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Kahler Differentials Conormal Sequences and Infinitesimal Lifting

1 · Prerequisites

2 · Summary

Kähler differentials linearise derivations. The page begins with derivations and the universal property that defines $Ω_{B / A}$ , proves existence by generators and relations for an arbitrary ring homomorphism, and records the resulting representation of the derivation functor, the freeness of $Ω_{A [x_{1}, \dots, x_{n}] / A}$ , the conormal sequence of a quotient and the Jacobian presentation it produces, and the transitivity sequence $C \otimes_{B} Ω_{B / A} \to Ω_{C / A} \to Ω_{C / B} \to 0$ . Localization and scalar base change are shown to commute with $Ω$ , and the first arrow of each sequence is deliberately not asserted to be injective.

The same package is then sheafified on schemes: relative differentials are built by gluing the affine constructions, their universal property is a bijection onto derivations of the structure sheaf, and on $Spec B \to Spec A$ they are computed by the sheaf attached to the module $Ω_{B / A}$ , for which the affine module-sheaf universal property is supplied locally. From there the page develops the conormal sequence of a closed immersion, the transitivity sequence of composable morphisms, base change $g^{*} Ω_{X / S} ≅ Ω_{X^{'} / S^{'}}$ , the cotangent space $Ω_{X / k} \otimes_{O_{X, x}} κ (x)$ at a $k$ -rational point together with its identification with $m_{x} / m_{x}^{2}$ , the bijection between dual-number points and tangent vectors, and the map induced by a morphism on differentials with its identity and chain rules.

The final part reads infinitesimal lifting off the differentials. Formally unramified, formally smooth and formally étale morphisms are defined, the diagonal ideal is identified through $J / J^{2} ≅ Ω_{B / A}$ , and $Ω_{X / S} = 0$ is proved equivalent to formal unramifiedness. An unramified morphism of finite type is characterised by its diagonal being an open immersion; conversely, the finite-type field lemma (which, like the residue-extension lemma $κ (x) / κ (s)$ finite separable with $m_{s} O_{X, x} = m_{x}$ , assumes the Axiom of Choice and states its exact uses) feeds the structure theorem for unramified morphisms with locally finite type. Smoothness of relative dimension $d$ is recorded as smoothness together with locally free $Ω$ of rank $d$ ; the page closes with two remarks: the left map of the conormal sequence need not be injective, and differentials detect infinitesimal thickening but not every singularity on their own.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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

Derivation of an algebra

Definition

Let $A \overset{φ}{\to} B$ be a homomorphism of commutative rings (Commutative ring), so that $B$ is an $A$ -algebra, and let $M$ be a $B$ -module (Unital left and right modules over a ring; unqualified module means left module). An $A$ -derivation of $B$ into $M$ is a map $D ⁣ : B \to M$ satisfying, for all $b, b^{'} \in B$ and all $a \in A$ , the three laws

$D (b + b^{'}) = D (b) + D (b^{'}), D (φ (a)) = 0, D (b b^{'}) = b D (b^{'}) + b^{'} D (b) .$

The first law says that $D$ is additive; the second that $D$ is $A$ -constant (it kills the image of $A$ ); the third is the Leibniz rule. The set of all such maps is written ${Der}_{A} (B, M)$ . It is a $B$ -module under the pointwise operations $(D + D^{'}) (b) : = D (b) + D^{'} (b)$ and $(c \cdot D) (b) : = c D (b)$ : the sum and scalar multiples are again additive $A$ -constant maps satisfying Leibniz, because each law is linear in $D$ , and the zero map is a derivation.

Three conventions are part of the definition.

No finiteness. Nothing is assumed about $B$ as an $A$ -algebra: it need not be finitely generated, finitely presented, or flat, and $A$ need not be Noetherian. The definitions used later on this page are the same ones used for the earlier algebraic-differentials interface of this track.
$A$ -linearity, not $B$ -linearity. Every $A$ -derivation is $A$ -linear in the sense that $D (φ (a) b) = φ (a) D (b)$ for $a \in A$ , $b \in B$ : by Leibniz, $D (φ (a) b) = φ (a) D (b) + b D (φ (a))$ and the second term vanishes. A derivation is in general not $B$ -linear, and this failure is exactly what the Leibniz rule measures; it also shows $D (1) = 0$ , since $1 = φ (1_{A})$ makes $1$ $A$ -constant.
Functored variables. For a fixed ring map $A \to B$ and a $B$ -linear map $h ⁣ : M \to N$ of $B$ -modules, composition $D \mapsto h \circ D$ is a $B$ -module map ${Der}_{A} (B, M) \to {Der}_{A} (B, N)$ . Consequently ${Der}_{A} (B, -)$ is a functor from $B$ -modules to $B$ -modules, and the Leibniz rule is preserved by postcomposition with any module map.

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

Universal Kähler differential module

Definition

Let $A \overset{φ}{\to} B$ be a homomorphism of commutative rings and let ${Der}_{A} (B, -)$ be the derivation functor of Derivation of an algebra. A Kähler differential module for $A \to B$ is a pair $(Ω_{B / A}, d)$ consisting of a $B$ -module $Ω_{B / A}$ and an $A$ -derivation $d ⁣ : B \to Ω_{B / A}$ such that for every $B$ -module $M$ the assignment

$g ⟼ g \circ d, {Hom}_{B} (Ω_{B / A}, M) ⟶ {Der}_{A} (B, M),$

is a bijection, and such that these bijections are natural in $M$ : for every $B$ -linear map $h ⁣ : M \to N$ the square

$\begin{array}{ccc} {Hom}_{B} (Ω_{B / A}, M) & \overset{g \mapsto g \circ d}{\to} & {Der}_{A} (B, M) \\ ↓ h \circ - & ↓ h \circ - \\ {Hom}_{B} (Ω_{B / A}, N) & \overset{g \mapsto g \circ d}{\to} & {Der}_{A} (B, N) \end{array}$

commutes. In other words, $Ω_{B / A}$ represents the covariant functor $M \mapsto {Der}_{A} (B, M)$ on $B$ -modules, and $d$ is the universal $A$ -derivation of $B$ over $A$ ; the element $d b$ is the image of $b$ under it. Whether such a pair exists for a given $A \to B$ is not part of the definition; when it does, the pair is uniquely determined up to a unique compatible isomorphism, as the next paragraph records.

Uniqueness. If $(Ω, d)$ and $(Ω^{'}, d^{'})$ are both Kähler differential modules for the same ring map $A \to B$ , the universal property of the first applied to the derivation $d^{'}$ produces a unique $B$ -linear $u ⁣ : Ω \to Ω^{'}$ with $u \circ d = d^{'}$ , and the property of the second applied to $d$ produces a unique $B$ -linear $v ⁣ : Ω^{'} \to Ω$ with $v \circ d^{'} = d$ . Then $(v \circ u) \circ d = v \circ d^{'} = d$ and $(u \circ v) \circ d^{'} = d^{'}$ , while the identity maps of $Ω$ and $Ω^{'}$ have the same property; the injectivity clause of the universal property applied twice gives $v \circ u = {i d}_{Ω}$ and $u \circ v = {i d}_{Ω^{'}}$ . So $u$ is an isomorphism with inverse $v$ , and $u$ is the only $B$ -linear map from $Ω$ to $Ω^{'}$ compatible with the two universal derivations. In particular $Ω_{B / A}$ is determined by the ring map $A \to B$ up to canonical isomorphism, which is what justifies writing it as $Ω_{B / A}$ without further qualification.

Functoriality in ring maps. Given a commutative square of ring maps $A \to B$ , $A^{'} \to B^{'}$ , $u : A \to A^{'}$ and $v : B \to B^{'}$ , and universal pairs for its two horizontal maps, regard $Ω_{B^{'} / A^{'}}$ as a $B$ -module through $v$ . The composite $d^{'} \circ v$ is an $A$ -derivation: it is additive, satisfies Leibniz with this module action, and kills $A$ since the square commutes. Universality gives a unique $B$ -linear map $Ω_{B / A} \to Ω_{B^{'} / A^{'}}$ sending $d b$ to $d^{'} (v (b))$ . For identity squares this is the identity; for composable squares the composite has the prescribed values on $d b$ and so equals the map of the composite square by uniqueness. This proves functoriality in ring maps separately from naturality in the target module $M$ .

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

Existence and generators of Kähler differentials

Statement

Let $A \overset{φ}{\to} B$ be a homomorphism of commutative rings. Let $F$ be the free $B$ -module on the set underlying $B$ , with basis written $[b]$ for $b \in B$ , let $R \subseteq F$ be the $B$ -submodule generated by all elements

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

for $b, b^{'} \in B$ and $a \in A$ , and put $Ω_{B / A} : = F / R$ with $d ⁣ : B \to Ω_{B / A}$ , $d b : = [b] + R$ . Then $(Ω_{B / A}, d)$ is a Kähler differential module for $A \to B$ in the sense of Universal Kähler differential module: for every $B$ -module $M$ the assignment $g \mapsto g \circ d$ is a bijection ${Hom}_{B} (Ω_{B / A}, M) \to {Der}_{A} (B, M)$ , natural in $M$ . In particular a Kähler differential module exists for every ring homomorphism $A \to B$ , with no finiteness hypothesis on $B$ over $A$ , and $Ω_{B / A}$ is generated as a $B$ -module by the classes $d b$ of the elements of $B$ .

Facts & Assumptions

Given: A homomorphism $A \overset{φ}{\to} B$ of commutative rings, the free $B$ -module $F$ on the set underlying $B$ , the submodule $R \subseteq F$ of the three relator families, and the pair $(Ω_{B / A}, d)$ with $Ω_{B / A} = F / R$ and $d b = [b] + R$ .

[F1]

Universal algebraic differentials and A-derivations: the module of algebraic differentials is $Ω_{B / A} = F / R$ for the free $B$ -module $F$ on the set underlying $B$ with basis symbol $[b]$ , modulo the submodule generated by $[b + b^{'}] - [b] - [b^{'}]$ , $[b b^{'}] - b [b^{'}] - b^{'} [b]$ and $[φ (a)]$ , and $d b = [b] + R$ is its universal $A$ -derivation.

[F2]

Derivation of an algebra: an $A$ -derivation of $B$ into a $B$ -module $M$ is an additive, $A$ -constant map satisfying $D (b b^{'}) = b D (b^{'}) + b^{'} D (b)$ , and ${Der}_{A} (B, M)$ is the $B$ -module of all such maps.

[F3]

Universal Kähler differential module: $(Ω_{B / A}, d)$ is a Kähler differential module for $A \to B$ when $g \mapsto g \circ d$ is a bijection ${Hom}_{B} (Ω_{B / A}, M) \to {Der}_{A} (B, M)$ for every $B$ -module $M$ , naturally in $M$ .

Proof

1.1

The pair of [F1] is a $B$ -module with an $A$ -derivation. The free module $F$ is a $B$ -module and $R$ is a $B$ -submodule by construction, so $Ω_{B / A} = F / R$ is a $B$ -module and $d$ is a map $B \to Ω_{B / A}$ . Each of the three relator families lies in $R$ , hence vanishes in the quotient: $[b + b^{'}] - [b] - [b^{'}] \in R$ gives $d (b + b^{'}) = d b + d b^{'}$ , the element $[b b^{'}] - b [b^{'}] - b^{'} [b] \in R$ gives $d (b b^{'}) = b d b^{'} + b^{'} d b$ , and $[φ (a)] \in R$ gives $d (φ (a)) = 0$ . So $d$ is additive, $A$ -constant and satisfies Leibniz, that is, $d \in {Der}_{A} (B, Ω_{B / A})$ by [F2].

F1F2algebra

2.1

Every derivation descends to a map out of $Ω_{B / A}$ . Let $M$ be a $B$ -module and $D \in {Der}_{A} (B, M)$ . Since $F$ is free with basis ${[b] : b \in B}$ , there is a unique $B$ -linear map $G ⁣ : F \to M$ with $G ([b]) = D (b)$ for all $b \in B$ . By [F2] the map $D$ is additive, $A$ -constant and satisfies Leibniz, so $G$ kills each relator: $G ([b + b^{'}] - [b] - [b^{'}]) = D (b + b^{'}) - D (b) - D (b^{'}) = 0$ , similarly $G ([b b^{'}] - b [b^{'}] - b^{'} [b]) = D (b b^{'}) - b D (b^{'}) - b^{'} D (b) = 0$ , and $G ([φ (a)]) = D (φ (a)) = 0$ ; here we used that $G$ is $B$ -linear, so that $G (b [b^{'}]) = b G ([b^{'}]) = b D (b^{'})$ . The three families generate $R$ as a $B$ -submodule and [F1] presents $Ω_{B / A} = F / R$ , so $G$ factors through a $B$ -linear $\overset{ˉ}{G} ⁣ : Ω_{B / A} \to M$ with $\overset{ˉ}{G} ([b] + R) = D (b)$ , that is, $\overset{ˉ}{G} \circ d = D$ .

step 1.1F1F2algebra

3.1

The construction of step 2.1 is the unique inverse. Let $h ⁣ : Ω_{B / A} \to M$ be $B$ -linear with $h \circ d = D$ for some $D \in {Der}_{A} (B, M)$ . Evaluating on $d b$ gives $h (d b) = h ([b] + R) = D (b) = \overset{ˉ}{G} (d b)$ for every $b \in B$ , where $\overset{ˉ}{G}$ is the map produced in step 2.1. The classes $[b] + R = d b$ range over the images of a basis of the free module $F$ , so they generate $Ω_{B / A} = F / R$ as a $B$ -module, and two $B$ -linear maps agreeing on a generating set are equal; hence $h = \overset{ˉ}{G}$ . Therefore $g \mapsto g \circ d$ is a bijection ${Hom}_{B} (Ω_{B / A}, M) \to {Der}_{A} (B, M)$ for every $B$ -module $M$ .

step 2.1F1algebra

4.1

Naturality in $M$ . Let $t ⁣ : M \to N$ be $B$ -linear and let $h \in {Hom}_{B} (Ω_{B / A}, M)$ . Then $t \circ (h \circ d) = (t \circ h) \circ d$ as maps $B \to N$ , since both sides send $b$ to $t (h (d b))$ , and $t \circ h$ is again $B$ -linear, so the assignment of step 3.1 carries $h$ followed by $t$ to the derivation $t \circ (h \circ d)$ ; this is exactly the commutativity required of the bijections in [F3].

step 3.1F3algebra

5.1

Conclusion. Steps 1.1, 2.1 and 3.1 verify both clauses of [F3] for the pair $(Ω_{B / A}, d)$ of [F1], and step 4.1 verifies naturality, so that pair is a Kähler differential module for $A \to B$ . The construction used only the free module on the set $B$ and the submodule generated by the three relator families, so it exists for every ring homomorphism $A \to B$ with no finiteness hypothesis, and step 3.1 exhibits the classes $d b$ as a generating set of $Ω_{B / A}$ over $B$ .

step 1.1step 2.1step 3.1step 4.1F1F3∎

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

Derivations are maps out of Ω

Statement

Let $A \to B$ be a homomorphism of commutative rings, let $(Ω_{B / A}, d)$ be a Kähler differential module for it (Universal Kähler differential module), which exists by Existence and generators of Kähler differentials, and let $M$ be a $B$ -module. Then composition with $d$ is an isomorphism of $B$ -modules

${Hom}_{B} (Ω_{B / A}, M) \tilde{\to} {Der}_{A} (B, M), g ⟼ g \circ d,$

natural in $M$ : for every $B$ -linear $t ⁣ : M \to N$ the two composites ${Hom}_{B} (Ω_{B / A}, M) \to {Der}_{A} (B, N)$ obtained by applying $t$ before and after the isomorphism agree. Equivalently, $Ω_{B / A}$ represents the covariant functor $M \mapsto {Der}_{A} (B, M)$ on $B$ -modules.

Facts & Assumptions

Given: A ring homomorphism $A \to B$ , a Kähler differential module $(Ω_{B / A}, d)$ for it, and a $B$ -module $M$ .

[F1]

Existence and generators of Kähler differentials: for the module $Ω_{B / A} = F / R$ presented by the free $B$ -module on the symbols $[b]$ modulo the additive, Leibniz and $A$ -constant relators, and for every $B$ -module $M$ , the assignment $g \mapsto g \circ d$ is a bijection ${Hom}_{B} (Ω_{B / A}, M) \to {Der}_{A} (B, M)$ , natural in $M$ .

[F2]

Universal Kähler differential module: a Kähler differential module for $A \to B$ is a pair $(Ω_{B / A}, d)$ with $d$ an $A$ -derivation of $B$ into $Ω_{B / A}$ such that $g \mapsto g \circ d$ is a bijection ${Hom}_{B} (Ω_{B / A}, M) \to {Der}_{A} (B, M)$ for every $B$ -module $M$ , and such that these bijections are natural in $M$ .

[F3]

Derivation of an algebra: ${Der}_{A} (B, M)$ is a $B$ -module under pointwise addition and scalar multiplication, and for $B$ -linear $t ⁣ : M \to N$ composition $D \mapsto t \circ D$ is a $B$ -module map ${Der}_{A} (B, M) \to {Der}_{A} (B, N)$ .

Proof

1.1

Bijectivity. By [F1] the pair $(Ω_{B / A}, d)$ is a Kähler differential module for $A \to B$ , so [F2] gives, for every $B$ -module $M$ , that $g \mapsto g \circ d$ is a bijection ${Hom}_{B} (Ω_{B / A}, M) \to {Der}_{A} (B, M)$ ; the same statement holds for any Kähler differential module, since any two are related by a unique compatible isomorphism identifying the two assignments.

F1F2

1.2

Additivity and $B$ -linearity of the bijection. Both sides are $B$ -modules: ${Hom}_{B} (Ω_{B / A}, M)$ under pointwise operations, and ${Der}_{A} (B, M)$ under the operations of [F3]. For $g, g^{'} \in {Hom}_{B} (Ω_{B / A}, M)$ and $c \in B$ one has $(g + g^{'}) \circ d = g \circ d + g^{'} \circ d$ and $(c \cdot g) \circ d = c \cdot (g \circ d)$ as maps $B \to M$ , because evaluation at any $b$ gives $c g (d b)$ on both sides. Hence $g \mapsto g \circ d$ is a homomorphism of $B$ -modules.

F2F3algebra

2.1

Naturality. Let $t ⁣ : M \to N$ be $B$ -linear. By [F3] the composite $t \circ D$ is a derivation for every $D \in {Der}_{A} (B, M)$ and the assignment $D \mapsto t \circ D$ is $B$ -linear; moreover $t \circ (g \circ d) = (t \circ g) \circ d$ for every $B$ -linear $g ⁣ : Ω_{B / A} \to M$ , since both sides send $b$ to $t (g (d b))$ . Thus applying $t$ after the isomorphism agrees with applying $t$ before it, and the isomorphism of step 1.2 is natural in $M$ : the $B$ -module $Ω_{B / A}$ represents the functor $M \mapsto {Der}_{A} (B, M)$ by [F2].

step 1.2F2F3∎

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

Polynomial differentials are free

Statement

Let $A$ be a commutative ring and let $P = A [x_{1}, \dots, x_{n}]$ be the polynomial algebra on finitely many indeterminates, $n \geq 0$ . Then:

$Ω_{P / A}$ is a free $P$ -module with basis $d x_{1}, \dots, d x_{n}$ ; for $n = 0$ this says $Ω_{A / A} = 0$ ;
for every $P$ -module $M$ and every $n$ -tuple $(m_{1}, \dots, m_{n}) \in M^{n}$ there is exactly one $A$ -derivation $D ⁣ : P \to M$ with $D (x_{i}) = m_{i}$ ;
writing $\partial / \partial x_{i}$ for the derivation with $\partial x_{j} / \partial x_{i} = δ_{i j}$ , one has $d f = \sum_{i = 1}^{n} (\partial f / \partial x_{i}) d x_{i}$ for every $f \in P$ .

Neither statement assumes anything of $A$ beyond commutativity, and the correspondence is natural in $M$ .

Facts & Assumptions

Given: A commutative ring $A$ , an integer $n \geq 0$ , the polynomial algebra $P = A [x_{1}, \dots, x_{n}]$ , and a $P$ -module $M$ .

[F1]

Derivations are maps out of Ω: for every $P$ -module $N$ , composition with the universal derivation is a natural $P$ -module isomorphism ${Hom}_{P} (Ω_{P / A}, N) ≅ {Der}_{A} (P, N)$ .

[F2]

Derivation of an algebra: an $A$ -derivation of $P$ into $M$ is an additive $A$ -constant map satisfying the Leibniz rule, and ${Der}_{A} (P, M)$ is a $P$ -module under pointwise operations.

[F3]

Universal property of a polynomial ring on an arbitrary family of indeterminates: for commutative rings $R, S$ , a ring homomorphism $φ ⁣ : R \to S$ and a family $(s_{i})_{i \in I}$ in $S$ , there is a unique ring homomorphism $R [x_{i} : i \in I] \to S$ restricting to $φ$ on $R$ and sending $x_{i}$ to $s_{i}$ .

Proof

1.1

Sections of a square-zero thickening. Let $E (M)$ be the commutative ring whose underlying abelian group is $P \oplus M$ with product $(p, m) (p^{'}, m^{'}) = (p p^{'}, p m^{'} + p^{'} m)$ , made into an $A$ -algebra by $a \mapsto (φ (a), 0)$ . The first projection $π ⁣ : E (M) \to P$ is an $A$ -algebra homomorphism with kernel the square-zero ideal $M$ . If $s ⁣ : P \to E (M)$ is an $A$ -algebra homomorphism with $π \circ s = {i d}_{P}$ , write $s (f) = (f, D_{s} (f))$ ; additivity of $s$ gives $D_{s} (f + g) = D_{s} (f) + D_{s} (g)$ , the identity $π \circ s = i d$ and $A$ -linearity give $D_{s} (φ (a)) = 0$ , and multiplicativity $s (f g) = s (f) s (g)$ , expanded with $m m^{'} = 0$ in $E (M)$ , gives $D_{s} (f g) = f D_{s} (g) + g D_{s} (f)$ ; conversely these three laws make the formula $s (f) = (f, D_{s} (f))$ multiplicative and unital. So sections of $π$ over the identity correspond bijectively to the elements of ${Der}_{A} (P, M)$ by [F2].

