}$ as standard smooth over $R$ ; since $u v \notin n$ , this witnesses that $R \to T$ is standard smooth at $n$ [F1]. As $n$ was arbitrary, a composite of locally standard smooth maps is locally standard smooth, which completes clause 2.

F1F4F7step 2.1step 5.1∎

TheoremStatement: Literature-sourcedProof: AI-adaptedaudited 2026-09-27Open item page →

Submersion criterion for locally standard smooth morphisms

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field, let $X$ and $Y$ be $k$ -schemes that are locally standard smooth over $k$ at the points considered below (Standard smooth presentations and locally standard smooth maps), and let $f ⁣ : X \to Y$ be a morphism of $k$ -schemes of finite type (Locally finite type and finite type morphisms). Let $x \in X$ be a $k$ -rational point and put $y = f (x)$ , assumed $k$ -rational as well, so that $κ (x) = κ (y) = k$ (The residue field at a point of an affine scheme). Write $A = O_{Y, y}$ , $S = O_{X, x}$ , with maximal ideals $m_{y} \subseteq A$ , $m_{x} \subseteq S$ , and let $f^{*} ⁣ : A \to S$ be the induced local homomorphism. Let $X_{y} = X \times_{Y} Spec κ (y)$ be the scheme-theoretic fibre (Scheme-theoretic fibre), whose local ring at $x$ is $S / m_{y} S$ . Then:

Submersion criterion. $f$ is locally standard smooth at $x$ — that is, there are affine opens $Spec C \subseteq X$ and $Spec D \subseteq Y$ with $x \in Spec C$ , $f (Spec C) \subseteq Spec D$ and $D \to C$ standard smooth at the prime of $C$ corresponding to $x$ — if and only if the $k$ -linear map $f^{*} ⁣ : m_{y} / m_{y}^{2} \to m_{x} / m_{x}^{2}$ induced by $f^{*}$ is injective.
Flatness and fibres. If the equivalent conditions of clause 1 hold and $m = \dim S$ , $n = \dim A$ , then $S$ is flat over $A$ , that is $f$ is flat at $x$ , and $S / m_{y} S$ is a regular local ring of dimension $m - n$ ; in other words the fibre $X_{y}$ is regular at $x$ of dimension $m - n$ . Any standard smooth chart of $f$ at $x$ has relative dimension $m - n$ .

The two schemes are only required to be locally standard smooth at $x$ and $y$ , not globally; $f$ is of finite type as assumed above, and no hypothesis is imposed on the base field. The Axiom of Choice is used through the local-flatness, regular-parameter and geometric-regularity suppliers cited below.

Facts & Assumptions

Given: A field $k$ , $k$ -schemes $X, Y$ locally standard smooth over $k$ at a $k$ -rational point $x$ and at its image $y = f (x)$ , a finite-type morphism of $k$ -schemes $f ⁣ : X \to Y$ , the local rings $A = O_{Y, y}$ , $S = O_{X, x}$ with maximal ideals $m_{y}, m_{x}$ and residue fields $k$ , the induced local homomorphism $f^{*} ⁣ : A \to S$ , and the Axiom of Choice.

[F1]

Standard smooth presentations and locally standard smooth maps: a standard smooth presentation of an $R$ -algebra $S$ consists of $n \geq c \geq 0$ , $f_{1}, \dots, f_{c} \in R [x_{1}, \dots, x_{n}]$ and $g$ with $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ such that some $c \times c$ Jacobian minor has image a unit of $S$ ; $n - c$ is the relative dimension, the invertible minor may be assumed leading, and a further principal localisation may be absorbed. For a finitely presented $R$ -algebra map $R \to S$ and a prime $q \in Spec S$ , standard smooth at $q$ means that $S_{h}$ has a standard smooth presentation over $R$ for some $h \notin q$ ; locally standard smooth means this holds at every prime.

[F2]

Standard smooth algebras are finitely presented and flat: under the Axiom of Choice, a standard smooth $R$ -algebra is a finitely presented $R$ -algebra and is flat over $R$ , for every commutative ring $R$ .