F2algebra

2.1

Every tuple of values is realised. Let $m_{1}, \dots, m_{n} \in M$ . By [F3] applied to $φ ⁣ : A \to E (M)$ and the family $(x_{i}, m_{i}) \in E (M)$ , $i = 1, \dots, n$ , there is a unique $A$ -algebra homomorphism $θ ⁣ : P \to E (M)$ with $θ (x_{i}) = (x_{i}, m_{i})$ and $θ (φ (a)) = (φ (a), 0)$ . The composite $π \circ θ ⁣ : P \to P$ is an $A$ -algebra endomorphism of $P$ with $x_{i} \mapsto x_{i}$ , so by the uniqueness clause of [F3] it is the identity; hence $θ (f) = (f, D (f))$ for the map $D ⁣ : P \to M$ given by the second coordinate, and $D (x_{i}) = m_{i}$ . By step 1.1 the map $D$ is an $A$ -derivation of $P$ into $M$ .

step 1.1F3

3.1

Uniqueness of the values on generators. If $D^{'} \in {Der}_{A} (P, M)$ satisfies $D^{'} (x_{i}) = m_{i}$ for all $i$ , then $f \mapsto (f, D^{'} (f))$ is an $A$ -algebra homomorphism $P \to E (M)$ by step 1.1, it agrees with $θ$ on $A$ and on each $x_{i}$ , and hence equals $θ$ by the uniqueness clause of [F3]; therefore $D^{'} = D$ . So for every $P$ -module $M$ the evaluation map ${Der}_{A} (P, M) \to M^{n}$ , $D \mapsto (D (x_{1}), \dots, D (x_{n}))$ , is a bijection; it is $P$ -linear, and natural in $M$ because a $P$ -linear $t ⁣ : M \to N$ sends $D$ to $t \circ D$ with values $t (D (x_{i}))$ .

step 2.1F2F3

4.1

Freeness. Composing the natural bijections of [F1] and of step 3.1 gives natural bijections ${Hom}_{P} (Ω_{P / A}, M) ≅ M^{n} ≅ {Hom}_{P} (P^{n}, M)$ for every $P$ -module $M$ . The image of the identity of $P^{n}$ is a $P$ -linear map $φ ⁣ : Ω_{P / A} \to P^{n}$ , and its inverse image is a $P$ -linear map $ψ ⁣ : P^{n} \to Ω_{P / A}$ with $φ \circ ψ = i d$ and $ψ \circ φ = i d$ : both identities are checked on generating sets, the standard basis of $P^{n}$ and, by the explicit construction of the bijection in step 3.1, the elements $d x_{i}$ . Hence $ψ$ is an isomorphism sending $e_{i}$ to $d x_{i}$ , so $Ω_{P / A}$ is free with basis $d x_{1}, \dots, d x_{n}$ . For $n = 0$ we have $P = A$ and $M^{0} = 0$ , so step 3.1 says that every $A$ -derivation of $A$ into any $A$ -module is zero, and [F1] gives $Ω_{A / A} = 0$ .

step 3.1F1∎

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

Conormal exact sequence for an algebra quotient

Statement

Let $A \to P$ be a homomorphism of commutative rings, let $I \subseteq P$ be an ideal and let $B = P / I$ , with quotient map $π ⁣ : P \to B$ . Then the sequence of $B$ -modules

$I / I^{2} ⟶ B \otimes_{P} Ω_{P / A} ⟶ Ω_{B / A} ⟶ 0$

is exact, where $I / I^{2}$ is regarded as a $B$ -module and the first map sends the class of $i \in I$ to $1 \otimes d i$ , while the second is induced by $d_{P / A}$ and $π$ . No injectivity of the first arrow is asserted; it fails in general, and the failure is recorded on the examples page.

Facts & Assumptions

Given: A ring homomorphism $A \to P$ , an ideal $I \subseteq P$ and the quotient $B = P / I$ with quotient map $π$ .

[F1]

Derivations are maps out of Ω: for every ring map $R \to S$ with Kähler differential module $(Ω_{S / R}, d)$ and every $S$ -module $N$ , composition with $d$ is a natural $S$ -module isomorphism ${Hom}_{S} (Ω_{S / R}, N) ≅ {Der}_{R} (S, N)$ .

[F2]

Existence and generators of Kähler differentials: a Kähler differential module exists for every ring map, $Ω_{S / R}$ is generated as an $S$ -module by the elements $d s$ , and the representability statement of [F1] holds for it.

[F3]

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

[F4]

Derivation of an algebra: an $A$ -derivation is additive, $A$ -constant and satisfies the Leibniz rule; ${Der}_{A} (S, N)$ is an $S$ -module under pointwise operations.

Proof

1.1

The second map exists and is surjective. Regard $Ω_{B / A}$ as a $P$ -module along $π$ . The composite $P \overset{π}{\to} B \overset{d_{B / A}}{\to} Ω_{B / A}$ is an $A$ -derivation of $P$ into $Ω_{B / A}$ : it is additive, kills $A$ , and satisfies Leibniz because $π$ is a ring map and $d_{B / A}$ is a derivation. By [F1] it corresponds to a $P$ -linear map $u ⁣ : Ω_{P / A} \to Ω_{B / A}$ with $u (d p) = d_{B / A} (π (p))$ . For $i \in I$ we have $u (d i) = d_{B / A} (0) = 0$ , and $P$ -linearity gives $u (i ω) = π (i) u (ω) = 0$ , so $u$ kills the submodule $I Ω_{P / A} \subseteq Ω_{P / A}$ . By [F3] applied to $I \to P \to B \to 0$ tensored with $Ω_{P / A}$ we have $B \otimes_{P} Ω_{P / A} ≅ Ω_{P / A} / I Ω_{P / A}$ , so $u$ induces a $B$ -linear map $β ⁣ : B \otimes_{P} Ω_{P / A} \to Ω_{B / A}$ with $β (1 \otimes d p) = d_{B / A} (π (p))$ . It is surjective: every $b \in B$ is $π (p)$ for some $p \in P$ , and the elements $d_{B / A} (b)$ generate $Ω_{B / A}$ over $B$ by [F2].

F1F2F3F4

1.2

The first map is well defined. The assignment $i \mapsto 1 \otimes d i$ defines a $P$ -linear map $I \to B \otimes_{P} Ω_{P / A}$ , and it kills $I^{2}$ : for $i, j \in I$ , $1 \otimes d (i j) = 1 \otimes (i d j + j d i) = i (1 \otimes d j) + j (1 \otimes d i) = 0$ in the $B$ -module $B \otimes_{P} Ω_{P / A}$ , because the classes of $i$ and $j$ in $B$ are zero. Hence it induces a $B$ -linear map $α ⁣ : I / I^{2} \to B \otimes_{P} Ω_{P / A}$ with $α ([i]) = 1 \otimes d i$ .

F4algebra

2.1

The composite vanishes. For $i \in I$ , $β (α ([i])) = β (1 \otimes d i) = d_{B / A} (π (i)) = d_{B / A} (0) = 0$ ; thus $β$ factors through the cokernel $Q : = coker α = (B \otimes_{P} Ω_{P / A}) / α (I / I^{2})$ , giving a surjective $B$ -linear map $\overset{ˉ}{β} ⁣ : Q \to Ω_{B / A}$ .

step 1.1step 1.2

3.1

A left inverse for $\overset{ˉ}{β}$ . Let $D ⁣ : P \to Q$ send $p$ to the class of $1 \otimes d p$ ; it is the composite of the $A$ -derivation $p \mapsto 1 \otimes d p$ with the $B$ -linear quotient map, hence an $A$ -derivation, and it kills $I$ because the class of $1 \otimes d i$ is $α ([i]) = 0$ for $i \in I$ . Since $Q$ is a $B$ -module, $D$ is constant on cosets of $I$ and satisfies Leibniz, so it descends to an $A$ -derivation $\overset{ˉ}{D} ⁣ : B \to Q$ : any $b$ has a lift $p$ , and $\overset{ˉ}{D} (b) : = D (p)$ is well defined because $D$ kills $I$ . Applying [F1] to the ring map $A \to B$ gives a $B$ -linear map $ℓ ⁣ : Ω_{B / A} \to Q$ with $ℓ (d_{B / A} (b)) = \overset{ˉ}{D} (b)$ for all $b \in B$ .

step 2.1F1F4

4.1

$ℓ$ is inverse to $\overset{ˉ}{β}$ . For $p \in P$ we have $ℓ (\overset{ˉ}{β} ([1 \otimes d p])) = ℓ (d_{B / A} (π (p))) = \overset{ˉ}{D} (π (p)) = D (p) = [1 \otimes d p]$ , and the classes $[1 \otimes d p]$ generate $Q$ over $B$ because the $d p$ generate $Ω_{P / A}$ , so $ℓ \circ \overset{ˉ}{β} = {i d}_{Q}$ . Conversely, for $b \in B$ with lift $p$ , $\overset{ˉ}{β} (ℓ (d_{B / A} (b))) = \overset{ˉ}{β} ([1 \otimes d p]) = d_{B / A} (b)$ , and the $d_{B / A} (b)$ generate $Ω_{B / A}$ by [F2], so $\overset{ˉ}{β} \circ ℓ = i d$ . Hence $\overset{ˉ}{β}$ is an isomorphism, $\ker β = im α$ , and with $β$ surjective the displayed sequence is exact.

step 2.1step 3.1F2∎

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

Jacobian presentation of Ω

Statement

Let $A$ be a commutative ring, let $P = A [x_{1}, \dots, x_{n}]$ and let $I = (f_{1}, \dots, f_{r}) \subseteq P$ be the ideal generated by finitely many elements, with quotient $B = P / I$ . Then $Ω_{B / A}$ is the cokernel of the $B$ -linear map $B^{r} \to B^{n}$ whose $j$ -th column is the vector of partial derivatives $(\partial f_{j} / \partial x_{i})_{i = 1}^{n}$ , that is,

$Ω_{B / A} ≅ B^{n} / \sum_{j = 1}^{r} B \cdot (\partial f_{j} / \partial x_{1}, \dots, \partial f_{j} / \partial x_{n}) .$

This is a presentation of $Ω_{B / A}$ by $r$ relations and is not by itself a smoothness criterion: it carries no flatness or fibre hypothesis, and the number $r$ of generators of $I$ is not asserted to be minimal.

Facts & Assumptions

Given: A commutative ring $A$ , the polynomial algebra $P = A [x_{1}, \dots, x_{n}]$ , elements $f_{1}, \dots, f_{r} \in P$ , the ideal $I = (f_{1}, \dots, f_{r})$ and $B = P / I$ .

[F1]

Polynomial differentials are free: $Ω_{P / A}$ is free with basis $d x_{1}, \dots, d x_{n}$ , and for the derivations $\partial / \partial x_{i}$ with $\partial x_{j} / \partial x_{i} = δ_{i j}$ one has $d f = \sum_{i} (\partial f / \partial x_{i}) d x_{i}$ for every $f \in P$ .

[F2]

Conormal exact sequence for an algebra quotient: for every ideal $J \subseteq P$ and $C = P / J$ the sequence $J / J^{2} \to C \otimes_{P} Ω_{P / A} \to Ω_{C / A} \to 0$ is exact, the first map sending the class of $x \in J$ to $1 \otimes d x$ .

[F3]

Derivation of an algebra: an $A$ -derivation is additive, kills $A$ and satisfies Leibniz. The particular universal derivation on $P$ and the free $P$ -basis $d x_{1}, \dots, d x_{n}$ of $Ω_{P / A}$ come from [F1].

Proof

1.1

The middle term. By [F1] the elements $d x_{1}, \dots, d x_{n}$ form a $P$ -basis of $Ω_{P / A}$ . Since extension of scalars along $P \to B$ carries a free module with basis $d x_{i}$ to the free $B$ -module with basis $1 \otimes d x_{i}$ , there is an isomorphism of $B$ -modules $B \otimes_{P} Ω_{P / A} \to B^{n}$ sending $1 \otimes d x_{i}$ to the standard basis vector $e_{i}$ .

F1algebra

1.2

The conormal term. In $I / I^{2}$ , every class is a $B$ -linear combination of the classes $[f_{1}], \dots, [f_{r}]$ : an element of $I$ has the form $\sum_{j} p_{j} f_{j}$ with $p_{j} \in P$ , and by bilinearity of the class map $[x y] = x [y]$ for $x \in P$ , $y \in I$ , its class is $\sum_{j} (p_{j} + I) [f_{j}]$ .

F2algebra

2.1

The Jacobian columns. By [F1] the universal derivation, which obeys the laws of [F3], satisfies $d f_{j} = \sum_{i} (\partial f_{j} / \partial x_{i}) d x_{i}$ , so the first map of the conormal sequence of [F2] sends $[f_{j}]$ to $1 \otimes d f_{j} = \sum_{i} (\partial f_{j} / \partial x_{i}) (1 \otimes d x_{i})$ , which under the identification of step 1.1 is the $j$ -th column $(\partial f_{j} / \partial x_{1}, \dots, \partial f_{j} / \partial x_{n})$ of the Jacobian matrix.

step 1.1F1F2F3

3.1

Conclusion. By the exactness of [F2] applied to the quotient $B = P / I$ , the module $Ω_{B / A}$ is the cokernel of the first map $I / I^{2} \to B \otimes_{P} Ω_{P / A}$ , which by steps 1.2 and 2.1 is the $B$ -linear map with the Jacobian columns on a generating set of $I / I^{2}$ ; under step 1.1 this is the displayed presentation $Ω_{B / A} ≅ B^{n} / \sum_{j} B \cdot (\partial f_{j} / \partial x_{i})_{i}$ . Since $A$ , $n$ and the generating family $f_{1}, \dots, f_{r}$ were arbitrary, the presentation holds without additional hypotheses, and no smoothness conclusion is drawn from it.

step 1.2step 2.1F2∎

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

Transitivity sequence for differential modules

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 sends $c \otimes d b$ to $c d_{C / A} (b)$ and the second is induced by $d_{C / B}$ . The first arrow is not asserted to be injective, and it fails to be injective in general; exactness on the left is not part of the statement.

Facts & Assumptions

Given: Ring homomorphisms $A \to B \to C$ of commutative rings.

[F1]

[F2]

Existence and generators of Kähler differentials: a Kähler differential module exists for every ring map, and it is generated as a module by the elements $d s$ .

[F3]

Derivation of an algebra: an $A$ -derivation is additive, $A$ -constant and satisfies the Leibniz rule; an $A$ -derivation $D : C \to M$ that kills the image of $B$ is a $B$ -derivation, since $D (b c) = b D (c) + c D (b) = b D (c)$ .

Proof

1.1

The first map. The composite $B \to C \overset{d_{C / A}}{\to} Ω_{C / A}$ is an $A$ -derivation of $B$ into the $C$ -module $Ω_{C / A}$ ; by [F1] it corresponds to a $B$ -linear map $Ω_{B / A} \to Ω_{C / A}$ with $d b \mapsto d_{C / A} (b)$ . Its extension of scalars along $B \to C$ is the $C$ -linear map $γ ⁣ : C \otimes_{B} Ω_{B / A} \to Ω_{C / A}$ with $γ (c \otimes d b) = c d_{C / A} (b)$ .

F1F3

1.2

The second map. The universal $B$ -derivation $d_{C / B} ⁣ : C \to Ω_{C / B}$ is also an $A$ -derivation, so [F1] applied to $A \to C$ gives a $C$ -linear map $δ ⁣ : Ω_{C / A} \to Ω_{C / B}$ with $δ (d_{C / A} (c)) = d_{C / B} (c)$ . It is surjective because the elements $d_{C / B} (c)$ generate $Ω_{C / B}$ over $C$ by [F2].

F1F2

2.1

The composite vanishes. For $c \in C$ and $b \in B$ , $δ (γ (c \otimes d b)) = c d_{C / B} (b) = 0$ , since $d_{C / B}$ is $B$ -constant: $b$ is the image of an element of $B$ , so $d_{C / B} (b) = 0$ in the definition of a $B$ -derivation of $C$ . Hence there is an induced $C$ -linear map $\overset{ˉ}{δ} ⁣ : Q \to Ω_{C / B}$ out of $Q : = coker γ$ , and it is surjective by step 1.2.

step 1.1step 1.2F3

3.1

A left inverse for $\overset{ˉ}{δ}$ . The map $D ⁣ : C \to Q$ sending $c$ to the class of $d_{C / A} (c)$ is the composite of the $A$ -derivation $d_{C / A}$ with the $C$ -linear quotient map, hence an $A$ -derivation, and it kills $B$ because $d_{C / A} (b)$ is the class of $γ (1 \otimes d b)$ , which is zero in $Q$ . As $D$ is $A$ -linear and kills $B$ , it satisfies $D (b c) = b D (c)$ for $b \in B$ , $c \in C$ by the Leibniz rule, so $D$ is a $B$ -derivation; [F1] applied to the ring map $B \to C$ gives a $C$ -linear map $ℓ ⁣ : Ω_{C / B} \to Q$ with $ℓ (d_{C / B} (c)) = [d_{C / A} (c)]$ .

step 2.1F1F3

4.1

$ℓ$ is inverse to $\overset{ˉ}{δ}$ . For all $c \in C$ we have $\overset{ˉ}{δ} (ℓ (d_{C / B} (c))) = \overset{ˉ}{δ} ([d_{C / A} (c)]) = d_{C / B} (c)$ and $ℓ (\overset{ˉ}{δ} ([d_{C / A} (c)])) = ℓ (d_{C / B} (c)) = [d_{C / A} (c)]$ by the defining property of $ℓ$ in step 3.1. The elements $d_{C / B} (c)$ generate $Ω_{C / B}$ and the classes $[d_{C / A} (c)]$ generate $Q$ over $C$ by [F2], so $\overset{ˉ}{δ} \circ ℓ = i d$ and $ℓ \circ \overset{ˉ}{δ} = i d$ . Hence $\overset{ˉ}{δ}$ is an isomorphism, $\ker δ = im γ$ , and with $δ$ surjective the displayed sequence is exact.

step 2.1step 3.1F2∎

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

Kähler differentials commute with localization

Statement

Let $A \overset{φ}{\to} B$ be a homomorphism of commutative rings, let $U \subseteq B$ be a multiplicative subset and let $V \subseteq A$ be a multiplicative subset with $φ (V) \subseteq U$ . Then the canonical $(U^{- 1} B)$ -linear map induced by the localization map $λ ⁣ : B \to U^{- 1} B$ , namely

$U^{- 1} Ω_{B / A} ⟶ Ω_{U^{- 1} B / V^{- 1} A}, \frac{d b}{u} ⟼ \frac{d (b / 1)}{u},$

is an isomorphism of $U^{- 1} B$ -modules. The subsets $V = {1}$ and $U = {1}$ are allowed; in the second case the map is the identity on $Ω_{B / A}$ . No finiteness hypothesis is imposed on $B$ over $A$ , and the result is not asserted for an arbitrary ring homomorphism $B \to C$ that is not a localization.

Facts & Assumptions

Given: A ring homomorphism $A \to B$ , a multiplicative subset $U \subseteq B$ and a multiplicative subset $V \subseteq A$ with $φ (V) \subseteq U$ .

[F1]

[F2]

Localisation of a module at a multiplicative subset: the localization $U^{- 1} M$ of an $R$ -module $M$ consists of the classes $m / u$ with $u \in U$ , the canonical map is $m \mapsto m / 1$ , and the elements of $U^{- 1} Ω$ are exactly the classes $ω / u$ .

[F3]

Universal property of localisation for modules: for an $R$ -linear map $f ⁣ : M \to N$ with $N$ an $S^{- 1} R$ -module, there is a unique $S^{- 1} R$ -linear $\tilde{f} ⁣ : S^{- 1} M \to N$ with $\tilde{f} (m / s) = (1 / s) f (m)$ .

[F4]

Universal property of localisation: maps that invert $S$ factor uniquely through $S^{- 1} R$ : if $f ⁣ : R \to A$ sends every $s \in S$ to a unit, there is a unique unital ring homomorphism $\tilde{f} ⁣ : S^{- 1} R \to A$ with $\tilde{f} \circ λ_{S} = f$ , given by $\tilde{f} (r / s) = f (r) f (s)^{- 1}$ .

[F5]

Derivation of an algebra: derivations are additive, constant on the base and satisfy the Leibniz rule, and these three laws characterise ring sections of the square-zero extension $C \oplus N$ by $(c, n) (c^{'}, n^{'}) = (c c^{'}, c n^{'} + c^{'} n)$ .

[F6]

Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions: $U^{- 1} B$ is a commutative ring, $λ ⁣ : B \to U^{- 1} B$ is a ring homomorphism, each $u \in U$ maps to a unit $1 / u$ , and $V^{- 1} A$ is defined likewise.

Proof

1.1

The canonical map. Since $φ (V) \subseteq U$ , [F4] extends $A \to U^{- 1} B$ uniquely to a ring map $V^{- 1} A \to U^{- 1} B$ , and $λ ⁣ : B \to U^{- 1} B$ is an $A$ -algebra homomorphism, and the composite $B \overset{λ}{\to} U^{- 1} B \overset{d^{'}}{\to} Ω_{U^{- 1} B / V^{- 1} A}$ is an $A$ -derivation of $B$ into $Ω_{U^{- 1} B / V^{- 1} A}$ : it is additive, kills $φ (A)$ , and satisfies Leibniz. By [F1] it corresponds to a $B$ -linear $ρ ⁣ : Ω_{B / A} \to Ω_{U^{- 1} B / V^{- 1} A}$ with $ρ (d b) = d^{'} (b / 1)$ , and by [F3] applied to the canonical map $λ_{Ω} ⁣ : Ω_{B / A} \to U^{- 1} Ω_{B / A}$ the map $ρ$ factors uniquely through a $U^{- 1} B$ -linear map $α ⁣ : U^{- 1} Ω_{B / A} \to Ω_{U^{- 1} B / V^{- 1} A}$ with $α (ω / u) = (1 / u) ρ (ω)$ ; in particular $α (d b / u) = d^{'} (b / 1) / u$ . This is the canonical map of the statement.

F1F2F3F4F6

1.2

A derivation of the localization. Let $E : = U^{- 1} B \oplus U^{- 1} Ω_{B / A}$ with the product $(x, ω) (x^{'}, ω^{'}) = (x x^{'}, x ω^{'} + x^{'} ω)$ , a commutative ring in which the second summand is an ideal of square zero, and let $s ⁣ : B \to E$ , $s (b) : = (b / 1, d b)$ . Then $s$ is a unital ring homomorphism: it is additive, and multiplicativity is exactly the Leibniz rule $d (b b^{'}) = b d b^{'} + b^{'} d b$ of [F5]. For $u \in U$ the element $s (u) = (u / 1, d u)$ is a unit of $E$ with inverse $(1 / u, - d u / u^{2})$ , since $(u / 1) (- d u / u^{2}) + (1 / u) d u = - d u / u + d u / u = 0$ . By [F4] there is a unique unital ring homomorphism $\tilde{s} ⁣ : U^{- 1} B \to E$ with $\tilde{s} \circ λ = s$ ; writing $\tilde{s} (x) = (s_{1} (x), D (x))$ , the first coordinate $s_{1}$ is a unital ring homomorphism $U^{- 1} B \to U^{- 1} B$ with $s_{1} (λ (b)) = b / 1$ , so $s_{1} = i d$ by the uniqueness clause of [F4] applied to the identity. Hence $\tilde{s} (x) = (x, D (x))$ .

F5F6F4algebra

2.1

The second coordinate is a derivation. Multiplicativity of $\tilde{s}$ in the square-zero extension gives $D (x x^{'}) = x D (x^{'}) + x^{'} D (x)$ , and additivity of $\tilde{s}$ gives $D (x + x^{'}) = D (x) + D (x^{'})$ . For $a \in A$ we have $\tilde{s} (λ (φ (a))) = s (φ (a)) = (φ (a) / 1, 0)$ , so $D$ kills $λ \circ φ (A)$ ; since $D$ also kills $λ (φ (v))$ for $v \in V$ and $D$ satisfies Leibniz with $D (1) = 0$ , it kills the inverse of each such unit, hence the image of $V^{- 1} A \to U^{- 1} B$ . So $D$ is a $V^{- 1} A$ -derivation of $U^{- 1} B$ into the $U^{- 1} B$ -module $U^{- 1} Ω_{B / A}$ , and by [F1] it corresponds to a $U^{- 1} B$ -linear map $β ⁣ : Ω_{U^{- 1} B / V^{- 1} A} \to U^{- 1} Ω_{B / A}$ with $β (d^{'} x) = D (x)$ .

step 1.2F1F5

3.1

The two maps are inverse. For $b \in B$ and $u \in U$ , multiplicativity of $\tilde{s}$ gives $D (b / u) = D (λ (b) \cdot (1 / u)) = (1 / u) D (b) + (b / 1) D (1 / u)$ , and $D (1 / u) = - d u / u^{2}$ because $0 = D (1) = D ((u / 1) (1 / u)) = (1 / u) D (u) + (u / 1) D (1 / u)$ with $D (u) = d u$ from $\tilde{s} (λ (u)) = (u / 1, d u)$ ; hence $D (b / u) = (u d b - b d u) / u^{2}$ . Therefore $α (β (d^{'} (b / u))) = α ((u d b - b d u) / u^{2}) = (u d^{'} (b / 1) - b d^{'} (u / 1)) / u^{2} = d^{'} (b / u)$ , the last equality being the derivation identity for the fraction $b / u$ with $u$ invertible, obtained from the Leibniz rule and $d^{'} (u \cdot (1 / u)) = 0$ . Since the elements $d^{'} (b / u)$ generate $Ω_{U^{- 1} B / V^{- 1} A}$ over $U^{- 1} B$ , this gives $α \circ β = i d$ . Conversely $β (α (d b / u)) = β (d^{'} (b / 1) / u) = (1 / u) D (b / 1) = d b / u$ for all $b \in B$ , $u \in U$ , and the elements $d b / u$ generate $U^{- 1} Ω_{B / A}$ over $U^{- 1} B$ by [F2], so $β \circ α = i d$ . Hence $α$ is an isomorphism. Taking $V = {1}$ gives $V^{- 1} A = A$ and taking $U = {1}$ makes $λ$ the identity, so both degenerate cases are covered by the same computation.

step 1.1step 2.1F2F6∎

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

Kähler differentials commute with scalar base change

Statement

Let $A \to B$ and $A \to A^{'}$ be homomorphisms of commutative rings, and put $B^{'} = B \otimes_{A} A^{'}$ , so that $B \to B^{'}$ , $b \mapsto b \otimes 1$ , is a ring map and $B^{'}$ is an $A^{'}$ -algebra. Then the canonical $B^{'}$ -linear map

$Ω_{B / A} \otimes_{B} B^{'} ⟶ Ω_{B^{'} / A^{'}}, d b \otimes a^{'} ⟼ a^{'} d (b \otimes 1),$

is an isomorphism. It is natural in the base-change data $A \to A^{'}$ , and it does not assert that $Ω_{B / A}$ is unchanged under an arbitrary ring map $B \to C$ that is not one of these base-change maps.

Facts & Assumptions

Given: Ring homomorphisms $A \to B$ and $A \to A^{'}$ , the ring $B^{'} = B \otimes_{A} A^{'}$ and the canonical map $b \mapsto b \otimes 1$ .

[F1]

[F2]

Universal property of the tensor product for balanced maps into abelian groups: for a balanced map $b ⁣ : M \times N \to X$ out of a right $R$ -module $M$ and a left $R$ -module $N$ there is a unique group homomorphism $\overline{b} ⁣ : M \otimes_{R} N \to X$ with $\overline{b} (m \otimes n) = b (m, n)$ .

[F3]

Universal mapping property of the tensor product of commutative algebras: $B^{'} = B \otimes_{A} A^{'}$ is the coproduct of the two commutative $A$ -algebras, so there is a unique $A$ -algebra structure in which $b \mapsto b \otimes 1$ and $a^{'} \mapsto 1 \otimes a^{'}$ are $A$ -algebra maps, and the pure tensors $b \otimes a^{'}$ generate $B^{'}$ as an $A^{'}$ -algebra.

[F4]

Derivation of an algebra: derivations are additive, constant on the base and satisfy the Leibniz rule; a $B^{'}$ -module map out of $Ω_{B^{'} / A^{'}}$ is determined by its values on a generating set of $Ω_{B^{'} / A^{'}}$ .

Proof

1.1

Restriction and extension of derivations. Let $M$ be a $B^{'}$ -module. Restriction along $b \mapsto b \otimes 1$ sends a derivation in ${Der}_{A^{'}} (B^{'}, M)$ to an element of ${Der}_{A} (B, M)$ , because the composite is additive, $A$ -constant and satisfies Leibniz. Conversely, given $D \in {Der}_{A} (B, M)$ , the map $β_{D} ⁣ : B \times A^{'} \to M$ , $β_{D} (b, a^{'}) : = a^{'} D (b)$ , is $A$ -bilinear: it is additive in each variable and $β_{D} (α b, a^{'}) = a^{'} α D (b) = β_{D} (b, α a^{'})$ for $α \in A$ . By [F2] it factors through a group homomorphism $\tilde{D} ⁣ : B^{'} \to M$ with $\tilde{D} (b \otimes a^{'}) = a^{'} D (b)$ ; this is $A^{'}$ -linear because $\tilde{D} ((b \otimes a^{'}) a^{''}) = a^{'} a^{''} D (b) = a^{''} \tilde{D} (b \otimes a^{'})$ , and it is a derivation, since $\tilde{D} ((b \otimes a^{'}) (b^{'} \otimes c^{'})) = a^{'} c^{'} D (b b^{'}) = a^{'} c^{'} (b D (b^{'}) + b^{'} D (b)) = (b \otimes a^{'}) \tilde{D} (b^{'} \otimes c^{'}) + (b^{'} \otimes c^{'}) \tilde{D} (b \otimes a^{'})$ . Also $\tilde{D} (1 \otimes a^{'}) = a^{'} D (1) = 0$ , so $\tilde{D}$ is $A^{'}$ -constant. The two assignments are inverse: restriction of $\tilde{D}$ gives $b \mapsto D (b)$ , and an extension of a restricted derivation agrees with $\tilde{D}$ on the pure tensors $b \otimes a^{'}$ , which generate $B^{'}$ over $A^{'}$ by [F3]. So restriction is a natural bijection ${Der}_{A^{'}} (B^{'}, M) ≅ {Der}_{A} (B, M)$ for every $B^{'}$ -module $M$ .

F2F3F4

1.2

The canonical map. The composite $B \to B^{'} \overset{d^{'}}{\to} Ω_{B^{'} / A^{'}}$ is an $A$ -derivation of $B$ into the $B^{'}$ -module $Ω_{B^{'} / A^{'}}$ , so by [F1] it corresponds to a $B$ -linear $ρ ⁣ : Ω_{B / A} \to Ω_{B^{'} / A^{'}}$ with $ρ (d b) = d^{'} (b \otimes 1)$ . The map $Ω_{B / A} \times B^{'} \to Ω_{B^{'} / A^{'}}$ , $(ω, b^{'}) \mapsto b^{'} ρ (ω)$ , is $B$ -balanced, so by [F2] it factors through a group homomorphism $α ⁣ : Ω_{B / A} \otimes_{B} B^{'} \to Ω_{B^{'} / A^{'}}$ with $α (d b \otimes a^{'}) = a^{'} d^{'} (b \otimes 1)$ ; it is $B^{'}$ -linear by construction.

F1F2F4

2.1

The inverse map. Let $N : = Ω_{B / A} \otimes_{B} B^{'}$ , a $B^{'}$ -module, and let $D ⁣ : B \to N$ be $D (b) : = d b \otimes 1$ ; this is an $A$ -derivation, since $b \mapsto d b$ is one and $- \otimes 1$ is additive. By step 1.1 there is a unique $A^{'}$ -derivation $\tilde{D} ⁣ : B^{'} \to N$ with $\tilde{D} (b \otimes 1) = D (b)$ and $\tilde{D} (b \otimes a^{'}) = a^{'} (d b \otimes 1)$ . By [F1] applied to $A^{'} \to B^{'}$ it corresponds to a $B^{'}$ -linear map $β ⁣ : Ω_{B^{'} / A^{'}} \to N$ with $β (d^{'} (b \otimes a^{'})) = a^{'} (d b \otimes 1)$ .

step 1.1F1F4

3.1

The maps are inverse. On the one hand $β (α (d b \otimes a^{'})) = β (a^{'} d^{'} (b \otimes 1)) = a^{'} (d b \otimes 1) = d b \otimes a^{'}$ , and the elements $d b \otimes a^{'}$ generate $Ω_{B / A} \otimes_{B} B^{'}$ over $B^{'}$ because the elements $d b$ generate $Ω_{B / A}$ over $B$ ; hence $β \circ α = i d$ . On the other hand $α (β (d^{'} (b \otimes a^{'}))) = α (a^{'} (d b \otimes 1)) = a^{'} d^{'} (b \otimes 1) = d^{'} (b \otimes a^{'})$ , where the last equality uses $b \otimes a^{'} = (b \otimes 1) (1 \otimes a^{'})$ , the Leibniz rule and $d^{'} (1 \otimes a^{'}) = 0$ ; since the elements $d^{'} (b \otimes a^{'})$ generate $Ω_{B^{'} / A^{'}}$ over $B^{'}$ by [F3] and [F4], we get $α \circ β = i d$ . Hence $α$ is an isomorphism. The construction used only the given base-change maps, so no statement is made about an arbitrary ring map $B \to C$ .

step 1.2step 2.1F3F4∎

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

Sheaf of relative Kähler differentials

Definition

Let $f ⁣ : X \to S$ be a morphism of schemes (Morphisms of schemes), so that $f$ is in particular a morphism of ringed spaces and comes with a map of sheaves of rings $f^{♯} ⁣ : f^{- 1} O_{S} \to O_{X}$ (Inverse image presheaf and inverse image sheaf); the pair $(X, f)$ is an $S$ -scheme (Schemes and morphisms over a base). Let $F$ be a sheaf of $O_{X}$ -modules (Modules on a ringed space).

$S$ -derivations. An $S$ -derivation of $O_{X}$ into $F$ is a morphism of sheaves of abelian groups $D ⁣ : O_{X} \to F$ such that for all local sections $a, a^{'}$ of $O_{X}$ over a common open set $W \subseteq X$ one has

$D (a + a^{'}) = D (a) + D (a^{'}), D (a a^{'}) = a D (a^{'}) + a^{'} D (a),$

and such that $D$ annihilates the image of $f^{♯} ⁣ : f^{- 1} O_{S} \to O_{X}$ : the composite $f^{- 1} O_{S} \overset{f^{♯}}{\to} O_{X} \overset{D}{\to} F$ is the zero map. Here the products and sums are taken in the rings $O_{X} (W)$ . The set of all such $D$ is written ${Der}_{S} (O_{X}, F)$ ; it is a $Γ (X, O_{X})$ -module under the pointwise operations, and for a morphism $F \to G$ of $O_{X}$ -modules, postcomposition $D \mapsto α \circ D$ maps ${Der}_{S} (O_{X}, F)$ to ${Der}_{S} (O_{X}, G)$ . For $S$ -schemes and $S$ -morphisms the condition is that $D$ kills $f^{- 1} O_{S}$ ; when $S$ is fixed one simply says derivation.

Construction of $Ω_{X / S}$ . Consider the presheaf of $O_{X}$ -modules

$P ⁣ : W ⟼ Ω_{O_{X} (W) / (f^{- 1} O_{S}) (W)},$

where $W \subseteq X$ is open, the ring $(f^{- 1} O_{S}) (W)$ maps to $O_{X} (W)$ by $f_{W}^{♯}$ , and $Ω_{O_{X} (W) / (f^{- 1} O_{S}) (W)}$ is the Kähler differential module of that ring map (Universal Kähler differential module), which exists by Existence and generators of Kähler differentials. For $W^{'} \subseteq W$ the restriction $P (W) \to P (W^{'})$ is the unique $O_{X} (W)$ -linear map induced, via the universal property of $P (W)$ , by the derivation $O_{X} (W) \to O_{X} (W^{'}) \to P (W^{'})$ , where $P (W^{'})$ is viewed as an $O_{X} (W)$ -module by restriction of scalars; the restriction maps compose, so $P$ is a presheaf of $O_{X}$ -modules. Define

$Ω_{X / S} : = a P,$

the sheafification of $P$ (Sheafification of a presheaf); this is a sheaf of $O_{X}$ -modules by defining scalar multiplication on local representatives in the double-plus construction, with equality on germs making the operation well defined. The universal derivations $d_{W} ⁣ : O_{X} (W) \to P (W)$ are compatible with the restriction maps by construction, so they define a morphism of presheaves and hence a morphism of sheaves

$d_{X / S} ⁣ : O_{X} ⟶ Ω_{X / S},$

the universal $S$ -derivation of $X$ over $S$ , and $d_{X / S}$ is an $S$ -derivation because each $d_{W}$ is an $A_{W}$ -derivation for $A_{W} = (f^{- 1} O_{S}) (W)$ and the maps $A_{W} \to O_{X} (W)$ are the structure maps $f_{W}^{♯}$ .

Local descriptions. Two descriptions are used constantly and are recorded here for orientation; both are proved from the universal property in Universal property of relative differential sheaves and Affine charts recover the algebraic module of differentials.

Functor of points form. For every $O_{X}$ -module $F$ there is a natural bijection ${Hom}_{O_{X}} (Ω_{X / S}, F) ≅ {Der}_{S} (O_{X}, F)$ , $g \mapsto g \circ d_{X / S}$ ; this is the universal property that characterizes $Ω_{X / S}$ .
Affine charts. If $U = Spec B \subseteq X$ is an affine open subscheme and $f (U) \subseteq V = Spec A$ for an affine open $V \subseteq S$ , then the map $B \to Ω_{X / S} (U)$ , $b \mapsto d_{X / S} (b)$ exhibits $Ω_{X / S} (U)$ as $Ω_{B / A}$ , compatibly with the universal derivations, and restriction to a basic open $D (g) \subseteq U$ corresponds to the localization $Ω_{B / A} \to Ω_{B_{g} / A}$ .

Affine module convention. The affine description (2) identifies $Ω_{X / S}$ on each affine chart with the sheaf attached to $Ω_{B / A}$ , with localization as restriction. This local description is the part used below; it needs no finiteness, flatness or separatedness hypothesis on $f$ and also applies to the identity $X \to X$ .

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

Universal property of relative differential sheaves

Statement

Let $f ⁣ : X \to S$ be a morphism of schemes and let $Ω_{X / S} = a P$ with universal derivation $d_{X / S} ⁣ : O_{X} \to Ω_{X / S}$ (Sheaf of relative Kähler differentials). For every $O_{X}$ -module $F$ , composition with $d_{X / S}$ is a bijection ${Hom}_{O_{X}} (Ω_{X / S}, F) ⟶ {Der}_{S} (O_{X}, F), α ⟼ α \circ d_{X / S},$ natural in $F$ . No quasi-coherence or finiteness is assumed on $F$ , and no condition is imposed on $f$ ; the sheaf $Ω_{X / S}$ is determined up to unique compatible isomorphism by this property.

Facts & Assumptions

Given: A morphism of schemes $f ⁣ : X \to S$ , the presheaf $P (W) = Ω_{O_{X} (W) / (f^{- 1} O_{S}) (W)}$ of Sheaf of relative Kähler differentials, and an $O_{X}$ -module $F$ .

[F1]

Sheaf of relative Kähler differentials: $Ω_{X / S} = a P$ , the universal derivations $d_{W}$ assemble to $d_{X / S}$ , and an $S$ -derivation $O_{X} \to F$ is a morphism of sheaves of abelian groups that is additive, satisfies the Leibniz rule on sections over every open $W$ , and kills the image of $f^{♯} ⁣ : f^{- 1} O_{S} \to O_{X}$ .

[F2]

Sheafification is left adjoint to the inclusion of sheaves into presheaves: every morphism of presheaves $φ ⁣ : P \to F$ with $F$ a sheaf factors uniquely through the canonical map $P \to a P$ .

[F3]

Derivations are maps out of Ω: for a ring map $R \to T$ and every $T$ -module $N$ , composition with the universal derivation is an isomorphism ${Hom}_{T} (Ω_{T / R}, N) ≅ {Der}_{R} (T, N)$ .

[F4]

Modules on a ringed space: an $O_{X}$ -module has section groups that are modules over the section rings, restriction is linear after restricting scalars, and morphisms are linear on every open set.

[F5]

Derivation of an algebra: an $R$ -derivation $T \to N$ is additive, kills the image of $R$ and satisfies the Leibniz rule; sums and scalar multiples of derivations are derivations.

Proof

technique · direct

1.1

Sheafification adjunction. Restriction along $P \to a P = Ω_{X / S}$ gives a bijection ${Hom}_{O_{X}} (Ω_{X / S}, F) ≅ {Hom}_{O_{X} -presheaf} (P, F)$ : [F2] gives the factorization on underlying presheaves, and the factor is $O_{X}$ -linear because sections of $a P$ are locally represented by sections of $P$ , with scalar multiplication defined on those representatives by [F1]. Linearity therefore holds locally and hence globally. A morphism of presheaves $P \to F$ is exactly a compatible family of $O_{X} (W)$ -linear maps $φ_{W} ⁣ : P (W) \to F (W)$ .

F1F2F4

1.2

Algebraic universal property on each open. Since $O_{X} (W)$ is an $(f^{- 1} O_{S}) (W)$ -algebra, [F3] turns $φ_{W}$ into the derivation $D_{W} : = φ_{W} \circ d_{W} ⁣ : O_{X} (W) \to F (W)$ , an $(f^{- 1} O_{S}) (W)$ -derivation by [F5]; conversely every such derivation arises from exactly one $φ_{W}$ . Compatibility of the family $(φ_{W})$ under restriction is equivalent to compatibility of the family $(D_{W})$ , because the restriction maps of $Ω_{X / S}$ are defined so that $d_{W^{'}} \circ ρ = ρ \circ d_{W}$ for $W^{'} \subseteq W$ .

F1F3F5

2.1

Compatible families of derivations are $S$ -derivations. The families $(D_{W})$ in step 1.2 are in canonical bijection with morphisms of sheaves of abelian groups $D ⁣ : O_{X} \to F$ that are additive and satisfy Leibniz on every open and kill the image of each $(f^{- 1} O_{S}) (W)$ ; by gluing, the last condition is exactly $D \circ f^{♯} = 0$ , so these are precisely the $S$ -derivations of [F1]. The two passes are inverse because a derivation determines its components $D_{W}$ , and $φ_{W}$ is recovered from $D_{W}$ by the universal property of $P (W)$ .

F1step 1.2

3.1

Conclusion. Composing the bijections of steps 1.1, 1.2 and 2.1 gives the displayed bijection $α \mapsto α \circ d_{X / S}$ , since $α$ corresponds to the composite of its components with $d_{W}$ and $d_{X / S}$ is assembled from the $d_{W}$ by [F1]. Each step is natural in $F$ : a morphism $F \to G$ of $O_{X}$ -modules composes with $φ_{W}$ and with $D_{W}$ , so the bijection is compatible with postcomposition, and by the Yoneda lemma $Ω_{X / S}$ is determined up to unique compatible isomorphism.