[F3]

Fibres of standard smooth algebras are regular of relative dimension: under the Axiom of Choice, for a standard smooth $R$ -algebra $S ≅ (R [x_{1}, \dots, x_{n}] / (f_{1}, \dots, f_{c}))_{g}$ with leading minor a unit, a prime $p \in Spec R$ and a field extension $K / κ (p)$ , every local ring $(F_{K})_{Q}$ of $F_{K} = (S \otimes_{R} κ (p)) \otimes_{κ (p)} K$ is regular local of dimension $ht (Q^{'}) - c$ , where $Q^{'} \subseteq K [x_{1}, \dots, x_{n}]$ corresponds to $Q$ , and every irreducible component of $Spec F_{K}$ has dimension $n - c$ ; for $c = 0$ this says that a localisation of a polynomial ring over a field is regular local of dimension $ht (Q^{'})$ .

[F4]

Separable residue and the cotangent sequence of a local algebra: let $R$ be a Noetherian local $k$ -algebra with maximal ideal $m$ and residue field $κ$ , finitely generated and separably generated over $k$ . Then $0 \to m / m^{2} \to Ω_{R / k} \otimes_{R} κ \to Ω_{κ / k} \to 0$ is short exact, the first map sending the class of $x$ to $d x \otimes 1$ ; if $κ / k$ is finite separable then $Ω_{κ / k} = 0$ and that map is an isomorphism $m / m^{2} ≅ Ω_{R / k} \otimes_{R} κ$ .

[F5]

Transitivity sequence for differentials: for homomorphisms $A \to B \to C$ of commutative rings the sequence $C \otimes_{B} Ω_{B / A} \to Ω_{C / A} \to Ω_{C / B} \to 0$ of $C$ -modules is exact, the first map being the extension of scalars of $d_{B / A}$ .

[F6]

Localization, base change and functoriality of differentials: for ring maps $A \to A^{'}$ there is a natural isomorphism $A^{'} \otimes_{A} Ω_{B / A} ≅ Ω_{B \otimes_{A} A^{'} / A^{'}}$ , and for multiplicative sets $U \subseteq B$ , $V \subseteq A$ with the image of $V$ in $B$ contained in $U$ there is a $U^{- 1} B$ -module isomorphism $U^{- 1} Ω_{B / A} ≅ Ω_{U^{- 1} B / V^{- 1} A}$ .

[F7]

Differentials of a polynomial quotient and the Jacobian cokernel: for $P = A [x_{1}, \dots, x_{n}]$ , $Ω_{P / A}$ is free on $d x_{1}, \dots, d x_{n}$ ; if $B = P / I$ then $I / I^{2} \to B \otimes_{P} Ω_{P / A} \to Ω_{B / A} \to 0$ is exact; and if $I = (f_{1}, \dots, f_{c})$ then $Ω_{B / A} ≅ B^{n} / \sum_{j} B \cdot (\partial_{i} f_{j})_{i}$ is the cokernel of the Jacobian matrix, so that with an invertible $c \times c$ minor $Ω_{B / A}$ is free of rank $n - c$ .

[F8]

[F9]

regular local rings are domains and cohen macaulay: under the Axiom of Choice, a regular local ring $R$ of dimension $d$ is a domain and Cohen–Macaulay, and for every regular system of parameters $(x_{1}, \dots, x_{d})$ the tuple is $R$ -regular and $R / (x_{1}, \dots, x_{c})$ is regular local of dimension $d - c$ for all $0 \leq c \leq d$ .

[F10]

regular local regular quotient ideal is parameter generated: under the Axiom of Choice, for a regular local ring $(R, m, k)$ of dimension $d$ and an ideal $I \subseteq m$ , the quotient $R / I$ is regular if and only if $\dim_{k} ((I + m^{2}) / m^{2}) = d - \dim (R / I)$ , equivalently if and only if $I$ is generated by an initial part of a regular system of parameters.