F1step 1.1step 1.2step 2.1∎

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

The sheaf attached to a module on an affine scheme

Statement

Let $B$ be a commutative ring, let $M$ be a $B$ -module and let $X = Spec B$ with structure sheaf $O_{X}$ (The underlying space of an affine spectrum). Let $P_{M}$ be the presheaf of abelian groups $P_{M} (U) : = M \otimes_{B} O_{X} (U),$ with restrictions ${i d}_{M} \otimes ρ_{V \subseteq U}$ induced by those of $O_{X}$ ; it is a presheaf of $O_{X}$ -modules. Its sheafification $\tilde{M} : = a P_{M}$ is a sheaf of $O_{X}$ -modules, the sheaf attached to $M$ , and for every $O_{X}$ -module $F$ the map ${Hom}_{O_{X}} (\tilde{M}, F) ⟶ {Hom}_{B} (M, F (X)), φ ⟼ φ_{X} \circ ε,$ where $ε : = η_{P_{M}, X} \circ ι_{M}$ and $ι_{M} ⁣ : M \overset{≅}{\to} P_{M} (X)$ is the canonical identification from $B ≅ O_{X} (X)$ , is a bijection, natural in $M$ and in $F$ . Its inverse sends a $B$ -linear $g ⁣ : M \to F (X)$ to the unique morphism whose component over $U$ , after precomposition with $P_{M} (U) \to \tilde{M} (U)$ , is $M \otimes_{B} O_{X} (U) ⟶ F (U), m \otimes a ⟼ a \cdot g (m) ∣_{U} .$ A $B$ -linear map $M \to N$ induces a morphism $\tilde{M} \to \tilde{N}$ , so $M \mapsto \tilde{M}$ is a functor; it is right exact, and $\tilde{B} ≅ O_{X}$ with $Γ (X, \tilde{B}) = B$ . No finiteness assumption is made on $M$ or on $B$ .

Facts & Assumptions

Given: A commutative ring $B$ , a $B$ -module $M$ , the affine scheme $X = Spec B$ and the presheaf $P_{M}$ .

[F1]

Sheafification of a presheaf: for a presheaf $F$ the sheafification $a F$ is a sheaf equipped with a morphism $η_{F} ⁣ : F \to a F$ ; it is the double plus construction using germ-compatible local presentations (The plus construction for a presheaf).

[F2]

Sheafification is left adjoint to the inclusion of sheaves into presheaves: for every morphism of presheaves $φ ⁣ : F \to G$ with $G$ a sheaf there is a unique morphism of sheaves $\overline{φ} ⁣ : a F \to G$ with $φ = \overline{φ} \circ η_{F}$ .

[F3]

Modules on a ringed space: an $O_{X}$ -module is a sheaf of abelian groups whose section groups are $O_{X} (U)$ -modules compatibly with restriction, and a morphism of $O_{X}$ -modules is $O_{X} (U)$ -linear on every open set.

[F4]

Universal property of the tensor product for balanced maps into abelian groups: for a $B$ -bilinear map $β ⁣ : M \times N \to P$ into a $B$ -module there is a unique $B$ -linear $\overline{β} ⁣ : M \otimes_{B} N \to P$ with $\overline{β} (m \otimes n) = β (m, n)$ .

[F5]

Global functions on Spec A recover A: the canonical map $B \to Γ (X, O_{X})$ is an isomorphism.

[F6]

Tensoring is right exact: tensoring an exact sequence of $B$ -modules with any $B$ -module preserves its cokernel and surjectivity.

Proof

technique · direct

1.1

$P_{M}$ is a presheaf of $O_{X}$ -modules. The restriction $M \otimes_{B} O_{X} (U) \to M \otimes_{B} O_{X} (V)$ is ${i d}_{M} \otimes ρ$ for $V \subseteq U$ , it is additive and functorial, and $a \cdot (m \otimes a^{'}) = m \otimes a a^{'}$ shows that it is $O_{X} (U)$ -linear after restricting scalars along $O_{X} (U) \to O_{X} (V)$ ; the module structure on $a P_{M}$ is constructed as follows. For a presheaf of $O_{X}$ -modules $P$ , a scalar $a \in O_{X} (U)$ acts on a plus-section represented by $(U_{i}, s_{i})$ by $(U_{i}, a ∣_{U_{i}} s_{i})$ . This is independent of the representative because equality of germs is preserved by multiplication. Addition is defined on the common refinement of two covers. All module laws and compatibility with restriction follow on these local representatives from the corresponding laws in $P$ . Apply this construction twice to obtain the module structure on $a P_{M}$ ; its unit map is linear. Moreover every section of $a P$ is locally represented by a section of $P$ , by refining twice the presentations in [F1].

F1F3

1.2

Morphisms of presheaves of $O_{X}$ -modules $φ ⁣ : P_{M} \to F$ with $F$ a sheaf are the compatible families of $O_{X} (U)$ -linear maps $φ_{U} ⁣ : M \otimes_{B} O_{X} (U) \to F (U)$ , and these are in canonical bijection with $B$ -linear maps $g ⁣ : M \to F (X)$ : the map $g$ is recovered as $φ_{X} (- \otimes 1)$ , while for a given $g$ the formulas $ψ_{U} (m \otimes a) : = a \cdot g (m) ∣_{U}$ define a family that is well defined and $B$ -bilinear in $(m, a)$ , hence $O_{X} (U)$ -linear by [F4], and compatible with restrictions by the compatibility of the restrictions of $F$ . The two assignments are inverse because the values on the elements $m \otimes 1$ determine an $O_{X} (U)$ -linear map on all of $M \otimes_{B} O_{X} (U)$ .

F3F4

2.1

By [F2] a linear presheaf map $P_{M} \to F$ extends uniquely as a morphism of sheaves. This extension is linear: locally write a section as $η (s)$ using step 1.1, and then $\overline{φ} (a η (s)) = \overline{φ} (η (a s)) = φ (a s) = a φ (s)$ ; additivity is checked on a common local presentation in the same way. Equality of sheaf sections is local. Conversely precomposition of a linear sheaf map with the linear unit is linear. Thus [F2] restricts to the module morphisms, and step 1.2 gives a bijection ${Hom}_{O_{X}} (\tilde{M}, F) ≅ {Hom}_{B} (M, F (X))$ ; under it, $φ$ corresponds to $φ_{X} \circ η_{P_{M}, X} \circ ι_{M} = φ_{X} \circ ε$ , using the identification $ι_{M}$ from [F5]. The displayed formula is the component of the presheaf map $P_{M} \to F$ , hence the composite of the induced sheaf morphism with $P_{M} (U) \to \tilde{M} (U)$ . Naturality in $M$ and in $F$ is immediate from the formula $m \otimes a \mapsto a \cdot g (m) ∣_{U}$ , and a $B$ -linear $u ⁣ : M \to N$ induces $u \otimes i d ⁣ : P_{M} \to P_{N}$ and hence a morphism $\tilde{M} \to \tilde{N}$ .

F2F5step 1.1step 1.2

3.1

For $M = B$ one has $P_{B} (U) = B \otimes_{B} O_{X} (U) ≅ O_{X} (U)$ , so $\tilde{B} ≅ a O_{X} = O_{X}$ because $O_{X}$ is already a sheaf, and $Γ (X, \tilde{B}) ≅ B$ by [F5]. For an exact sequence $M^{'} \to M \to M^{''} \to 0$ , [F6] makes the corresponding sequence of presheaves $P_{M^{'}} \to P_{M} \to P_{M^{''}} \to 0$ objectwise right exact; sheafification, as the left adjoint supplied by [F2], preserves its cokernel and gives ${\tilde{M}}^{'} \to \tilde{M} \to {\tilde{M}}^{''} \to 0$ exact.

F2F5F6step 1.1∎

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

Affine charts recover the algebraic module of differentials

Statement

Let $A \to B$ be a homomorphism of commutative rings, let $f ⁣ : X = Spec B \to S = Spec A$ be the induced morphism of affine schemes, and let $\tilde{Ω}$ be the sheaf attached to the $B$ -module $Ω_{B / A}$ (The sheaf attached to a module on an affine scheme). Then there is a unique isomorphism of $O_{X}$ -modules $\tilde{Ω} ⟶ Ω_{X / S}, ε (d b) ⟼ d_{X / S} (b),$ and it is natural in the ring map $A \to B$ , in particular compatible with restriction to a further affine open. Consequently, for every $g \in B$ , $Γ (X, Ω_{X / S}) ≅ Ω_{B / A}, Ω_{X / S} (D (g)) ≅ Ω_{B_{g} / A},$ compatibly with $d_{X / S}$ and with the localization maps $Ω_{B / A} \to Ω_{B_{g} / A}$ ; for $g = 1$ the two displays agree. No finiteness, flatness or separatedness hypothesis is imposed on $A \to B$ .

Facts & Assumptions

Given: A ring map $A \to B$ , the induced morphism $X = Spec B \to S = Spec A$ and an $O_{X}$ -module $F$ .

[F1]

The sheaf attached to a module on an affine scheme: the sheaf $\tilde{M}$ attached to a $B$ -module $M$ satisfies ${Hom}_{O_{X}} (\tilde{M}, F) ≅ {Hom}_{B} (M, F (X))$ naturally in $M$ and $F$ , the bijection being $φ \mapsto φ_{X} \circ ε$ .

[F2]

Universal property of relative differential sheaves: ${Hom}_{O_{X}} (Ω_{X / S}, F) ≅ {Der}_{S} (O_{X}, F)$ via $α \mapsto α \circ d_{X / S}$ , naturally in $F$ , for every $O_{X}$ -module $F$ .

[F3]

Derivations are maps out of Ω: for a ring map $R \to T$ and a $T$ -module $N$ , ${Hom}_{T} (Ω_{T / R}, N) ≅ {Der}_{R} (T, N)$ via precomposition with the universal derivation.

[F4]

Kähler differentials commute with localization: for a multiplicative subset $U \subseteq B$ the canonical map $U^{- 1} Ω_{B / A} \to Ω_{U^{- 1} B / A}$ is an isomorphism of $U^{- 1} B$ -modules; for $U = {1, g, g^{2}, \dots}$ this reads $Ω_{B / A} \otimes_{B} B_{g} ≅ Ω_{B_{g} / A}$ , compatible with the universal derivations.

[F5]

Global functions on Spec A recover A: the canonical map $B \to Γ (X, O_{X})$ is an isomorphism, so $O_{X} (X) = B$ and global sections of any $O_{X}$ -module are a $B$ -module.

[F6]

Sections and restrictions on distinguished opens of an affine scheme: $Γ (D (g), O_{X}) = B_{g}$ , and for $D (h) \subseteq D (g)$ the restriction is the canonical localization map $B_{g} \to B_{h}$ .

[F7]

Sheaf of relative Kähler differentials: an $S$ -derivation $O_{X} \to F$ kills the image of $f^{♯} ⁣ : f^{- 1} O_{S} \to O_{X}$ ; in particular it kills the image of $A \to (f^{- 1} O_{S}) (X) \to O_{X} (X) = B$ under the structure map.

Proof

technique · direct

1.1

Restriction of derivations. Let $D ⁣ : O_{X} \to F$ be an $S$ -derivation. Its global component $D_{X} ⁣ : B \to F (X)$ is additive and satisfies Leibniz, and it kills the image of $A$ , because $A$ maps into $O_{X} (X) = B$ through $(f^{- 1} O_{S}) (X)$ and $D$ kills that image by [F7]. So $D \mapsto D_{X}$ is a map ${Der}_{S} (O_{X}, F) \to {Der}_{A} (B, F (X))$ .

F5F7

1.2

Localizing a derivation of global sections. Conversely let $D_{0} ⁣ : B \to F (X)$ be an $A$ -derivation. For every $g$ the composite $B \overset{D_{0}}{\to} F (X) \to F (D (g))$ is an $A$ -derivation, so by [F3] it corresponds to a $B$ -linear map $Ω_{B / A} \to F (D (g))$ , which by [F4] is the same as a $B_{g}$ -linear map $ε_{g} ⁣ : Ω_{B_{g} / A} \to F (D (g))$ ; put $D_{g} : = ε_{g} \circ d_{B_{g} / A} ⁣ : B_{g} \to F (D (g))$ . These maps are compatible with restriction to a smaller basic open, since both restrictions are induced by the same $A$ -derivation composite $B \to F (D (g h))$ and [F4] is compatible with the universal derivations.

F3F4

2.1

Gluing. For an open $W \subseteq X$ and $a \in O_{X} (W)$ , the elements $D_{g} (a ∣_{D (g)})$ , indexed by basic opens $D (g) \subseteq W$ , are compatible on intersections $D (g h)$ by step 1.2, so they glue to a unique element $D_{W} (a) \in F (W)$ . The resulting $D_{W}$ are additive and satisfy Leibniz because this can be checked on a basic-open cover. They kill $f^{- 1} O_{S}$ locally: a germ in the image of $f^{- 1} O_{S}$ at $x \in D (g)$ comes from a section of $O_{S}$ on an open neighbourhood of $f (x)$ ; after shrinking to an affine neighbourhood of $f (x)$ and then to a basic open around $x$ , that section is a fraction of elements of $A$ . The derivation $D_{g}$ kills $A$ , and the Leibniz rule applied to an inverse shows it kills such fractions. Vanishing at every stalk implies the sheaf composite $f^{- 1} O_{S} \to F$ is zero.

F4step 1.2

3.1

The two constructions are inverse. If $D$ is an $S$ -derivation with global component $D_{0} = D_{X}$ , then for each $g$ the map $ε_{g}$ of step 1.2 is the composite $Ω_{B / A} \to Ω_{B_{g} / A} \to F (D (g))$ induced by $D_{0}$ and restriction, so $D_{g}$ agrees with $D$ on $B_{g}$ ; by the sheaf property, the derivation produced in step 2.1 equals $D$ . Conversely the derivation produced from $D_{0}$ has global component $D_{0}$ , since its component on $D (g)$ restricts from $D_{0}$ . Hence restriction of global sections is a bijection ${Der}_{S} (O_{X}, F) ≅ {Der}_{A} (B, F (X)),$ natural in $F$ .

step 1.1step 1.2step 2.1

4.1

The comparison isomorphism. By [F1] with $M = Ω_{B / A}$ and [F3], ${Hom}_{O_{X}} (\tilde{Ω}, F) ≅ {Hom}_{B} (Ω_{B / A}, F (X)) ≅ {Der}_{A} (B, F (X))$ , and by step 3.1 and [F2] the last group is ${Hom}_{O_{X}} (Ω_{X / S}, F)$ . All identifications are natural in $F$ , so the Yoneda lemma produces a unique isomorphism $\tilde{Ω} \to Ω_{X / S}$ ; tracking the universal elements (the identity of $\tilde{Ω}$ corresponds to the derivation $b \mapsto ε (d b)$ and the identity of $Ω_{X / S}$ to $d_{X / S}$ ) shows that the isomorphism sends $ε (d b)$ to $d_{X / S} (b)$ . Naturality in the ring map $A \to B$ follows from the functoriality of [F1] in $M$ and of [F3].

F1F2F3step 3.1

5.1

Sections over affine and basic opens. The sheaf attached to $Ω_{B / A}$ is computed from its values on the distinguished-open basis: the assignment $D (g) \mapsto Ω_{B / A} \otimes_{B} B_{g} = Ω_{B_{g} / A}$ with the localization maps as restrictions is a sheaf on the basis (the localization exactness makes fractions glue; see Localisation of a module at a multiplicative subset and [F4]) and extends to the sheaf $\tilde{Ω}$ with those values and restrictions, exactly as Sections and restrictions on distinguished opens of an affine scheme records for $O_{X}$ itself. Hence $Γ (X, \tilde{Ω}) = Ω_{B / A}$ and $Ω_{X / S} (D (g)) ≅ Ω_{B_{g} / A}$ under step 4.1, compatible with $d_{X / S}$ by the characterization of that isomorphism and with the localization maps because those are the restriction maps of $\tilde{Ω}$ .

F4F6step 4.1∎

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

Conormal sequence for a closed immersion

Statement

Let $S$ be a scheme, let $Y$ be an $S$ -scheme and let $i ⁣ : X \to Y$ be a morphism of $S$ -schemes which is a closed immersion (Closed immersions of schemes, Schemes and morphisms over a base). Let

$I : = \ker (O_{Y} ⟶ i_{*} O_{X})$

be its ideal sheaf (Ideal sheaves), let $I^{2} \subseteq O_{Y}$ be the image of the multiplication map $I \otimes_{O_{Y}} I \to O_{Y}$ (Tensor product of sheaves of modules, Kernel sheaves are objectwise, while cokernels and images are sheafified), and put $Q : = coker (I^{2} ↪ I)$ as a sheaf on $Y$ . In the sequence below the notation $I / I^{2}$ means $i^{- 1} Q$ , a sheaf on $X$ (Inverse image presheaf and inverse image sheaf). The ideal $i^{- 1} I$ annihilates it, so its $i^{- 1} O_{Y}$ -action factors through $i^{- 1} O_{Y} / i^{- 1} I ≅ O_{X}$ . This last identification follows on stalks from the closed immersion: $O_{X, x} = O_{Y, i (x)} / I_{i (x)}$ . Then the sequence of $O_{X}$ -modules

$I / I^{2} \overset{α}{\to} i^{*} Ω_{Y / S} \overset{β}{\to} Ω_{X / S} ⟶ 0$

is exact, where $α$ sends the class of a local section $t$ of $I$ to $1 \otimes d_{Y / S} (t)$ , and $β$ is the pullback of the universal $S$ -derivation, characterised by $β (1 \otimes d_{Y / S} (g)) = d_{X / S} (g \circ i)$ for local sections $g$ of $O_{Y}$ . The map $α$ is not asserted to be injective, and it is not injective in general; no finiteness, flatness or separatedness hypothesis is imposed on $i$ or on the structure morphisms.

Facts & Assumptions

Given: A scheme $S$ , an $S$ -scheme $Y$ , and a closed immersion of $S$ -schemes $i ⁣ : X \to Y$ .

[F1]

Closed immersions of schemes: a morphism $i ⁣ : X \to Y$ is a closed immersion if its underlying map is a homeomorphism onto a closed subset and $O_{Y} \to i_{*} O_{X}$ is surjective.

[F2]

Pullback of a module along a morphism of ringed spaces: the pullback of an $O_{Y}$ -module $G$ is $i^{*} G = O_{X} \otimes_{i^{- 1} O_{Y}} i^{- 1} G$ ; the canonical map $i^{- 1} G \to i^{*} G$ , $s \mapsto 1 \otimes s$ , is $i^{- 1} O_{Y}$ -linear, and $O_{X}$ is an $i^{- 1} O_{Y}$ -algebra, so a local section $g$ of $O_{Y}$ acts on $i^{*} G$ as $g \circ i$ .

[F3]

Quasi-coherent ideals and closed subschemes: for a closed immersion $i ⁣ : Z ↪ X$ and an affine open $U = Spec A \subseteq X$ , the kernel of $O_{U} \to i_{*} O_{Z \cap U}$ is the ideal sheaf associated to an ideal $I_{U} \subseteq A$ .

[F4]

Affine charts recover the algebraic module of differentials: for a ring map $A \to B$ with induced morphism $Spec B \to Spec A$ , the sheaf $Ω_{X / S}$ has $Γ (Spec B, Ω_{X / S}) = Ω_{B / A}$ compatibly with $d_{X / S}$ , and restriction to a basic open $D (g)$ corresponds to the localization $Ω_{B / A} \to Ω_{B_{g} / A}$ .

[F5]

Conormal exact sequence for an algebra quotient: for a ring map $A \to P$ , an ideal $I \subseteq P$ and $B = P / I$ , the sequence $I / I^{2} \to B \otimes_{P} Ω_{P / A} \to Ω_{B / A} \to 0$ is exact, the first map sending the class of $t$ to $1 \otimes d t$ and the second sending $1 \otimes d p$ to $d (p + I)$ .

[F6]

A sequence of abelian sheaves is exact exactly when it is exact on every stalk: a sequence of sheaves of abelian groups is exact if and only if all its stalk sequences are exact.

[F7]

Universal property of relative differential sheaves: for every $O_{X}$ -module $F$ , composition with $d_{X / S}$ is a natural bijection ${Hom}_{O_{X}} (Ω_{X / S}, F) ≅ {Der}_{S} (O_{X}, F)$ .

[F8]

Pullback of modules is left adjoint to pushforward: for a morphism of ringed spaces $f$ there is a natural bijection ${Hom}_{O_{X}} (f^{*} G, F) ≅ {Hom}_{O_{Y}} (G, f_{*} F)$ ; the map corresponding to $u ⁣ : G \to f_{*} F$ sends $1 \otimes s$ to the germ $u (s)$ .

[F9]

Localisation of modules is exact: localization of modules at a prime is exact, so an exact sequence of $B$ -modules remains exact after applying $- \otimes_{B} B_{p}$ .

[F10]

Polynomial differentials are free: for $P = A [x_{1}, \dots, x_{n}]$ the module $Ω_{P / A}$ is free on $d x_{1}, \dots, d x_{n}$ .

[F11]

Kernel sheaves are objectwise, while cokernels and images are sheafified and The stalk of a presheaf at a point: images and cokernels of morphisms of sheaves are computed by sheafifying the objectwise constructions, and the stalk at a point is the filtered colimit of the sections over the open neighbourhoods of that point.

Proof

technique · direct

1.1

The map $α$ . For a local section $t$ of $I$ over an open $V \subseteq Y$ let $α (t) \in (i^{*} Ω_{Y / S}) (i^{- 1} V)$ be the image of $d_{Y / S} (t)$ under the canonical map $i^{- 1} Ω_{Y / S} \to i^{*} Ω_{Y / S}$ of [F2], i.e. $1 \otimes d_{Y / S} (t)$ . For a local section $g$ of $O_{Y}$ over $V$ one has $d (g t) = g d t + t d g$ , hence $1 \otimes d (g t) = (g \circ i) (1 \otimes d t) + (t \circ i) (1 \otimes d g) = (g \circ i) (1 \otimes d t)$ because $t$ lies in the kernel of $O_{Y} \to i_{*} O_{X}$ , so that $t \circ i = 0$ ; thus these formulas define an $O_{Y}$ -linear map $I \to i_{*} i^{*} Ω_{Y / S}$ . For local sections $t, t^{'}$ of $I$ one has $1 \otimes d (t t^{'}) = (t \circ i) (1 \otimes d t^{'}) + (t^{'} \circ i) (1 \otimes d t) = 0$ , so $α$ kills $I^{2}$ , and since $I$ annihilates both $I / I^{2}$ and the pullback (a local section $t$ of $I$ acts on $i^{*} Ω_{Y / S}$ as $t \circ i = 0$ ), the descended formulas on germs define a map $i^{- 1} Q \to i^{*} Ω_{Y / S}$ . It is linear over $i^{- 1} O_{Y}$ and hence over its quotient $O_{X}$ , so this is the required $O_{X}$ -linear $α$ .

F1F2given

1.2

The map $β$ . Since $Y$ and $X$ are $S$ -schemes and $i$ is an $S$ -morphism, the composite $D ⁣ : O_{Y} \to i_{*} O_{X} \to i_{*} Ω_{X / S}$ of the structure map $i^{♯}$ with $i_{*} d_{X / S}$ is additive, satisfies Leibniz for the $O_{Y}$ -module structure of $i_{*} Ω_{X / S}$ transported along $i^{♯}$ , and kills the image of $O_{S}$ : the image of $g^{- 1} O_{S} \to O_{Y} \to i_{*} O_{X}$ is the image of $f^{- 1} O_{S} \to O_{X}$ for the structure morphism $f ⁣ : X \to S$ , which $d_{X / S}$ annihilates. Hence $D$ is an $S$ -derivation of $O_{Y}$ into $i_{*} Ω_{X / S}$ , and [F7] applied to the $S$ -scheme $Y$ gives a unique $O_{Y}$ -linear map $u ⁣ : Ω_{Y / S} \to i_{*} Ω_{X / S}$ with $u (d_{Y / S} (g)) = d_{X / S} (g \circ i)$ . Let $β ⁣ : i^{*} Ω_{Y / S} \to Ω_{X / S}$ be the $O_{X}$ -linear map corresponding to $u$ under the adjunction [F8]; it satisfies $β (1 \otimes d_{Y / S} (g)) = d_{X / S} (g \circ i)$ by the description of the correspondence, and it is unique with this property.

F7F8given

2.1

The composite vanishes. For a local section $t$ of $I$ one has $β (α (t)) = β (1 \otimes d t) = d_{X / S} (t \circ i) = d_{X / S} (0) = 0$ by the characterisations of steps 1.1 and 1.2, so the image of $α$ is contained in the kernel of $β$ .

step 1.1step 1.2

2.2

Affine charts. Let $V = Spec P \subseteq Y$ be an affine open with image in an affine open $W = Spec A \subseteq S$ , and take $U = i^{- 1} (V)$ . Since $i$ is a closed immersion, [F3] identifies $U$ with $Spec B$ for $B = P / I$ , where $I = I (V)$ , and $U \to V$ with the quotient morphism. By [F4], $Ω_{X / S} ∣_{U}$ and $Ω_{Y / S} ∣_{V}$ are attached to $Ω_{B / A}$ and $Ω_{P / A}$ . The ideal sheaf $I ∣_{V}$ is attached to $I$ by [F3]; on every principal open $D (h) \subseteq V$ , multiplication has image $(I^{2})_{h}$ , so $I^{2} ∣_{V}$ is attached to $I^{2}$ and $I / I^{2}$ on $U$ is attached to $I / I^{2}$ . For the pullback, let $x \in U$ correspond to $p \subseteq B$ and $i (x)$ to $q \subseteq P$ . By [F2] and the stalk construction [F11], $(i^{*} Ω_{Y / S})_{x} = Ω_{Y / S, i (x)} \otimes_{O_{Y, i (x)}} O_{X, x} ≅ (Ω_{P / A})_{q} \otimes_{P_{q}} B_{p} ≅ (B \otimes_{P} Ω_{P / A})_{p}$ . The last equality follows by localising the tensor product; it does not identify $i^{- 1} O_{V}$ with $O_{U}$ . Thus the pullback is the sheaf attached to $B \otimes_{P} Ω_{P / A}$ . On these stalks the maps of steps 1.1 and 1.2 agree with [F5]: $α$ sends $[t]$ to $1 \otimes d t$ and $β$ sends $1 \otimes d g$ to $d (g + I)$ .

F2F3F4F5F11step 1.1step 1.2

3.1

Surjectivity of $β$ . By [F6] surjectivity of a morphism of sheaves may be checked on stalks. Every point of $X$ lies in a chart as in step 2.2, and on that chart $β$ becomes the second map of the exact sequence [F5], which is surjective; forming the stalk at a point of the chart is a filtered colimit of localizations and preserves surjectivity. Hence $β$ is surjective.

F5F6step 2.2

3.2

Exactness at the middle term. Let $x \in X$ , choose a chart $U = Spec B$ and $V = Spec P$ as in step 2.2 with $x$ corresponding to a prime $p \subseteq B$ and $i (x)$ to $p \cap P \supseteq I$ . By step 2.2 the stalks of the three sheaves at $x$ are $(I / I^{2}) \otimes_{B} B_{p}$ , $(B \otimes_{P} Ω_{P / A}) \otimes_{B} B_{p}$ and $Ω_{B / A} \otimes_{B} B_{p}$ , and the stalk maps are the localizations of the maps of [F5] at $p$ . Applying $- \otimes_{B} B_{p}$ to the exact sequence [F5] and using [F9], the stalk sequence is exact at the middle term, so $im α_{x} = \ker β_{x}$ . Since $x$ was arbitrary, [F6] gives $im α = \ker β$ as subsheaves of $i^{*} Ω_{Y / S}$ .

F5F6F9step 2.2

3.3

Failure of injectivity. Take $A = k$ a field, $P = k [x]$ , $I = (x^{2})$ and $B = k [x] / (x^{2})$ , so that $I / I^{2} = (x^{2}) / (x^{4})$ , in which the class of $x^{3}$ is nonzero. By [F10] we have $B \otimes_{P} Ω_{P / A} = B d x$ with $d x$ a free generator, and $α ([x^{3}]) = 1 \otimes d (x^{3}) = 3 x^{2} (1 \otimes d x) = 0$ , because $x^{2} = 0$ in $B$ : the class of $x^{3}$ lies in the kernel of $α$ and is nonzero. Hence the left map of the conormal sequence is not injective in general, and in particular no injectivity is claimed.

F5F10step 2.2

4.1

Conclusion. Steps 1.1 and 1.2 construct $O_{X}$ -linear maps $α$ and $β$ with the asserted descriptions, step 2.1 shows $β \circ α = 0$ , step 3.1 shows that $β$ is surjective and step 3.2 that its kernel is exactly the image of $α$ ; step 3.3 exhibits a case where $α$ has nonzero kernel. Hence the displayed sequence of $O_{X}$ -modules is exact and its left map is not generally injective, with no finiteness, flatness or separatedness hypothesis used anywhere.

step 2.1step 3.1step 3.2step 3.3∎

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

Transitivity sequence for schemes

Statement

Let $X \overset{f}{\to} Y \overset{h}{\to} S$ be morphisms of schemes. Then the sequence of $O_{X}$ -modules

$f^{*} Ω_{Y / S} \overset{γ}{\to} Ω_{X / S} \overset{δ}{\to} Ω_{X / Y} ⟶ 0$

is exact, where $γ$ is characterised by $γ (1 \otimes d_{Y / S} (g)) = d_{X / S} (g \circ f)$ for local sections $g$ of $O_{Y}$ , and $δ$ is characterised by $δ (d_{X / S} (c)) = d_{X / Y} (c)$ for local sections $c$ of $O_{X}$ . The maps $γ$ and $δ$ are natural in the morphisms $f$ and $h$ , and the first arrow is not asserted to be injective: it fails to be injective in general. No finiteness, flatness or separatedness hypothesis is imposed.

Facts & Assumptions

Given: Morphisms of schemes $f ⁣ : X \to Y$ and $h ⁣ : Y \to S$ .

[F1]

Sheaf of relative Kähler differentials: for a morphism $Z \to T$ of schemes there is an $O_{Z}$ -module $Ω_{Z / T}$ with universal $T$ -derivation $d_{Z / T} ⁣ : O_{Z} \to Ω_{Z / T}$ , and an $S$ -derivation of $O_{X}$ kills the image of the structure map from $O_{S}$ .

[F2]

Universal property of relative differential sheaves: for every $O_{X}$ -module $F$ , composition with $d_{X / S}$ is a natural bijection ${Hom}_{O_{X}} (Ω_{X / S}, F) ≅ {Der}_{S} (O_{X}, F)$ , and likewise over $Y$ .

[F3]

Pullback of a module along a morphism of ringed spaces: $f^{*} G = O_{X} \otimes_{f^{- 1} O_{Y}} f^{- 1} G$ , the canonical map $f^{- 1} G \to f^{*} G$ , $s \mapsto 1 \otimes s$ , is $f^{- 1} O_{Y}$ -linear, and a local section $g$ of $O_{Y}$ acts on $f^{*} G$ as $g \circ f$ .

[F4]

Pullback of modules is left adjoint to pushforward: there is a natural bijection ${Hom}_{O_{X}} (f^{*} G, F) ≅ {Hom}_{O_{Y}} (G, f_{*} F)$ ; the map corresponding to $u ⁣ : G \to f_{*} F$ sends $1 \otimes s$ to $u (s)$ .

[F5]

Transitivity sequence for differential modules: for ring maps $A \to B \to C$ the sequence $C \otimes_{B} Ω_{B / A} \to Ω_{C / A} \to Ω_{C / B} \to 0$ is exact, with the first map $c \otimes d b \mapsto c d_{C / A} (b)$ and the second induced by $d_{C / B}$ .

[F6]

Affine charts recover the algebraic module of differentials: for a ring map $A \to B$ with induced morphism $Spec B \to Spec A$ , the sections of $Ω_{B / A}$ over a basic open are $Ω_{B_{g} / A}$ , compatibly with the universal derivations and with localization.

[F7]

A sequence of abelian sheaves is exact exactly when it is exact on every stalk: a sequence of sheaves of abelian groups is exact if and only if every stalk sequence is exact.

[F8]

Localisation of modules is exact: localization at a prime is exact.

[F9]

The stalk of a presheaf at a point: the stalk is the filtered colimit of the sections over a basis of neighbourhoods.

[F10]

Polynomial differentials are free and Jacobian presentation of Ω: $Ω_{k [x] / k}$ is free on $d x$ , and for $B = P / I$ the module $Ω_{B / k}$ is presented as the cokernel of the Jacobian map on $I / I^{2}$ .

Proof

technique · direct

1.1

The map $γ$ . The composite $D ⁣ : O_{Y} \to f_{*} O_{X} \overset{f_{*} d_{X / S}}{\to} f_{*} Ω_{X / S}$ is an $S$ -derivation: it is additive, satisfies Leibniz for the $O_{Y}$ -module structure of $f_{*} Ω_{X / S}$ transported along $f^{♯}$ , and kills the image of $O_{S}$ because [F1] applied to $X \to S$ says that $d_{X / S}$ annihilates it. By [F2] applied to the $S$ -scheme $Y$ there is a unique $O_{Y}$ -linear $u ⁣ : Ω_{Y / S} \to f_{*} Ω_{X / S}$ with $u (d_{Y / S} (g)) = d_{X / S} (g \circ f)$ , and by [F4] there is a unique $O_{X}$ -linear $γ ⁣ : f^{*} Ω_{Y / S} \to Ω_{X / S}$ with $γ (1 \otimes d_{Y / S} (g)) = d_{X / S} (g \circ f)$ , where $f^{*} Ω_{Y / S} = O_{X} \otimes_{f^{- 1} O_{Y}} f^{- 1} Ω_{Y / S}$ is the pullback of [F3] and the elements $1 \otimes s$ generate it; the value on $1 \otimes s$ is $u (s)$ by the description of the adjunction.

F1F2F3F4

1.2

The map $δ$ . The universal $Y$ -derivation $d_{X / Y} ⁣ : O_{X} \to Ω_{X / Y}$ annihilates the image of $O_{Y}$ , hence also the image of $O_{S}$ under $O_{S} \to O_{Y} \to O_{X}$ ; so it is an $S$ -derivation, and [F2] over $S$ gives a unique $O_{X}$ -linear $δ ⁣ : Ω_{X / S} \to Ω_{X / Y}$ with $δ (d_{X / S} (c)) = d_{X / Y} (c)$ . It is surjective because the sections $d_{X / Y} (c)$ generate $Ω_{X / Y}$ over $O_{X}$ by [F1].

F1F2given

2.1

The composite vanishes. For a local section $g$ of $O_{Y}$ one has $δ (γ (1 \otimes d g)) = δ (d_{X / S} (g \circ f)) = d_{X / Y} (g \circ f) = 0$ , because $g \circ f$ is the image of a section of $O_{Y}$ ; hence $im γ \subseteq \ker δ$ .

step 1.1step 1.2

2.2

Affine charts. Let $V = Spec B \subseteq Y$ be an affine open whose image lies in $W = Spec A \subseteq S$ , and let $U = Spec C \subseteq X$ be an affine open with $f (U) \subseteq V$ . The structure maps give $A \to B \to C$ . By [F6], $Ω_{X / S} ∣_{U}$ and $Ω_{X / Y} ∣_{U}$ are attached to $Ω_{C / A}$ and $Ω_{C / B}$ . For $x \in U$ , let $p \subseteq C$ correspond to $x$ and $q = p \cap B$ to $f (x)$ . The pullback definition [F3] and stalk construction [F9] give $(f^{*} Ω_{Y / S})_{x} ≅ Ω_{Y / S, f (x)} \otimes_{O_{Y, f (x)}} O_{X, x} ≅ (Ω_{B / A})_{q} \otimes_{B_{q}} C_{p} ≅ (C \otimes_{B} Ω_{B / A})_{p}$ . Consequently $f^{*} Ω_{Y / S} ∣_{U}$ is the sheaf attached to $C \otimes_{B} Ω_{B / A}$ . These stalk identifications use tensor products after taking inverse-image stalks; no equality between $f^{- 1} O_{Y}$ and $O_{X}$ is needed. The maps $γ$ and $δ$ become the maps of [F5] because their values on $d b$ and $d_{C / A} (c)$ are those of steps 1.1 and 1.2.

F3F5F6F9step 1.1step 1.2

3.1

Exactness at $Ω_{X / S}$ . Let $x \in X$ and take a chart as in step 2.2 with $x$ corresponding to a prime $p \subseteq C$ . By step 2.2 the stalks of the three sheaves at $x$ are $(C \otimes_{B} Ω_{B / A}) \otimes_{C} C_{p}$ , $Ω_{C / A} \otimes_{C} C_{p}$ and $Ω_{C / B} \otimes_{C} C_{p}$ , and the stalk maps are the localizations at $p$ of the maps of [F5]. Applying $- \otimes_{C} C_{p}$ to the exact sequence [F5] and using [F8], the stalk sequence is exact at the middle term, so $im γ_{x} = \ker δ_{x}$ . As $x$ was arbitrary, [F7] gives $im γ = \ker δ$ and the sequence of the statement is exact at $Ω_{X / S}$ ; combined with step 1.2 and step 2.1 this is the asserted exactness.

F5F7F8step 1.2step 2.1step 2.2

3.2

Failure of injectivity of the first arrow. Let $k$ be a field of characteristic $\neq 2$ , let $A = k$ , $B = k [x]$ , $C = k [x] / (x^{2})$ , so that $Ω_{B / A}$ is free on $d x$ by [F10] and $Ω_{C / A}$ is the cokernel of $I / I^{2} \to C \otimes_{B} Ω_{B / A}$ for $I = (x^{2})$ . By [F10] the module $C \otimes_{B} Ω_{B / A} = C d x$ has the two $k$ -linearly independent elements $1 \otimes d x$ and $x \otimes d x$ , while $d (x^{2}) = 2 x d x$ shows that $x \otimes d x$ lies in the kernel of $C \otimes_{B} Ω_{B / A} \to Ω_{C / A}$ ; since $2 \neq 0$ in $k$ , the element $1 \otimes d x$ does not, so this map has a nonzero kernel and $γ$ is not injective in general.

F5F10step 2.2

4.1

Conclusion. Steps 1.1 and 1.2 construct $γ$ and $δ$ with the stated properties, step 2.1 shows that the composite vanishes, step 3.1 identifies the kernel of $δ$ with the image of $γ$ and makes $δ$ surjective by step 1.2, and step 3.2 shows that $γ$ need not be injective. Hence the displayed sequence is exact and the first arrow is not injective in general. Naturality in $f$ and $h$ follows because $γ$ and $δ$ are determined by the universal properties of [F2] and [F4] applied to the morphisms $f$ and $h$ , which are natural in those morphisms, and no finiteness, flatness or separatedness assumption was used.

step 1.1step 1.2step 2.1step 3.1step 3.2∎

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

Relative differentials commute with scheme base change

Statement

Let $X \to S$ be a morphism of schemes and let $S^{'} \to S$ be a morphism, with fibre product $X^{'} = X \times_{S} S^{'}, g ⁣ : X^{'} ⟶ X, X^{'} ⟶ S^{'} .$ Then the canonical map $g^{*} Ω_{X / S} ⟶ Ω_{X^{'} / S^{'}}, 1 \otimes d_{X / S} (a) ⟼ d_{X^{'} / S^{'}} (a \circ g),$ is an isomorphism of $O_{X^{'}}$ -modules. It is natural in the base-change data and compatible with the universal derivations of $X / S$ and $X^{'} / S^{'}$ ; no flatness, finiteness, separatedness or tor-independence hypothesis is imposed on $S^{'} \to S$ or on $X \to S$ .

Facts & Assumptions

Given: Morphisms of schemes $X \to S$ and $S^{'} \to S$ , with fibre product $X^{'} = X \times_{S} S^{'}$ and projections $g ⁣ : X^{'} \to X$ , $X^{'} \to S^{'}$ .

[F1]

Kähler differentials commute with scalar base change: for ring maps $A \to B$ and $A \to A^{'}$ with $B^{'} = B \otimes_{A} A^{'}$ , the canonical $B^{'}$ -linear map $Ω_{B / A} \otimes_{B} B^{'} \to Ω_{B^{'} / A^{'}}$ , $d b \otimes a^{'} \mapsto a^{'} d (b \otimes 1)$ , is an isomorphism.

[F2]

Affine fibre products are spectra of tensor products: for affine opens $U = Spec B \subseteq X$ over $V = Spec A \subseteq S$ and $W = Spec A^{'} \subseteq S^{'}$ over $V$ , the open subscheme $U \times_{V} W \subseteq X^{'}$ is affine with ring $B \otimes_{A} A^{'}$ .

[F3]

Sheaf of relative Kähler differentials: $Ω_{X / S}$ carries the universal $S$ -derivation $d_{X / S}$ , an $S$ -derivation of $O_{X}$ kills the image of the structure map from $O_{S}$ , and $Ω_{X^{'} / S^{'}}$ is defined analogously.

[F4]

Universal property of relative differential sheaves: composition with $d_{X / S}$ is a natural bijection ${Hom}_{O_{X}} (Ω_{X / S}, F) ≅ {Der}_{S} (O_{X}, F)$ for every $O_{X}$ -module $F$ .

[F5]

Pullback of modules is left adjoint to pushforward: there is a natural bijection ${Hom}_{O_{X^{'}}} (g^{*} G, F) ≅ {Hom}_{O_{X}} (G, g_{*} F)$ , and the map corresponding to $u$ sends $1 \otimes s$ to $u (s)$ .

[F6]

Pullback of a module along a morphism of ringed spaces: $g^{*} G = O_{X^{'}} \otimes_{g^{- 1} O_{X}} g^{- 1} G$ , and a local section $a$ of $O_{X}$ acts on $g^{*} G$ as $a \circ g$ .

[F7]

Affine charts recover the algebraic module of differentials and The stalk of a presheaf at a point: on an affine chart, $Ω_{X / S}$ is the sheaf attached to the relevant algebraic module with the restriction maps given by localization, and stalks are filtered colimits of sections over basic opens.

Proof

technique · direct

1.1

The canonical map. The composite $D ⁣ : O_{X} \to g_{*} O_{X^{'}} \overset{g_{*} d_{X^{'} / S^{'}}}{\to} g_{*} Ω_{X^{'} / S^{'}}$ is an $S$ -derivation of $O_{X}$ into the $O_{X}$ -module $g_{*} Ω_{X^{'} / S^{'}}$ : it is additive, satisfies Leibniz for the $O_{X}$ -module structure transported along $g^{♯}$ , and kills the image of $O_{S}$ , because that image is mapped into the image of the $O_{S^{'}}$ -structure of $X^{'}$ , which $d_{X^{'} / S^{'}}$ annihilates by [F3]. By [F4] there is a unique $O_{X}$ -linear $u ⁣ : Ω_{X / S} \to g_{*} Ω_{X^{'} / S^{'}}$ with $u (d_{X / S} (a)) = d_{X^{'} / S^{'}} (a \circ g)$ , and by [F5] there is a unique $O_{X^{'}}$ -linear map $γ ⁣ : g^{*} Ω_{X / S} \to Ω_{X^{'} / S^{'}}$ sending $1 \otimes d_{X / S} (a)$ to $d_{X^{'} / S^{'}} (a \circ g)$ .

F3F4F5F6

2.1

Affine charts compute $γ$ . Let $x^{'} \in X^{'}$ have image $x \in X$ and $s \in S$ , and choose an affine open $V = Spec A \subseteq S$ containing $s$ ; then choose an affine open $U = Spec B \subseteq X$ containing $x$ with image in $V$ and an affine open $W = Spec A^{'} \subseteq S^{'}$ containing the image of $x^{'}$ with image in $V$ . By [F2], $U^{'} : = U \times_{V} W = Spec B^{'}$ for $B^{'} = B \otimes_{A} A^{'}$ is an open affine neighbourhood of $x^{'}$ . By [F7], $Ω_{X / S} ∣_{U}$ and $Ω_{X^{'} / S^{'}} ∣_{U^{'}}$ are attached to $Ω_{B / A}$ and $Ω_{B^{'} / A^{'}}$ . If $x^{'}$ corresponds to $p^{'} \subseteq B^{'}$ and $x$ to $p = p^{'} \cap B$ , the pullback definition [F6] and stalk construction [F7] give $(g^{*} Ω_{X / S})_{x^{'}} ≅ (Ω_{B / A})_{p} \otimes_{B_{p}} B_{p^{'}}^{'} ≅ (B^{'} \otimes_{B} Ω_{B / A})_{p^{'}}$ . Hence the pullback is the sheaf attached to $B^{'} \otimes_{B} Ω_{B / A}$ on $U^{'}$ . The map $γ$ sends $1 \otimes d b$ to $d (b \otimes 1)$ by step 1.1, so on these stalks it is the localisation of the canonical isomorphism of [F1].