[F11]

Local flatness criterion by regular parameters: under the Axiom of Choice, for a local homomorphism $(R, m) \to (S, n)$ of Noetherian local rings and a finite $S$ -module $M$ with ${Tor}_{1}^{R} (R / m, M) = 0$ , the module $M$ is flat over $R$ ( $M$ need not be finite over $R$ ); consequently, if $R$ and $S$ are regular local and the images in $S$ of a regular system of parameters of $R$ extend to a regular system of parameters of $S$ , then $S$ is flat over $R$ .

[F12]

Locally standard smooth iff flat with geometrically regular fibres: under the Axiom of Choice, for a ring map $R \to S$ of finite presentation and $q \in Spec S$ with $p = q \cap R$ , the map is standard smooth at $q$ if and only if $R_{p} \to S_{q}$ is flat and the fibre $S \otimes_{R} κ (p)$ is geometrically regular at $q$ ; and for a finite-type $k$ -algebra $A$ that is locally standard smooth over $k$ , the relative dimension of a standard smooth chart at a $k$ -rational prime $q$ equals $\dim A_{q}$ .

[F13]

Base change and composition of standard smooth presentations: base change of a standard smooth presentation along any ring map $R \to R^{'}$ yields a standard smooth $R^{'}$ -presentation with the same parameters and relative dimension, standard smoothness at a prime is stable under such base change, and composing standard smooth presentations over $R \to S \to T$ yields a standard smooth $R$ -presentation of $T$ with relative dimension the sum of the two relative dimensions; composition is likewise standard smooth at a prime.

[F14]

Maximal ideals of an affine domain have full height, A polynomial ring in n variables over a field has dimension n: for a field $k$ , $\dim k [x_{1}, \dots, x_{n}] = n$ and every maximal ideal of a finite-type $k$ -domain has height equal to the dimension of that domain; in particular a maximal ideal $Q^{'} \subseteq k [x_{1}, \dots, x_{n}]$ satisfies $ht (Q^{'}) = n$ .

[F15]

Geometrically regular algebras and geometrically regular fibres: for a finitely presented $R$ -algebra $S$ , $q \in Spec S$ with $p = q \cap R$ , the fibre $S \otimes_{R} κ (p)$ is geometrically regular at $q$ when for every field extension $K / κ (p)$ and every prime of $(S \otimes_{R} κ (p)) \otimes_{κ (p)} K$ lying over the image of $q$ the local ring there is regular.

[F16]

Scheme-theoretic fibre, Base change of objects, morphisms and properties, Universal mapping property of the tensor product of commutative algebras, Existence of all scheme fibre products, Prime ideals of a localization are exactly the primes disjoint from the denominator set, Localising twice is localising once at the multiplicative set generated by both denominator sets, Localisation commutes with kernels images and cokernels: the fibre $X_{y}$ is $X \times_{Y} Spec κ (y)$ ; over affine charts $Spec C$ , $Spec D$ it is computed by the coproduct $C \otimes_{D} κ (p)$ with $p = q \cap D$ , so that its local ring at the point induced by $q$ is $C_{q} \otimes_{D_{p}} κ (p) = S / m_{y} S$ ; primes of a localisation $A_{g}$ are the primes of $A$ not containing $g$ , localisation commutes with quotients and cokernels, and iterated localisation is localisation at the product of the inverted elements.

[F17]

Flatness is transitive under a flat change of rings, Every localization is flat, and localizing a flat module preserves flatness: a localisation is flat, a composite of flat ring homomorphisms is flat, and a base change along a flat map is flat.

[F18]

Tensoring is right exact, Localisation of modules is extension of scalars: tensoring an exact sequence preserves right exactness, so a right-exact sequence stays right exact after tensoring with a module; for an ideal $I \subseteq B$ one has $(B / I) \otimes_{B} C ≅ C / I C$ .

[F19]

Every algebra of finite type over a Noetherian ring is a Noetherian ring, Every quotient and every localisation of a Noetherian ring is Noetherian, Every algebra of finite type over a Noetherian ring is finitely presented: a finite-type algebra over the field $k$ is Noetherian, as are its quotients and localisations; and a finite-type algebra over a Noetherian ring is finitely presented.

[F20]

The Axiom of Choice: every family of nonempty sets has a choice function; it is assumed in the statement and used through [F3], [F8], [F9], [F10], [F11] and [F12].

Proof

1.1

Setup. Since $f$ is of finite type, the point $x$ has an affine open neighbourhood $Spec C \subseteq X$ and $y$ has an affine open neighbourhood $Spec D \subseteq Y$ with $f (Spec C) \subseteq Spec D$ and $D \to C$ of finite type; shrinking $C$ we may suppose that $k \to C$ has a standard smooth presentation $C ≅ (k [x_{1}, \dots, x_{N}] / (f_{1}, \dots, f_{c}))_{g}$ with leading $c \times c$ minor a unit [F1], and shrinking $D$ that $k \to D$ has a standard smooth presentation $D ≅ (k [y_{1}, \dots, y_{M}] / (G_{1}, \dots, G_{b}))_{H}$ with leading $b \times b$ minor a unit. Let $q \subseteq C$ be the prime corresponding to $x$ and $p = q \cap D$ the prime corresponding to $y$ ; both are maximal with $C / q = D / p = k$ , since $x$ and $y$ are $k$ -rational points and the $k$ -algebra maps $k [x] / Q^{'} \to κ (x) = k$ and $k [y] / P^{'} \to κ (y) = k$ have finite-type domains, hence are isomorphisms. Put $A : = D_{p}$ and $S : = C_{q}$ , so that $A \to S$ is a local homomorphism of Noetherian local rings with residue field $k$ [F19], and put $F : = S / m_{y} S$ .

F1F19givenconstructF20

2.1

Converse direction: extending regular parameters. By [F3] applied over $k$ to the two charts of step 1.1, $A$ and $S$ are regular local rings; write $n = \dim A$ and $m = \dim S$ . Assume now that the map $m_{y} / m_{y}^{2} \to m_{x} / m_{x}^{2}$ induced by $f^{*}$ is injective. Choose a $k$ -basis of $m_{y} / m_{y}^{2}$ and lift it to $y_{1}, \dots, y_{n} \in m_{y}$ ; by [F8] the tuple $(y_{1}, \dots, y_{n})$ is a regular system of parameters of $A$ . Its images form an independent tuple of $n$ elements of the $k$ -vector space $m_{x} / m_{x}^{2}$ of dimension $m$ , which therefore extends to a $k$ -basis; lifting that basis so that the first $n$ lifts are $y_{1}, \dots, y_{n}$ and the remaining $m - n$ lifts are new elements gives $(x_{1}, \dots, x_{m}) \in m_{x}^{m}$ with $x_{i} = y_{i}$ for $i \leq n$ , and [F8] again makes it a regular system of parameters of $S$ .

F3F8step 1.1givenchooseconstruct

2.2

The local rings and the cotangent identifications. Applying [F3] to the two standard smooth presentations of step 1.1 over the base field $k$ with $p = (0)$ and $K = k$ , where the corresponding primes $Q^{'} \subseteq k [x_{1}, \dots, x_{N}]$ and $P^{'} \subseteq k [y_{1}, \dots, y_{M}]$ are maximal and hence of heights $N$ and $M$ by [F14], shows that $S$ is a regular local ring of dimension $N - c$ and that $A$ is a regular local ring of dimension $M - b$ ; by [F12] these integers are $m = \dim S$ and $n = \dim A$ , so $N - c = m$ and $M - b = n$ . Since the residue fields of $A$ and $S$ are the field $k$ , a finite separable extension of $k$ , [F4] gives isomorphisms $m_{y} / m_{y}^{2} ≅ Ω_{A / k} \otimes_{A} k$ and $m_{x} / m_{x}^{2} ≅ Ω_{S / k} \otimes_{S} k$ carrying the class of an element of the maximal ideal to $d □ \otimes 1$ .