F1F2F6F7step 1.1

3.1

$γ$ is an isomorphism. Every point $x^{'} \in X^{'}$ lies in a chart $U^{'}$ as in step 2.1, on which $γ ∣_{U^{'}}$ is the isomorphism of [F1]; a morphism of sheaves whose restriction to each member of an open cover is an isomorphism is an isomorphism (equivalently, its stalk maps are isomorphisms), and forming stalks of the sheaves attached to $B^{'}$ -modules at points of $U^{'}$ is compatible with the identifications of step 2.1. Hence $γ$ is an isomorphism of $O_{X^{'}}$ -modules, with the asserted description on generators.

F1step 2.1

4.1

Naturality and hypotheses. The map $γ$ was produced from the universal properties of [F4] and the adjunction [F5] applied to the given morphisms $S^{'} \to S$ and $X \to S$ ; replacing the base-change data by a morphism of squares replaces $γ$ by the corresponding pullback of $γ$ , and on affine charts this is the naturality statement of [F1]. Only the existence of the fibre product and the affine descriptions of $Ω$ were used, so no flatness, finiteness, separatedness or tor-independence hypothesis enters.

F1F4F5step 2.1step 3.1∎

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

Relative cotangent and tangent spaces

Definition

Let $f ⁣ : X \to S$ be a morphism of schemes, let $x \in X$ be a point with image $s = f (x) \in S$ , and let $Ω_{X / S}$ be the sheaf of relative differentials (Sheaf of relative Kähler differentials), an $O_{X}$ -module.

Relative cotangent space. The relative cotangent space of $X$ over $S$ at $x$ is the $κ (x)$ -vector space

$Ω_{X / S, x} \otimes_{O_{X, x}} κ (x),$

where $O_{X, x}$ is the local ring and $κ (x) = O_{X, x} / m_{x}$ is the residue field of $x$ (The residue field at a point of an affine scheme) and the tensor product is formed along the residue map $O_{X, x} \to κ (x)$ ; equivalently it is the fibre of the $O_{X}$ -module $Ω_{X / S}$ at $x$ in the sense of the tensor product with the residue field. Its elements are written $ω \otimes 1$ and, for a local section $a$ of $O_{X}$ near $x$ , $d_{X / S} (a) \otimes 1$ is the relative cotangent vector of $a$ at $x$ .

Relative tangent space. The relative tangent space of $X$ over $S$ at $x$ is the $κ (x)$ -linear dual

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

Residue-field dependence. The residue field map $κ (s) \to κ (x)$ induced by $f$ is part of the data: the $κ (x)$ -module $Ω_{X / S, x} \otimes κ (x)$ is a vector space over $κ (x)$ , and the $κ (s)$ -structure obtained by restriction of scalars along $κ (s) \to κ (x)$ is used whenever the base field is fixed. No finiteness hypothesis is imposed: the spaces above may be infinite-dimensional over their residue fields, and the notation applies to any point of any morphism of schemes, including non-closed points and points of relative dimension $0$ .

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

Cotangent space at a rational point

Statement

Let $k$ be a field, let $X$ be a $k$ -scheme (Schemes and morphisms over a base) and let $x \in X$ be a $k$ -rational point, that is, a point whose residue field $κ (x)$ is $k$ under the canonical map $k \to κ (x)$ (The residue field at a point of an affine scheme). Write $R = O_{X, x}$ and $m = m_{x}$ , so that $R / m = k$ . Then the map

$m / m^{2} ⟶ Ω_{X / k} \otimes_{O_{X, x}} κ (x), [a] ⟼ d_{X / k} (a) \otimes 1,$

is an isomorphism of $k$ -vector spaces; here $[a]$ denotes the class of $a \in m$ modulo $m^{2}$ and the relative cotangent space is as in Relative cotangent and tangent spaces. The isomorphism is natural in pairs $(X, x)$ of $k$ -schemes with a $k$ -rational point. No analogous statement is made for a point whose residue field is a nontrivial extension of $k$ , not even a purely inseparable one.

Facts & Assumptions

Given: A field $k$ , a $k$ -scheme $X$ and a $k$ -rational point $x \in X$ with $R = O_{X, x}$ , $m = m_{x}$ and $R / m = k$ .

[F1]

Conormal exact sequence for an algebra quotient: for a ring map $A \to P$ with ideal $I \subseteq P$ and $B = P / I$ , the sequence $I / I^{2} \to B \otimes_{P} Ω_{P / A} \to Ω_{B / A} \to 0$ is exact, the first map sending the class of $t$ to $1 \otimes d t$ .

[F2]

Derivations are maps out of Ω: for a ring map $C \to D$ and every $D$ -module $N$ , composition with the universal derivation is a natural bijection ${Hom}_{D} (Ω_{D / C}, N) ≅ {Der}_{C} (D, N)$ . In particular $Ω_{k / k} = 0$ , since ${Der}_{k} (k, N) = 0$ for every $k$ -module $N$ .

[F3]

Affine charts recover the algebraic module of differentials and Kähler differentials commute with localization: on an affine chart $Spec B ∋ x$ the sections of $Ω_{X / k}$ over basic opens are $Ω_{B_{g} / k}$ , so passing to the stalk at $x$ gives $(Ω_{X / k})_{x} ≅ Ω_{R / k}$ and hence $Ω_{X / k} \otimes_{O_{X, x}} κ (x) ≅ k \otimes_{R} Ω_{R / k}$ .

[F4]

Relative cotangent and tangent spaces: the relative cotangent space at $x$ is $Ω_{X / k} \otimes_{O_{X, x}} κ (x)$ , an object over $κ (x) = k$ .

Proof

technique · direct

1.1

The conormal sequence at the point. Apply [F1] to the ring map $k \to R$ and the ideal $m \subseteq R$ with quotient $R / m = k$ : the sequence $m / m^{2} ⟶ k \otimes_{R} Ω_{R / k} ⟶ Ω_{k / k} ⟶ 0$ is exact, the first map sending $[a]$ to $1 \otimes d a$ , and the middle term is $k \otimes_{R} Ω_{R / k}$ with $k = R / m$ . By [F2] the last term vanishes, so the first map is surjective.

F1F2given

1.2

A retraction. Define $D ⁣ : R \to m / m^{2}$ by $D (a) : = [a - ε (a)]$ , where $ε ⁣ : R \to R / m = k$ is the residue map and $[\cdot]$ is the class modulo $m^{2}$ . Then $D$ is additive, kills $k$ since $ε$ is the identity on $k \subseteq R$ , and is a $k$ -derivation: $D (a b) - a D (b) - b D (a) = [- (a - ε (a)) (b - ε (b))] = 0$ in $m / m^{2}$ , because both $a - ε (a)$ and $b - ε (b)$ belong to $m$ . Here elements of $k$ are viewed in $R$ via its structure map, which splits $ε$ , and $R$ acts on $m / m^{2}$ through $ε$ . By [F2] applied to $k \to R$ there is an $R$ -linear $\tilde{D} ⁣ : Ω_{R / k} \to m / m^{2}$ with $\tilde{D} (d a) = D (a)$ ; since $m \cdot (m / m^{2}) = 0$ , it kills $m Ω_{R / k}$ and therefore factors through an $R$ -linear, hence $k$ -linear, map $ψ ⁣ : k \otimes_{R} Ω_{R / k} \to m / m^{2}$ .

F2given

2.1

The identification of the target. By [F3] applied to an affine chart of $X$ containing $x$ , the stalk of $Ω_{X / k}$ at $x$ is $Ω_{R / k}$ , so the relative cotangent space of [F4], namely the residue-field tensor product $Ω_{X / k} \otimes_{O_{X, x}} κ (x)$ of the statement, is $k \otimes_{R} Ω_{R / k}$ ; under this identification the element $d_{X / k} (a) \otimes 1$ for $a \in R$ corresponds to $1 \otimes d a$ . Hence the map of the statement is the first map of the exact sequence of step 1.1, and it is natural in $(X, x)$ because the identification is induced by the universal derivation and localization.

F1F3F4step 1.1

2.2

$ψ$ is a left inverse of the first map. For $a \in m$ one has $ψ (1 \otimes d a) = \tilde{D} (d a) = D (a) = [a]$ , because $ε (a) = 0$ . Hence $ψ$ is a left inverse of the map $[a] \mapsto 1 \otimes d a$ of step 1.1, which is therefore injective.

step 1.1step 1.2

3.1

Conclusion. The map of step 1.1 is surjective by step 1.1 and injective by step 2.2, hence an isomorphism $m / m^{2} ≅ k \otimes_{R} Ω_{R / k}$ ; by step 2.1 this is exactly the map of the statement, which is therefore an isomorphism of $k$ -vector spaces, natural in $(X, x)$ . Nothing was used about $x$ beyond $κ (x) = k$ , and the hypothesis is essential to the argument: for a point with $κ (x) \neq k$ the residue map is not a $k$ -algebra section of $R \to κ (x)$ in general, and no such retraction is constructed.

step 2.1step 2.2∎

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

Tangent vectors as dual-number points

Statement

Let $f ⁣ : X \to S$ be a morphism of schemes and let $x \in X$ with image $s = f (x) \in S$ . Write $κ = κ (x)$ and let $D_{κ} = Spec κ [ϵ] / (ϵ^{2})$ be the dual-numbers scheme (The affine scheme of dual numbers), regarded as an $S$ -scheme through the canonical point $Spec κ \to X \to S$ . Consider $S$ -morphisms $τ ⁣ : D_{κ} ⟶ X$ whose reduction is the canonical $κ$ -point $x$ , i.e. the composite of $τ$ with the closed immersion $Spec κ ↪ D_{κ}$ ( $ϵ \mapsto 0$ ) is the canonical morphism $Spec κ \to X$ . Then evaluation of the $ϵ$ -coefficient induces a natural bijection ${τ ⁣ : D_{κ} \to X over S reducing to x} ≅ {Hom}_{κ} (Ω_{X / S} \otimes_{O_{X, x}} κ (x), κ (x)) = T_{X / S, x}$ with the relative tangent space at $x$ (Relative cotangent and tangent spaces). If $x$ is $k$ -rational for a field $k$ and $S = Spec k$ , the bijection reads ${τ} ≅ {Hom}_{k} (m_{x} / m_{x}^{2}, k)$ , recovering the classical description of the tangent space as the dual of $m_{x} / m_{x}^{2}$ . The $ϵ$ -coefficient of $τ$ is a $κ$ -linear functional whose vanishing on $Ω_{X / S} \otimes κ$ exactly means that $τ$ is the constant (reduction) morphism.

Facts & Assumptions

Given: A morphism $f ⁣ : X \to S$ , a point $x \in X$ with $s = f (x)$ , and the dual-numbers scheme $D_{κ} = Spec κ [ϵ] / (ϵ^{2})$ .

[F1]

The residue field at a point of an affine scheme and Relative cotangent and tangent spaces: $R : = O_{X, x}$ , $m = m_{x}$ , $κ = R / m$ , and $T_{X / S, x} = {Hom}_{κ} (Ω_{X / S} \otimes_{O_{X, x}} κ, κ)$ .

[F2]

Morphisms of schemes are local on compatible open covers: morphisms of schemes may be constructed and compared after passing to an affine chart around a point of the source; a morphism from a one-point scheme into $X$ with image $x$ factors through an affine open containing $x$ .

[F3]

The affine scheme of dual numbers: $D_{κ} = Spec κ [ϵ] / (ϵ^{2})$ is affine with ring $κ [ϵ] / (ϵ^{2})$ , whose maximal ideal $(ϵ)$ is nilpotent and whose quotient by $ϵ$ is $κ$ .

[F4]

Derivations are maps out of Ω: for a ring map $A \to R$ and $R$ -module $N$ there is a natural bijection ${Hom}_{R} (Ω_{R / A}, N) ≅ {Der}_{A} (R, N)$ .

[F5]

Kähler differentials commute with localization and Affine charts recover the algebraic module of differentials: $Ω_{X / S} \otimes_{O_{X, x}} κ ≅ Ω_{R / A} \otimes_{R} κ$ for $A = O_{S, s}$ .

[F6]

Cotangent space at a rational point: for a $k$ -rational point $x$ of a $k$ -scheme there is a natural isomorphism $m / m^{2} ≅ Ω_{X / k} \otimes κ (x)$ .

Proof

technique · direct

1.1

Dual-number points are local homomorphisms. Let $τ ⁣ : D_{κ} \to X$ be an $S$ -morphism reducing to $x$ . Since $D_{κ}$ is a one-point scheme with closed point mapping to $x$ , [F2] lets us work on an affine chart $U = Spec B ∋ x$ and shows that $τ$ corresponds to a ring map $B \to κ [ϵ] / (ϵ^{2})$ whose composite with $ϵ \mapsto 0$ is the restriction of the residue map $B \to κ$ . Passing to the local ring gives a well-defined ring map $φ ⁣ : R \to κ [ϵ] / (ϵ^{2})$ with $φ (m) \subseteq (ϵ)$ and $φ (a) \equiv a m o d m$ for $a \in R$ , and the $S$ -morphism condition says that $φ$ restricted to $A = O_{S, s}$ lands in $κ \subseteq κ [ϵ] / (ϵ^{2})$ . Conversely such a $φ$ determines $τ$ by the same description on an affine chart containing $x$ ; two charts give the same morphism by [F2].

F2F3given

2.1

Dual-number points are derivations. A ring map $φ ⁣ : R \to κ [ϵ] / (ϵ^{2})$ with $φ (a) \equiv a m o d m$ has the form $φ (a) = \overset{ˉ}{a} + ϵ D (a)$ with $\overset{ˉ}{a}$ the class of $a$ in $κ$ and a unique map $D ⁣ : R \to κ$ ; the map $φ$ is additive exactly when $D$ is, and $φ (a b) = φ (a) φ (b)$ for all $a, b$ is equivalent to the Leibniz rule $D (a b) = \overset{ˉ}{a} D (b) + \overset{ˉ}{b} D (a)$ , since $ϵ^{2} = 0$ . Moreover $φ ∣_{A}$ lands in $κ$ exactly when $D$ kills the image of $A$ . Hence passage to $D$ is a bijection between the ring maps of step 1.1 and the $A$ -derivations $D ⁣ : R \to κ$ ; the derivation is recovered from the product expansion of $φ$ , so the correspondence is natural in $(X, x)$ .

F3given

3.1

Derivations are tangent vectors. Evaluation gives ${Der}_{A} (R, κ) ≅ {Hom}_{R} (Ω_{R / A}, κ)$ by [F4] (with $N = κ$ , an $R$ -module through $R \to κ$ ), and restriction and extension of scalars along $R \to κ$ give ${Hom}_{R} (Ω_{R / A}, κ) ≅ {Hom}_{κ} (Ω_{R / A} \otimes_{R} κ, κ)$ . By [F5] the latter is ${Hom}_{κ} (Ω_{X / S} \otimes_{O_{X, x}} κ, κ)$ , which is the relative tangent space $T_{X / S, x}$ as recalled in [F1]. Composing the bijections of steps 1.1 and 2.1 with this identification gives the asserted bijection between the dual-number points reducing to $x$ and the relative tangent space.

F1F4F5step 1.1step 2.1

4.1

Naturality and the rational-point case. The correspondence of steps 1.1–3.1 is natural for morphisms of pointed $S$ -schemes with a fixed coefficient field $κ$ : if $h : X \to Y$ sends the chosen $κ$ -point over $x$ to a $κ$ -point over $y$ , composition sends a map $D_{κ} \to X$ to a map $D_{κ} \to Y$ . On local rings the $ϵ$ -coefficient is the derivation $D \circ h_{y}^{♯} : O_{Y, y} \to κ$ , matching pullback of cotangent vectors after tensoring $Ω_{Y / S, y}$ with $κ$ along $κ (y) \to κ$ . If the residue-field map $κ (y) \to κ (x)$ is an isomorphism, this is the usual map of relative tangent spaces; for a nontrivial residue-field extension, the target is instead the $κ$ -dual of the base-extended cotangent space, with no map from $D_{κ (x)}$ to $D_{κ (y)}$ assumed. In the case $S = Spec k$ and $x$ $k$ -rational, [F6] identifies $Ω_{X / k} \otimes κ (x)$ with $m_{x} / m_{x}^{2}$ , so the bijection becomes the classical tangent-space description. A dual-number point whose $ϵ$ -coefficient functional vanishes has $D = 0$ , hence $φ$ is the residue map and $τ$ is the constant morphism, and conversely.

F6step 3.1∎

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

Differential of an S-morphism

Statement

Let $S$ be a scheme and let $f ⁣ : X \to Y$ be a morphism of $S$ -schemes. Then the universal derivations of $X / S$ and $Y / S$ induce a unique $O_{X}$ -linear map $d f ⁣ : f^{*} Ω_{Y / S} ⟶ Ω_{X / S}, 1 \otimes d_{Y / S} (g) ⟼ d_{X / S} (g \circ f),$ the differential of $f$ . It satisfies:

(identity) for $f = {i d}_{X}$ the map $d f$ is the canonical identification ${i d}_{X}^{*} Ω_{X / S} ≅ Ω_{X / S}$ ;
(chain rule) for $X \overset{f}{\to} Y \overset{g}{\to} Z$ over $S$ the composite $f^{*} g^{*} Ω_{Z / S} \to f^{*} Ω_{Y / S} \to Ω_{X / S}$ , formed with the canonical identification $f^{*} g^{*} ≅ (g \circ f)^{*}$ , equals $d (g \circ f)$ ;
(fibres) at each $x \in X$ , with $y = f (x)$ , the map $d f$ induces a $κ (x)$ -linear map $(Ω_{Y / S, y} \otimes_{O_{Y, y}} κ (y)) \otimes_{κ (y)} κ (x) ⟶ Ω_{X / S, x} \otimes_{O_{X, x}} κ (x) .$ Dualising over $κ (x)$ gives a $κ (x)$ -linear tangent map $T_{X / S, x} ⟶ {Hom}_{κ (x)} ((Ω_{Y / S, y} \otimes_{O_{Y, y}} κ (y)) \otimes_{κ (y)} κ (x), κ (x)) .$ If $κ (y) = κ (x)$ , this target is $T_{Y / S, y}$ .

No finiteness, flatness or separatedness hypothesis is imposed.

Facts & Assumptions

Given: A scheme $S$ and a morphism $f ⁣ : X \to Y$ of $S$ -schemes.

[F1]

Transitivity sequence for schemes: the first arrow $γ ⁣ : f^{*} Ω_{Y / S} \to Ω_{X / S}$ of the transitivity sequence is the unique $O_{X}$ -linear map with $γ (1 \otimes d_{Y / S} (g)) = d_{X / S} (g \circ f)$ .

[F2]

Relative cotangent and tangent spaces and Pullback of a module along a morphism of ringed spaces: the relative cotangent space at $x$ is $Ω_{X / S, x} \otimes_{O_{X, x}} κ (x)$ , and the source stalk of $d f$ is $Ω_{Y / S, y} \otimes_{O_{Y, y}} O_{X, x}$ ; its fibre is the cotangent space at $y$ extended along $κ (y) \to κ (x)$ .

[F3]

Pullback of a module along a morphism of ringed spaces: the composite of pullbacks is canonically identified with the pullback along the composite, $f^{*} g^{*} ≅ (g \circ f)^{*}$ , by associativity of the sheaf tensor products defining pullback; on generators $1 \otimes 1 \otimes s$ the identification is the identity.

[F4]

Universal property of relative differential sheaves: a map out of $Ω$ is determined by its values on the universal differentials, since these generate the module.

[F5]

Sheaf of relative Kähler differentials: the modules $Ω_{X / S}$ and $Ω_{Y / S}$ and their universal derivations exist for arbitrary morphisms and kill the images of the structure maps from $O_{S}$ .

Proof

technique · direct

1.1

Construction. By [F1], applied to the $S$ -morphism $f$ , there is a unique $O_{X}$ -linear $d f ⁣ : f^{*} Ω_{Y / S} \to Ω_{X / S}$ with $d f (1 \otimes d_{Y / S} (g)) = d_{X / S} (g \circ f)$ for local sections $g$ of $O_{Y}$ ; it is obtained by applying the universal property [F4] to the $S$ -derivation $O_{Y} \to f_{*} Ω_{X / S}$ , $g \mapsto d_{X / S} (g \circ f)$ , and then the adjunction of Pullback of modules is left adjoint to pushforward, and it is natural in the data $(X, Y, f)$ by construction.

F1F4F5

2.1

Identity. For $f = {i d}_{X}$ the map sends $1 \otimes d_{X / S} (g)$ to $d_{X / S} (g)$ ; since the elements $d_{X / S} (g)$ generate $Ω_{X / S}$ over $O_{X}$ by [F4], this is the canonical identification ${i d}_{X}^{*} Ω_{X / S} ≅ Ω_{X / S}$ .

F4step 1.1

2.2

Chain rule. Let $X \overset{f}{\to} Y \overset{g}{\to} Z$ be morphisms of $S$ -schemes. Both $d (g \circ f)$ and the composite $d f \circ f^{*} (d g)$ are $O_{X}$ -linear maps $(g \circ f)^{*} Ω_{Z / S} \to Ω_{X / S}$ (the composite being formed with the identification [F3]), and on a generator $1 \otimes 1 \otimes d_{Z / S} (h)$ both take the value $d_{X / S} (h \circ g \circ f)$ : the composite because $d f (1 \otimes d_{Y / S} (h \circ g)) = d_{X / S} (h \circ g \circ f)$ and $d g (1 \otimes d_{Z / S} (h)) = d_{Y / S} (h \circ g)$ , and $d (g \circ f)$ by its definition. As the generators $1 \otimes 1 \otimes d_{Z / S} (h)$ generate the source over $O_{X}$ , the two maps agree.