F3F4F12F14step 1.1

2.3

Forward direction: charts give freeness, flatness and the fibre dimension. Assume $f$ is locally standard smooth at $x$ ; after shrinking the charts of step 1.1 we may suppose that $C$ carries a standard smooth presentation over $D$ of relative dimension $d : = N^{'} - c^{'}$ , say $C ≅ (D [x_{1}, \dots, x_{N^{'}}] / (f_{1}^{'}, \dots, f_{c^{'}}^{'}))_{g^{'}}$ with an invertible $c^{'} \times c^{'}$ Jacobian minor [F1]. By [F7] the $S$ -module $Ω_{S / A} ≅ S \otimes_{C} Ω_{C / D}$ [F6] is the cokernel of the Jacobian matrix $S^{c^{'}} \to S^{N^{'}}$ , hence is free of rank $d$ because the minor is a unit of $S$ . Base changing this presentation along $D \to A$ exhibits $S$ as a localisation of the standard smooth $A$ -algebra $(A [x_{1}, \dots, x_{N^{'}}] / (f_{1}^{'}, \dots, f_{c^{'}}^{'}))_{g^{'}}$ [F13], which is flat over $A$ by [F2]; localisation is flat and flatness is transitive [F17], so $S$ is flat over $A$ . Finally, $F = S / m_{y} S$ is the local ring of the fibre algebra $(k [x_{1}, \dots, x_{N^{'}}] / ({\overset{ˉ}{f}}_{1}^{'}, \dots, {\overset{ˉ}{f}}_{c^{'}}^{'}))_{{\overset{ˉ}{g}}^{'}}$ at the prime $Q^{'}$ corresponding to $x$ , which is maximal because its residue field is $κ (x) = k$ ; so [F3] and [F14] give $\dim F = ht (Q^{'}) - c^{'} = N^{'} - c^{'} = d$ .

F1F2F3F6F7F13F14F17step 1.1

3.1

Converse direction: flatness and a regular local fibre. The images in $S$ of the regular system of parameters $y_{1}, \dots, y_{n}$ of $A$ are the initial segment of the regular system of parameters $(x_{1}, \dots, x_{m})$ of $S$ from step 2.1, so the second assertion of [F11] shows that $S$ is flat over $A$ . By [F9] the tuple $(x_{1}, \dots, x_{n})$ is $S$ -regular and $S / (x_{1}, \dots, x_{n})$ is a regular local ring of dimension $m - n$ ; since $(x_{1}, \dots, x_{n}) = (y_{1}, \dots, y_{n}) S = m_{y} S$ , this quotient is $F = S / m_{y} S$ , the local ring of the fibre $X_{y}$ at $x$ [F16].

F9F11F16step 2.1

3.2

Forward direction: relative dimension $m - n$ and injectivity of the cotangent map. Composing the standard smooth presentation of $C$ over $D$ from step 2.3 with the standard smooth presentation of $D$ over $k$ from step 1.1 presents the finite-type $k$ -algebra $C$ as standard smooth over $k$ with relative dimension $n + d$ [F13]; its localisation at the $k$ -rational prime $q$ is $S$ , so [F12] identifies that relative dimension with $\dim S = m$ , whence $d = m - n$ and, by step 2.3, $\dim F = m - n$ . The transitivity sequence $Ω_{A / k} \otimes_{A} S \to Ω_{S / k} \to Ω_{S / A} \to 0$ of [F5] is right exact, and tensoring it with $S \to k$ yields, using [F4] and [F18], the exact sequence $m_{y} / m_{y}^{2} \to m_{x} / m_{x}^{2} \to Ω_{S / A} \otimes_{S} k \to 0$ , in which the first arrow is the map induced by $f^{*}$ ; since $Ω_{S / A} ≅ S^{d}$ the last term is a $k$ -vector space of dimension $d$ , so the image has dimension $m - d = n$ , which equals $\dim_{k} (m_{y} / m_{y}^{2}) = n$ by step 2.2 and forces the map to be injective.