F3F4step 1.1

2.3

Fibres and the tangent map. Fix $x \in X$ and put $y = f (x)$ . By [F2], the source stalk of $d f$ is $Ω_{Y / S, y} \otimes_{O_{Y, y}} O_{X, x}$ . Tensoring it with $κ (x)$ gives $Ω_{Y / S, y} \otimes_{O_{Y, y}} κ (x)$ , canonically $(Ω_{Y / S, y} \otimes_{O_{Y, y}} κ (y)) \otimes_{κ (y)} κ (x)$ , because $O_{Y, y} \to κ (x)$ factors through the residue field $κ (y)$ . Thus the fibre of $d f$ is the $κ (x)$ -linear cotangent map displayed in the statement. Dualising over $κ (x)$ gives the stated map from $T_{X / S, x}$ to the $κ (x)$ -dual of the extended cotangent space at $y$ . When the residue-field map is an isomorphism, this target is $T_{Y / S, y}$ ; without that hypothesis, the latter is only a $κ (y)$ -vector space and cannot be the target of a $κ (x)$ -linear map.

F2step 1.1

3.1

Conclusion. Step 1.1 gives the asserted map and its characterisation, steps 2.1 and 2.2 give the identity and chain rules, and step 2.3 gives the fibre and tangent maps; nothing beyond the universal property of $Ω$ and the functoriality of pullback and of extension of scalars was used, so no finiteness, flatness or separatedness hypothesis enters.

step 1.1step 2.1step 2.2step 2.3∎

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

Formally unramified morphism

Definition

Let $f ⁣ : X \to S$ be a morphism of schemes (Morphisms of schemes, Schemes and morphisms over a base).

Square-zero thickenings. A square-zero thickening of a scheme $T_{0}$ is a closed immersion $i ⁣ : T_{0} ↪ T$ (Closed immersions of schemes) whose ideal sheaf $I = \ker (O_{T} \to i_{*} O_{T_{0}})$ (Ideal sheaves) satisfies $I^{2} = 0$ , meaning that the product of any two local sections of $I$ over a common open set is zero. Such a thickening is an $S$ -thickening when $T$ is an $S$ -scheme and $i$ is an $S$ -morphism.

Formally unramified. The morphism $f$ is formally unramified if for every commutative diagram of schemes

$\begin{array}{ccc} T_{0} & \overset{a}{\to} & X \\ ↓ i & ↓ f \\ T & \overset{b}{\to} & S \end{array}$

in which $i ⁣ : T_{0} ↪ T$ is a square-zero thickening and the square is over $S$ — that is, $a$ and $b$ are compatible with $f$ — there is at most one $S$ -morphism $T \to X$ whose restriction to $T_{0}$ is $a$ . In other words, two $S$ -morphisms $T \to X$ agreeing on a square-zero closed subscheme agree everywhere.

The condition is a uniqueness condition only: no existence is required, no finite-type, finite-presentation, flatness or separatedness hypothesis is imposed on $f$ , and the test thickenings are required to be square-zero but are otherwise arbitrary, in particular not assumed to be affine or of finite type over $S$ . For the affine case $X = Spec B$ , $S = Spec A$ with $f$ induced by $A \to B$ , the condition is the algebraic one: for every $A$ -algebra $C$ with an ideal $I \subseteq C$ satisfying $I^{2} = 0$ , two $A$ -algebra maps $B \to C$ that agree modulo $I$ are equal.

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

Formally smooth morphism

Definition

Let $f ⁣ : X \to S$ be a morphism of schemes (Schemes and morphisms over a base) and let $i ⁣ : T_{0} ↪ T$ be a square-zero thickening, namely a closed immersion (Closed immersions of schemes) whose ideal sheaf $I = \ker (O_{T} \to i_{*} O_{T_{0}})$ (Ideal sheaves) satisfies $I^{2} = 0$ .

Formally smooth. The morphism $f$ is formally smooth if for every commutative $S$ -diagram

$\begin{array}{ccc} T_{0} & \overset{a}{\to} & X \\ ↓ i & ↓ f \\ T & \overset{b}{\to} & S \end{array}$

every point $t \in T$ has an open neighbourhood $U \subseteq T$ over which a lift exists: there is an $S$ -morphism $U \to X$ extending $a ∣_{T_{0} \cap U}$ . In other words, lifts exist Zariski locally on the test scheme $T$ , and there is no uniqueness requirement.

Equivalent formulation. Since the lifting problem is local on $T$ , it is equivalent to require that the sheaf-theoretic lifting problem ${Hom}_{S} (T, X) \to {Hom}_{S} (T_{0}, X)$ be surjective locally on $T$ ; equivalently, by the universal property of the fibre product, the projection $X \times_{S} T \to T$ admits a section locally on $T$ over the given morphism $T_{0} \to X \times_{S} T$ . No finite-type, finite-presentation or flatness hypothesis is imposed, and no uniqueness of lifts is asserted; in particular a formally smooth morphism need not be an open immersion or a submersion in any topological sense, and "formally smooth" is not by itself the same condition as "the relative differentials are locally free" nor as "smooth of finite presentation", which is treated elsewhere.

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

Formally etale morphism

Definition

A morphism of schemes $f ⁣ : X \to S$ (Schemes and morphisms over a base) is formally etale if it is both formally smooth (Formally smooth morphism) and formally unramified (Formally unramified morphism).

Uniqueness of the local lifts. Explicitly, $f$ is formally etale exactly when every commutative $S$ -diagram with a square-zero thickening $i ⁣ : T_{0} ↪ T$ admits lifts $T \to X$ extending the given $T_{0} \to X$ Zariski locally on $T$ , and any two such local lifts agree on the overlaps of their domains of definition: local existence is formal smoothness, and uniqueness is formal unramifiedness. Consequently the local lifts glue uniquely, by Morphisms of schemes are local on compatible open covers, to a single $S$ -morphism $T \to X$ extending the given morphism from $T_{0}$ . Thus for a formally etale morphism every square-zero lifting problem has a unique lift, and the unique lift is obtained by gluing the local ones.

No finite-type, finite-presentation or flatness hypothesis is part of the definition: those enter the notion of an etale morphism of schemes, which is a formally etale morphism that is additionally locally of finite presentation (and flat); the comparison with that finite-presentation notion is made on a later page and is not claimed here.

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

The diagonal ideal modulo its square is Omega

Statement

Let $A \to B$ be a homomorphism of commutative rings, let $μ ⁣ : B \otimes_{A} B ⟶ B, μ (b \otimes b^{'}) = b b^{'},$ be the multiplication, and let $J = \ker μ$ . Then $J / J^{2}$ is a $B$ -module through $b \cdot x : = (b \otimes 1) x = (1 \otimes b) x$ , and the map

$Ω_{B / A} ⟶ J / J^{2}, d b ⟼ [1 \otimes b - b \otimes 1],$

is an isomorphism of $B$ -modules, natural in the ring map $A \to B$ . Its inverse sends the class of $1 \otimes b - b \otimes 1$ to $d b$ .

Facts & Assumptions

Given: A ring map $A \to B$ , the ring $B \otimes_{A} B$ with multiplication $μ$ , and $J = \ker μ$ .

[F1]

Existence and generators of Kähler differentials and Derivations are maps out of Ω: $Ω_{B / A}$ exists and every $A$ -derivation $D ⁣ : B \to M$ into a $B$ -module factors uniquely as $D = α \circ d$ with $α$ $B$ -linear.

[F2]

Universal mapping property of the tensor product of commutative algebras and Universal property of the tensor product for balanced maps into abelian groups: $B \otimes_{A} B$ is the coproduct of the two $A$ -algebras $B$ , with the $A$ -algebra maps $b \mapsto b \otimes 1$ and $b^{'} \mapsto 1 \otimes b^{'}$ , and $A$ -bilinear maps on $B \times B$ correspond to $A - l i n e a r$ maps on $B \otimes_{A} B$ .

[F3]

Derivation of an algebra: an $A$ -derivation is additive, $A$ -constant and satisfies the Leibniz rule.

Proof

technique · direct

1.1

The class map is a derivation. Put $j_{b} : = 1 \otimes b - b \otimes 1 \in J$ and $D (b) : = [j_{b}] \in J / J^{2}$ . For $b, c \in B$ , expansion in $B \otimes_{A} B$ gives $j_{b c} - [(b \otimes 1) j_{c} + (c \otimes 1) j_{b}] = j_{b} j_{c} \in J^{2}$ , so $D (b c) = b D (c) + c D (b)$ . The factors $b \otimes 1$ and $1 \otimes b$ act identically on $J / J^{2}$ , since their difference lies in $J$ and $J (J / J^{2}) = 0$ . Also $D$ is additive and $D (a) = 0$ for $a \in A$ , since $1 \otimes a = a \otimes 1$ . Hence $D$ is an $A$ -derivation into the $B$ -module $J / J^{2}$ , and [F1] gives a unique $B$ -linear $α ⁣ : Ω_{B / A} \to J / J^{2}$ with $α (d b) = D (b)$ .

F1F3

2.1

$α$ is surjective. Every $x = \sum_{i} b_{i} \otimes c_{i} \in J$ satisfies $\sum_{i} b_{i} c_{i} = 0$ , hence $x = \sum_{i} (b_{i} \otimes 1) (1 \otimes c_{i} - c_{i} \otimes 1) = \sum_{i} (b_{i} \otimes 1) j_{c_{i}}$ in $B \otimes_{A} B$ , using $(b_{i} \otimes 1) (c_{i} \otimes 1) = b_{i} c_{i} \otimes 1$ and $\sum_{i} b_{i} c_{i} \otimes 1 = 0$ . Thus $J$ is generated as a left $B$ -module by the $j_{c}$ , and $J / J^{2}$ is generated by their classes $D (c)$ , which lie in the image of $α$ . Hence $α$ is surjective.

step 1.1F2

3.1

A left inverse. The assignment $(b, c) \mapsto b d c$ is $A$ -bilinear, so [F2] defines an $A$ -linear map $Ψ ⁣ : B \otimes_{A} B \to Ω_{B / A}$ with $Ψ (b \otimes c) = b d c$ . It is linear for the left $B$ -action $a \cdot (b \otimes c) = (a b) \otimes c$ . For $b, c \in B$ , expand $j_{b} j_{c} = 1 \otimes b c - c \otimes b - b \otimes c + b c \otimes 1$ . Then $Ψ (j_{b} j_{c}) = d (b c) - c d b - b d c + b c d 1 = 0$ by the Leibniz rule. By step 2.1, $J$ is generated as a left $B$ -module by the $j_{b}$ , so $J^{2}$ is generated as a left $B$ -module by their pairwise products; left $B$ -linearity of $Ψ$ therefore gives $Ψ (J^{2}) = 0$ . Restricting $Ψ$ to $J$ and passing to the quotient gives a $B$ -linear map $β ⁣ : J / J^{2} \to Ω_{B / A}$ with $β ([j_{b}]) = d b$ .

F2F3step 2.1

4.1

The maps are inverse. For $b \in B$ one has $β (α (d b)) = Ψ (j_{b}) = Ψ (1 \otimes b - b \otimes 1) = d b - b d 1 = d b$ . The differentials $d b$ generate $Ω_{B / A}$ , so $β \circ α = i d$ . Since $α$ is surjective by step 2.1, it follows also that $α \circ β = i d$ . Thus $α$ is an isomorphism. The formulas defining $j_{b}$ and $Ψ$ commute with maps of ring homomorphisms $A \to B$ , so the isomorphism is natural.

step 1.1step 2.1step 3.1∎

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

Formal unramifiedness iff Omega vanishes

Statement

Let $f ⁣ : X \to S$ be a morphism of schemes. Then $f$ is formally unramified (Formally unramified morphism) if and only if $Ω_{X / S} = 0$ (Sheaf of relative Kähler differentials). No finite-type, finite-presentation, flatness or separatedness hypothesis is imposed on $f$ , and no existence of lifts is asserted in either direction.

Facts & Assumptions

Given: A morphism of schemes $f ⁣ : X \to S$ .

[F1]

Formally unramified morphism: $f$ is formally unramified if for every square-zero thickening $i ⁣ : T_{0} ↪ T$ over $S$ and every $S$ -morphism $T_{0} \to X$ there is at most one $S$ -morphism $T \to X$ restricting to it; over affine opens $Spec B \to Spec A$ this says that two $A$ -algebra maps $B \to C$ into a ring $C$ with square-zero ideal $I$ that agree modulo $I$ are equal.

[F2]

The diagonal ideal modulo its square is Omega: for a ring map $A \to B$ and $J = \ker (B \otimes_{A} B \to B)$ one has $J / J^{2} ≅ Ω_{B / A}$ via $[1 \otimes b - b \otimes 1] \mapsto d b$ .

[F3]

Universal property of relative differential sheaves: for every $O_{X}$ -module $G$ the map $u \mapsto u \circ d_{X / S}$ is a bijection ${Hom}_{O_{X}} (Ω_{X / S}, G) ≅ {Der}_{S} (O_{X}, G)$ .

[F4]

Affine charts recover the algebraic module of differentials: for an affine open $Spec B \subseteq X$ lying over an affine open $Spec A \subseteq S$ one has $Γ (Spec B, Ω_{X / S}) = Ω_{B / A}$ ; hence $Ω_{X / S} = 0$ if and only if $Ω_{B / A} = 0$ for all such charts.

[F5]

Sheaf of relative Kähler differentials: an $S$ -derivation $O_{X} \to G$ is additive, satisfies Leibniz, and kills the image of the structure map from $O_{S}$ .

[F6]

Closed immersions of schemes and Schemes and morphisms over a base: a closed immersion with ideal sheaf $I = \ker (O_{T} \to i_{*} O_{T_{0}})$ is a square-zero thickening when $I^{2} = 0$ ; a morphism is determined by its map of structure sheaves, so two morphisms of schemes are equal exactly when their sheaf maps are.

Proof

technique · direct

1.1

Assume $Ω_{X / S} = 0$ ; we show that lifts are unique. Let $i ⁣ : T_{0} ↪ T$ be a square-zero thickening over $S$ and let $a_{0} ⁣ : T_{0} \to X$ be an $S$ -morphism with two $S$ -morphism lifts $a, b ⁣ : T \to X$ . Since $a$ and $b$ have the same underlying map on points, the direct images $a_{*} O_{T}$ and $b_{*} O_{T}$ are the same sheaf of rings $F = O_{T}$ pushed forward along this common map, and both $a^{♯}$ and $b^{♯}$ are maps $O_{X} \to F$ ; the difference $δ : = b^{♯} - a^{♯}$ is a morphism of sheaves of abelian groups valued in $G : = a_{*} I$ , where $I = \ker (O_{T} \to i_{*} O_{T_{0}})$ , because $a$ and $b$ agree on $T_{0}$ after composition with $O_{T} \to i_{*} O_{T_{0}}$ . The sheaf $G$ is an $O_{X}$ -module through $a^{♯}$ , and $δ$ is an $S$ -derivation: it is additive, and for local sections $x, y$ of $O_{X}$ one has $δ (x y) = b^{♯} (x) δ (y) + δ (x) a^{♯} (y) = a^{♯} (x) δ (y) + a^{♯} (y) δ (x)$ , since $b^{♯} (x) - a^{♯} (x) \in G$ and $G^{2} \subseteq a_{*} I^{2} = 0$ . It kills the image of $O_{S}$ because $a$ and $b$ are $S$ -morphisms. By [F3] with $G$ and $Ω_{X / S} = 0$ , ${Der}_{S} (O_{X}, G) = {Hom}_{O_{X}} (0, G) = 0$ , so $δ = 0$ , that is $b^{♯} = a^{♯}$ ; by [F6] $a = b$ . Hence $f$ is formally unramified.

F1F3F5F6

1.2

Assume $f$ formally unramified; we show $Ω_{B / A} = 0$ on every affine chart. Let $U = Spec B \subseteq X$ be affine over an affine open $V = Spec A \subseteq S$ , put $B^{'} = (B \otimes_{A} B) / J^{2}$ with $J = \ker (B \otimes_{A} B \to B)$ , and let $q ⁣ : B^{'} \to B$ be the quotient. The ideal $J / J^{2} = \ker q$ has square zero, so $Spec B \to Spec B^{'}$ is a square-zero thickening over $A$ ; the two $A$ -algebra maps $p_{1} (b) = b \otimes 1$ and $p_{2} (b) = 1 \otimes b$ from $B$ to $B^{'}$ both compose with $q$ to the identity, so the $S$ -morphisms $c_{i} ⁣ : Spec B^{'} \to Spec B ↪ X$ induced by $p_{i}$ agree on $Spec B$ . By [F1] applied to this thickening, $c_{1} = c_{2}$ , and therefore the maps on global sections agree: $p_{1} = p_{2}$ . Hence $b \otimes 1 = 1 \otimes b$ in $B^{'}$ for all $b \in B$ , that is $J / J^{2} = 0$ , and [F2] gives $Ω_{B / A} = 0$ .

F1F2F6

2.1

Conclusion. Step 1.1 proves that $Ω_{X / S} = 0$ implies that $f$ is formally unramified and step 1.2 that a formally unramified $f$ has $Ω_{B / A} = 0$ on every affine chart, hence $Ω_{X / S} = 0$ by [F4]. This proves the equivalence; nowhere were finiteness, flatness or separatedness used, and no lift was constructed, only used for uniqueness in step 1.1.

F1F4step 1.1step 1.2∎

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

Unramified morphism

Definition

A morphism of schemes $f ⁣ : X \to S$ is unramified if it is locally of finite type (Locally finite type and finite type morphisms) and formally unramified (Formally unramified morphism). By Formal unramifiedness iff Omega vanishes this is equivalent to asking that $f$ be locally of finite type and that $Ω_{X / S} = 0$ (Sheaf of relative Kähler differentials); either formulation may be used.

Convention: finite type versus finite presentation. The convention here is the one for which "unramified" requires only locally of finite type, in accordance with the Stacks Project. Some authors (and the older terminology of EGA) use the stronger convention, asking for local finite presentation instead of local finite type; a morphism with that stronger property is sometimes called G-unramified. The two notions coincide when the source and target are locally Noetherian, but not in general. This page uses the finite-type convention throughout; where a later page needs the finite-presentation notion, it says so explicitly.

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

An unramified morphism has an open diagonal

Statement

Let $f ⁣ : X \to S$ be a morphism of schemes and let $Δ = Δ_{X / S} ⁣ : X \to X \times_{S} X$ be its diagonal (The diagonal morphism).

If $f$ is unramified (Unramified morphism) then $Δ$ is an open immersion (Open immersions of schemes).
Conversely, if $f$ is locally of finite type and $Δ$ is an open immersion, then $f$ is unramified.

No separatedness hypothesis is needed in either direction: the diagonal of an unramified morphism need not be closed, and the diagonal need not be a closed immersion for the argument. Only local finite type is used, never finite presentation or flatness.

Facts & Assumptions

Given: A morphism $f ⁣ : X \to S$ with diagonal $Δ ⁣ : X \to X \times_{S} X$ .

[F1]

Unramified morphism and Formal unramifiedness iff Omega vanishes: $f$ is unramified if and only if $f$ is locally of finite type and $Ω_{X / S} = 0$ ; equivalently if and only if $f$ is locally of finite type and formally unramified.

[F2]

Locally finite type and finite type morphisms: over affine opens $Spec B \subseteq X$ and $Spec A \subseteq S$ with $f (Spec B) \subseteq Spec A$ , the induced ring map $A \to B$ exhibits $B$ as a finitely generated $A$ -algebra.

[F3]

The diagonal morphism and Affine fibre products are spectra of tensor products: on affine charts $U = Spec B$ over $V = Spec A$ the diagonal restricts to $U \to U \times_{V} U = Spec (B \otimes_{A} B)$ , the morphism induced by the multiplication $μ ⁣ : B \otimes_{A} B \to B$ ; it is a closed immersion by Closed immersions into affine schemes are quotient spectra because $μ$ is surjective, with $J = \ker μ$ .

[F4]

The diagonal ideal modulo its square is Omega: $J / J^{2} ≅ Ω_{B / A}$ .

[F5]

Determinant trick for Nakayama: if $M$ is a finitely generated module over a commutative ring $C$ and $I M = M$ for an ideal $I$ , then there is $a \in I$ with $(1 - a) M = 0$ .

[F6]

Open immersions of schemes: an open immersion identifies its source isomorphically with an open subscheme of its target. A morphism which is injective on points and restricts, over the members of an open cover of its source, to isomorphisms onto open subschemes of the target is an open immersion; morphisms glue by Morphisms of schemes are local on compatible open covers.

[F7]

Affine charts recover the algebraic module of differentials and An idempotent partitions the spectrum into complementary clopen subsets: $Ω_{X / S}$ vanishes if and only if $Ω_{B / A} = 0$ on every affine chart; an idempotent $e$ of a ring $C$ gives a clopen partition $Spec C = D (e) ⊔ D (1 - e)$ with $V (e) = D (1 - e)$ , and for an idempotent the restriction ring is $C_{1 - e} ≅ C / (e)$ .

Proof

technique · direct

1.1

Chart description. Let $U = Spec B \subseteq X$ be affine over an affine open $V = Spec A \subseteq S$ , and write $C = B \otimes_{A} B$ , $J = \ker μ$ for the multiplication $μ ⁣ : C \to B$ . By [F3] the diagonal restricts to the morphism $Δ_{U} ⁣ : U \to U \times_{V} U = Spec C$ induced by $μ$ , a closed immersion with ideal $J$ . The elements $1 \otimes b - b \otimes 1$ for $b$ in a generating set of $B$ over $A$ generate $J$ as a $C$ -module, so if $B$ is a finitely generated $A$ -algebra, $J$ is a finitely generated $C$ -module; and by [F4] $J / J^{2} ≅ Ω_{B / A}$ .