F3F4F5F12F13F18step 2.2step 2.3algebra

4.1

Converse direction: the fibre is geometrically regular. Write $Q^{'} \subseteq k [x_{1}, \dots, x_{N}]$ for the maximal ideal corresponding to $x$ in the presentation of step 1.1, and put $R^{'} : = k [x_{1}, \dots, x_{N}]_{Q^{'}}$ . The parameters $y_{1}, \dots, y_{n} \in A = D_{p}$ generate $p D_{p}$ . Since $D$ is Noetherian, after shrinking $Spec D$ around $y$ and its inverse-image chart around $x$ , we may represent every $y_{i}$ by an element of $D$ and arrange that $p D = (y_{1}, \dots, y_{n}) D$ on these charts: first clear their denominators outside $p$ , then invert an element outside $p$ annihilating the finite module $p D / (y_{1}, \dots, y_{n}) D$ . Absorb the corresponding principal localisations into the polynomial chart of $C$ . Write each image of $y_{i}$ in $C$ as $P_{i} / g^{e_{i}}$ with $P_{i} \in k [x_{1}, \dots, x_{N}]$ and $e_{i} \geq 0$ . Since $g$ is a unit of $C$ , the images of the numerators $P_{i}$ generate the same ideal as those of the $y_{i}$ , and each $P_{i}$ vanishes at $x$ . The finite-type fibre algebra $B : = C \otimes_{D} k = C / p C$ is then presented on this chart by $B ≅ (k [x_{1}, \dots, x_{N}] / (f_{1}, \dots, f_{c}, P_{1}, \dots, P_{n}))_{g}$ , and its local ring at $x$ is $F ≅ R^{'} / I$ for $I : = (f_{1}, \dots, f_{c}, P_{1}, \dots, P_{n}) R^{'}$ [F16]. The ring $R^{'}$ is regular local of dimension $N$ by [F3] with $c = 0$ and [F14], and $\dim F = m - n$ by step 3.1, so [F10] gives $\dim_{k} ((I + Q^{' 2}) / Q^{' 2}) = N - \dim F = N - m + n = c + n$ . The classes of the $c + n$ generators $f_{1}, \dots, f_{c}, P_{1}, \dots, P_{n}$ span that space, hence form a basis. By [F4] and [F7], their classes in $Q^{'} / Q^{' 2}$ are the $c + n$ rows of the Jacobian matrix evaluated at $Q^{'}$ , so some $(c + n) \times (c + n)$ minor $h$ does not lie in $Q^{'}$ . Localising the finite-type algebra $B$ at the image of $h$ gives a standard smooth $k$ -presentation with the displayed $c + n$ equations [F1]; this open chart contains $x$ . By [F3], after every field extension $K / k$ every local ring of $(B_{h}) \otimes_{k} K$ is regular. Every prime of the extended fibre lying over $x$ belongs to this chart because $h \notin Q^{'}$ , so the fibre $C \otimes_{D} k$ is geometrically regular at $x$ in the sense of [F15].

F1F3F4F7F10F14F15F16step 3.1algebra

5.1

Converse direction: concluding local standard smoothness. The $k$ -algebra map $D \to C$ of step 1.1 is of finite type, hence finitely presented because $D$ is a localisation of a finite-type $k$ -algebra and therefore Noetherian [F19]. Its localisation $A \to S$ is flat by step 3.1, and step 4.1 proves that the finite-type fibre $C \otimes_{D} κ (p)$ is geometrically regular at the point induced by $q$ (its local ring there is $F = S / m_{y} S$ ), so clause 1 of [F12] shows that $D \to C$ is standard smooth at $q$ ; that is exactly the assertion that $f$ is locally standard smooth at $x$ . Together with step 3.2 this proves the equivalence of clause 1, step 2.3 and step 3.1 give flatness and the regularity and dimension $m - n$ of the fibre local ring in both directions, and steps 3.2 and 2.3 show that a witnessing chart has relative dimension $m - n$ .

F12F19step 2.3step 3.2step 3.1step 4.1∎

5 · Examples, counterexamples and false statements

None yet.