F2F3F4given

2.1

Unramified implies finite generation and $Ω = 0$ on charts. If $f$ is unramified then by [F1] it is locally of finite type and $Ω_{X / S} = 0$ , so $B$ is a finitely generated $A$ -algebra and $Ω_{B / A} = 0$ on every chart; hence $J$ is a finitely generated $C$ -module with $J = J^{2}$ by [F4]. Applying [F5] inside the ring $C$ to the ideal $J$ and the $C$ -module $J$ , we find $e \in J$ with $(1 - e) J = 0$ ; then $e^{2} = e$ , since $e \in J$ , and $J = e C$ : indeed $j = e j \in e C$ for $j \in J$ and $e C \subseteq J$ as $J$ is an ideal.

F1F4F5step 1.1

2.2

Conversely, assume $f$ locally of finite type and $Δ$ an open immersion. Let $U = Spec B$ be an affine chart over $V = Spec A$ as in step 1.1. Since $Δ^{- 1} (U \times_{V} U) \supseteq U$ , the restriction $Δ_{U} ⁣ : U \to U \times_{V} U$ is again an open immersion, and by [F3] it is also the closed immersion induced by the surjection $μ ⁣ : C \to B$ with kernel $J$ . Its image is therefore open and closed in $Spec C$ .

F3F6step 1.1

3.1

The chart diagonal is an open immersion when $f$ is unramified. With $e$ as in step 2.1, $J = e C$ has radical $\sqrt{(e)}$ , so the image $V (J)$ of the closed immersion $Δ_{U}$ equals $V (e) = D (1 - e)$ , which is open in $Spec C$ by [F7]. Moreover $C / J = C / e C ≅ C / (e) ≅ C_{1 - e}$ is exactly the ring of the open subscheme $D (1 - e)$ , and $Δ_{U}$ is the morphism $C \to C / J$ over $Spec C$ ; hence $Δ_{U}$ identifies $U$ isomorphically with the open subscheme $D (1 - e)$ of $U \times_{V} U$ , so $Δ_{U}$ is an open immersion.

F3F6F7step 2.1

3.2

The image is a principal open. Let $Δ_{U} (U) = V (I)$ for the radical ideal $I$ of the closed image and $Spec C ∖ Δ_{U} (U) = V (K)$ . Since these closed sets are complementary, $I + K = C$ and $I K \subseteq nil (C)$ because $V (I K) = Spec C$ . Choose $i \in I$ and $k \in K$ with $i + k = 1$ , and choose $m \geq 1$ with $(i k)^{m} = 0$ . Expanding $1 = (i + k)^{2 m - 1}$ , every monomial is divisible by either $i^{m}$ or $k^{m}$ , so $1 = a i^{m} + b k^{m}$ for some $a, b \in C$ . Put $e = a i^{m}$ ; then $1 - e = b k^{m}$ and $e (1 - e) = a b (i k)^{m} = 0$ , hence $e^{2} = e$ . Since $e \in I$ and $1 - e \in K$ , we have $Δ_{U} (U) = V (I) \subseteq V (e) = D (1 - e) \subseteq Spec C ∖ V (K) = Δ_{U} (U)$ , so $Δ_{U} (U) = D (1 - e)$ .

F7step 2.2

4.1

$Δ$ is an open immersion when $f$ is unramified. The affine charts $U$ of step 3.1 cover $X$ , and for each of them $Δ ∣_{U} = Δ_{U}$ is an isomorphism onto the open subset $D (1 - e_{U})$ of $U \times_{V} U \subseteq X \times_{S} X$ . The diagonal is injective on points, since $Δ (x) = (x, x)$ determines $x$ , and an open immersion is exactly a morphism which is locally on the source an isomorphism onto an open subscheme and injective on points; by [F6] the local isomorphisms glue to an isomorphism of $X$ with the open subscheme $Δ (X) = ⋃_{U} Δ (U)$ of $X \times_{S} X$ . Hence $Δ$ is an open immersion.

F6step 3.1

4.2

The conormal module vanishes. The open immersion $Δ_{U}$ identifies $U$ with the open subscheme $D (1 - e)$ , whose ring is $C / (e)$ by [F7]; since the structure map of $Δ_{U}$ is $C \to C / J$ , the two descriptions of the same ring map give $J = (e)$ . Hence $J^{2} = (e^{2}) = (e) = J$ , so $J / J^{2} = 0$ and [F4] gives $Ω_{B / A} = 0$ . As the charts cover $X$ and $B$ was an arbitrary chart, [F7] gives $Ω_{X / S} = 0$ ; with $f$ locally of finite type, [F1] makes $f$ unramified.

F1F4F7step 3.2

5.1

Conclusion. Steps 2.1, 3.1 and 4.1 prove that an unramified $f$ has open diagonal, and steps 2.2, 3.2 and 4.2 prove that a locally finite type $f$ with open diagonal is unramified. No separatedness assumption was made: the diagonal is used as a closed immersion only on affine charts, where the multiplication $B \otimes_{A} B \to B$ is surjective, and the open condition comes from the idempotent splitting $J = e C$ .

step 4.1step 4.2∎

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

A field has only the zero ideal and itself, hence is Noetherian

Statement

Let $K$ be a field (Field). Then the only ideals of $K$ are the zero ideal $(0)$ and $K$ itself (Left, right and two-sided ideals, The ideal generated by a subset and principal ideals); consequently every ideal of $K$ is finitely generated, and $K$ is a Noetherian ring (Noetherian commutative rings and modules). No choice principle is used.

Facts & Assumptions

Given: A field $K$ and an ideal $I \subseteq K$ .

[F1]

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

[F2]

Left, right and two-sided ideals: an ideal $I \subseteq K$ is an additive subgroup closed under multiplication by elements of $K$ , so $r a \in I$ for all $r \in K$ , $a \in I$ ; hence $I = K$ as soon as $1 \in I$ .

[F3]

The ideal generated by a subset and principal ideals: for $a \in K$ the ideal $(a)$ is the intersection of all ideals containing $a$ ; in particular $(0) = {0}$ is generated by $0$ and $K = (1)$ is generated by $1$ , so both are generated by a single element.

[F4]

Noetherian commutative rings and modules: the ring $K$ is Noetherian if and only if every ideal of $K$ is finitely generated; the definition states the two conditions as equivalent.

Proof

technique · direct

1.1

A nonzero ideal is everything: if $I \neq (0)$ choose $a \in I$ with $a \neq 0$ ; by [F1] $a$ is invertible with inverse $a^{- 1} \in K$ , and since $I$ is closed under multiplication by elements of $K$ , $1 = a^{- 1} a \in I$ by [F2]; then $x = x \cdot 1 \in I$ for every $x \in K$ by [F2] again, so $I = K$ .

F1F2given

2.1

The ideal list: by step 1.1 every ideal of $K$ is either $(0)$ or $K$ ; the zero ideal is generated by the single element $0$ and $K = (1)$ is generated by the single element $1$ , so every ideal of $K$ is finitely generated.

step 1.1F3

3.1

Conclusion: by step 2.1 every ideal of $K$ is finitely generated, so [F4] makes $K$ a Noetherian ring. The argument used only the field axioms, the ideal axioms and the two-element list of ideals, so it invokes no choice principle.

step 2.1F4given∎

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

Finite-type field extensions with zero Ω

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k \subseteq L$ be fields with $L$ finitely generated over $k$ (Finitely generated field extensions $F (a_{1}, \dots, a_{r})$ ), and let $Ω_{L / k}$ be the Kähler differential module of $k \to L$ (Universal Kähler differential module).

If $Ω_{L / k} = 0$ , then $L / k$ is finite (The degree $[K : F] = \dim_{F} K$ of a finite field extension) and separable (Separable algebraic elements and separable extensions).
Conversely, if $L / k$ is finite and separable, then $Ω_{L / k} = 0$ .

The Axiom of Choice is used to obtain an algebraic closure of $k$ (Assuming Choice, every field has an algebraic closure), to select a $k$ -basis of the localisation $B_{s}$ and to produce maximal ideals, prime intersections and the Nakayama input inside the finite-type $K$ -algebra $C$ below; claim 2 is choice-free. Claim 1 assumes nothing about $char k$ and no separability beyond the vanishing of $Ω_{L / k}$ ; in particular no algebraicity of $L / k$ is assumed in advance.

Facts & Assumptions

Given: Fields $k \subseteq L$ with $L = k (y_{1}, \dots, y_{m})$ for some $m \geq 0$ and $y_{1}, \dots, y_{m} \in L$ , and the Kähler differential module $Ω_{L / k}$ of $k \to L$ .

[F1]

Finitely generated field extensions $F (a_{1}, \dots, a_{r})$ and Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions: $k (y_{1}, \dots, y_{m})$ is the smallest subfield of $L$ containing $k$ and the $y_{i}$ . The image $B = k [y_{1}, \dots, y_{m}]$ of the polynomial ring $k [x_{1}, \dots, x_{m}]$ under the homomorphism sending $x_{i}$ to $y_{i}$ (Universal property of $R [x]$ : a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism) is a subring of $L$ containing $k$ , it is a domain because $L$ is a field, and it is a finitely generated $k$ -algebra in the sense of Subalgebra generated by a subset, algebras of finite type, and module-finite algebras; since $L$ is the smallest subfield containing $k$ and the $y_{i}$ , the fraction field of $B$ is $L$ .

[F2]

Existence and generators of Kähler differentials, Jacobian presentation of Ω, A field has only the zero ideal and itself, hence is Noetherian, If $R$ is Noetherian then $R [x_{1}, \dots, x_{n}]$ is Noetherian for every $n \in N$ and Noetherian commutative rings and modules: a field is a Noetherian ring, so $k [x_{1}, \dots, x_{m}]$ is Noetherian and every ideal of it is finitely generated. Hence for $B = k [x_{1}, \dots, x_{m}] / I$ the ideal $I$ is generated by finitely many elements and $Ω_{B / k} ≅ B^{m} / \sum_{j = 1}^{r} B \cdot (\frac{\partial f_{j}}{\partial x_{1}}, \dots, \frac{\partial f_{j}}{\partial x_{m}}),$ a quotient of the free module $B^{m}$ , so $Ω_{B / k}$ is a finitely generated $B$ -module.

[F3]

Kähler differentials commute with localization: for a ring map $A \to B$ and multiplicative subsets $V \subseteq A$ , $U \subseteq B$ with $φ (V) \subseteq U$ , the canonical map $U^{- 1} Ω_{B / A} \to Ω_{U^{- 1} B / V^{- 1} A}$ is an isomorphism. With $V = {1}$ this gives $S^{- 1} Ω_{B / k} ≅ Ω_{S^{- 1} B / k}$ , and with $U = {1, s, s^{2}, \dots} = S_{s}$ it gives $(Ω_{B / k})_{s} ≅ Ω_{B_{s} / k}$ .

[F4]

A finite module that vanishes at a prime vanishes on some principal neighbourhood of that prime: if $M$ is a finitely generated module over a commutative ring and $p$ is a prime ideal with $M_{p} = 0$ , then there is $s \notin p$ with $M_{s} = 0$ , where $M_{s}$ is the localisation at ${1, s, s^{2}, \dots}$ .

[F5]

Assuming Choice, every field has an algebraic closure, An algebraically closed field: every nonconstant polynomial has a root in the field, Fields of characteristic zero, finite fields, and algebraically closed fields are perfect, A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective, Frobenius $x \mapsto x^{p}$ is an injective endomorphism in characteristic $p$ , and an automorphism for finite fields, The binomial theorem over an arbitrary commutative ring and A prime $p$ divides $(\binom{p}{k})$ for $0 < k < p$ : assuming Choice, $k$ has an algebraic closure $K$ , which is algebraically closed; every algebraically closed field and every field of characteristic zero is perfect, and a field of characteristic $p > 0$ is perfect exactly when its Frobenius map $x \mapsto x^{p}$ is surjective, in which case its $p^{e}$ -th power map is surjective for every $e \geq 0$ . In any commutative ring of characteristic $p$ the binomial theorem together with $p ∣ (\binom{p}{i})$ for $0 < i < p$ gives $(u + v)^{p} = u^{p} + v^{p}$ , hence $(u + v)^{p^{e}} = u^{p^{e}} + v^{p^{e}}$ as well.

[F6]

Kähler differentials commute with scalar base change: for ring maps $A \to B$ and $A \to A^{'}$ with $B^{'} = B \otimes_{A} A^{'}$ there is a canonical isomorphism $Ω_{B / A} \otimes_{B} B^{'} ≅ Ω_{B^{'} / A^{'}}$ .

[F7]

Principal localisation $R_{f} = {1, f, f^{2}, \dots}^{- 1} R$ , Subalgebra generated by a subset, algebras of finite type, and module-finite algebras and Presentations and localization under base extension: the principal localisation $B_{s} = S_{s}^{- 1} B$ has elements $b / s^{n}$ , and if $B$ is generated as a $k$ -algebra by $b_{1}, \dots, b_{N}$ then $B_{s}$ is generated as a $k$ -algebra by $b_{1}, \dots, b_{N}, s^{- 1}$ . For a finitely generated $k$ -algebra presented as $B = k [x_{1}, \dots, x_{m}] / I$ there is a ring isomorphism $B \otimes_{k} K ≅ K [x_{1}, \dots, x_{m}] / I K [x_{1}, \dots, x_{m}]$ ; consequently $B_{s} \otimes_{k} K$ is generated as a $K$ -algebra by the images of $y_{1}, \dots, y_{m}$ and of $s^{- 1}$ , hence is of finite type over $K$ , and $k [x] / (f) \otimes_{k} K ≅ K [x] / (f)$ for $f \in k [x]$ .

[F8]

Modules over a field are projective, flat, and injective, Every vector space has a basis, Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests, The regular module is a tensor unit: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ and Tensor products commute with arbitrary direct sums: assuming Choice, every module over a field is free and flat, and every vector space has a basis. A flat module $M$ over a commutative ring $R$ carries every injection $V ↪ W$ of $R$ -modules to an injection $V \otimes_{R} M ↪ W \otimes_{R} M$ . Moreover $R \otimes_{R} M ≅ M$ and $N \otimes_{R} ⨁_{i} M_{i} ≅ ⨁_{i} (N \otimes_{R} M_{i})$ .

[F9]

A maximal ideal of an affine algebra has finite residue field over the base field and A field is algebraically closed exactly when every nonconstant polynomial splits, equivalently when it has no nontrivial finite extension: in a finite-type $K$ -algebra every maximal ideal has residue field a finite extension of $K$ ; a field is algebraically closed exactly when it has no nontrivial finite extension.

[F10]

Cotangent space at a rational point, Affine charts recover the algebraic module of differentials, Relative cotangent and tangent spaces and Schemes and morphisms over a base: for a finite-type $K$ -algebra $C$ , regarded as the $K$ -scheme $Spec C$ , and a maximal ideal $m$ with $C / m = K$ , which is therefore a $K$ -rational point, the cotangent space is $m / m^{2} ≅ Ω_{(Spec C) / K} \otimes_{O_{Spec C, m}} κ (m) ≅ Ω_{C / K} \otimes_{C} (C / m) .$

[F11]

Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ , $R_{p}$ is local with unique maximal ideal $p R_{p}$ , Localisation of modules is exact, A localised module fraction is zero exactly when one denominator kills its numerator and $R_{p} / p R_{p} ≅ Frac (R / p)$ is the residue field at $p$ : for a prime $p$ of $C$ the localisation $C_{p}$ is a nonzero local ring with maximal ideal $p C_{p}$ , its residue field is $Frac (C / p)$ , an element $x$ satisfies $x / 1 = 0$ in $C_{p}$ exactly when $t x = 0$ for some $t \notin p$ , and localisation preserves short exact sequences.

[F12]

Assuming the Axiom of Choice, Nakayama's lemma, The Jacobson radical of a ring and A local ring is a nonzero commutative ring with a unique maximal ideal: assuming Choice, if $I \subseteq J (R)$ is an ideal of a commutative ring $R$ and $M$ is a finitely generated $R$ -module with $I M = M$ then $M = 0$ ; here $J (R)$ is the intersection of all maximal ideals, so in a local ring $J (R)$ is the unique maximal ideal.

[F13]

Prime ideals of a localization are exactly the primes disjoint from the denominator set: for a commutative ring $C$ , a multiplicative subset $S$ and the localisation map $λ ⁣ : C \to S^{- 1} C$ , contraction along $λ$ is a bijection from the prime ideals of $S^{- 1} C$ onto the prime ideals of $C$ disjoint from $S$ .

[F14]

In a nonzero commutative ring, every proper ideal is contained in a maximal ideal, Prime ideals and maximal ideals in a commutative ring and Correspondence theorem: ideals of $R / I$ correspond to ideals of $R$ containing $I$ : assuming Choice, every proper ideal of a nonzero commutative ring is contained in a maximal ideal, every maximal ideal is prime, and ideals of $C / p$ correspond to ideals of $C$ containing $p$ , so every prime of a nonzero ring $C$ is contained in a maximal ideal.

[F15]

The nilradical and reduced rings, The nilradical is the intersection of all prime ideals, A Noetherian ring has finitely many minimal prime ideals and Every algebra of finite type over a Noetherian ring is a Noetherian ring: assuming Choice, the nilradical of a commutative ring is the set of nilpotent elements and equals the intersection of all its prime ideals, and the ring is reduced exactly when that intersection is zero; a finite-type algebra over a Noetherian ring is Noetherian, and a Noetherian ring has only finitely many minimal prime ideals.

[F16]

Chinese remainder theorem for pairwise comaximal ideals: for pairwise comaximal ideals $I_{1}, \dots, I_{r}$ of a commutative ring with $r \geq 1$ the canonical map $C \to \prod_{i = 1}^{r} C / I_{i}$ is surjective with kernel $⋂_{i = 1}^{r} I_{i} = \prod_{i = 1}^{r} I_{i}$ .

[F17]

Rank-nullity: $\dim_{F} V = nullity T + rank T$ : for a linear map $T ⁣ : V \to W$ with $V$ finite-dimensional, $\dim_{k} V = nullity T + rank T$ ; in particular an injective $k$ -linear endomorphism of a finite-dimensional $k$ -vector space is surjective.

[F18]

Separable algebraic elements and separable extensions, Every finite field extension is algebraic, A finite extension generated by elements all but possibly one of which are separable is simple, The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element, For a nonconstant $p$ in $F [x]$ , the ideal $(p)$ is maximal and $F [x] / (p)$ is a field exactly when $p$ is irreducible and An irreducible polynomial over a field is separable exactly when its derivative is nonzero: an element is separable over the base when it is algebraic with separable minimal polynomial, and an extension is separable when all its elements are; every finite extension is algebraic; every finite separable extension is simple, so $L = k (α)$ for some $α \in L$ ; the minimal polynomial $f$ of $α$ is monic and irreducible, $f^{'} (α) = 0$ implies $f ∣ f^{'}$ , and $k [x] / (f) ≅ k [α]$ is a field; an irreducible polynomial is separable exactly when its derivative is nonzero.

[F19]

In characteristic $p$ , every irreducible polynomial is uniquely $g (x^{p^{e}})$ with $g$ irreducible and separable: let $char F = p > 0$ and let $f \in F [x]$ be nonconstant and irreducible. There are unique $e \in N$ and $g \in F [x]$ with $f (x) = g (x^{p^{e}})$ , $g$ irreducible and separable; the case $e = 0$ occurs exactly when $f$ is separable.

[F20]

Extension of scalars carries flat modules to flat modules, Associativity of tensor products for compatible bimodules and The tensor product of $R$ -algebras has multiplication $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ : extension of scalars carries flat modules to flat modules, and for fields $k \subseteq B^{'} \subseteq L$ and $k \subseteq K$ there is a canonical isomorphism of rings $(B^{'} \otimes_{k} K) \otimes_{B^{'}} L ≅ L \otimes_{k} K$ induced by $(b \otimes c) \otimes l \mapsto b l \otimes c$ , since tensor products of commutative algebras associate and commute with the multiplications.

[F21]

If $V$ has a spanning set with $n$ elements, then every linearly independent subset of $V$ is finite with at most $n$ elements; in particular $V$ has no linearly independent subset equinumerous with $N$ : if a vector space has a spanning set with $n$ elements, then every linearly independent subset of it is finite with at most $n$ elements.

Proof

technique · direct

1.1

Converse, setup. Assume that $L / k$ is finite and separable. By [F18] there is $α \in L$ with $L = k (α)$ (if $L = k$ take any $α \in k$ ). Let $f \in k [x]$ be the minimal polynomial of $α$ ; it is monic, irreducible, of degree $n \geq 1$ , and separable because $α$ is separable over $k$ . If $n = 1$ then $f^{'} = 1$ and $f^{'} (α) = 1 \neq 0$ . If $n \geq 2$ , then $f$ is irreducible and separable, so $f^{'} \neq 0$ by [F18]; as $\deg f^{'} < n = \deg f$ and $f$ is irreducible, $f ∤ f^{'}$ , so $f^{'} (α) \neq 0$ , since $f^{'} (α) = 0$ would give $f ∣ f^{'}$ by [F18]. In both cases $f^{'} (α) \neq 0$ .

givenF18

1.2

Forward, the finite-type model. Put $B : = k [y_{1}, \dots, y_{m}] \subseteq L$ . By [F1] the ring $B$ is a finitely generated $k$ -algebra, a domain, and $Frac (B) = L$ . Writing $B = k [x_{1}, \dots, x_{m}] / I$ for the kernel $I$ of the evaluation $x_{i} \mapsto y_{i}$ , [F2] shows that $I$ is finitely generated and that $Ω_{B / k}$ is a quotient of the free module $B^{m}$ ; hence $Ω_{B / k}$ is a finitely generated $B$ -module.

givenF1F2

1.3

Choice and the algebraic closure. Assume the Axiom of Choice (The Axiom of Choice). By [F5] there is an algebraic closure $K$ of $k$ , so $K$ is algebraically closed with $char K = char k$ ; by [F8] every vector space over a field has a basis and every module over a field is flat; and by [F14] every proper ideal of a nonzero ring lies in a maximal ideal.

givenF5F8F14

2.1

Converse, conclusion. The evaluation homomorphism $k [x] \to L$ , $x \mapsto α$ , has kernel $(f)$ by [F18], so $k [x] / (f) ≅ k [α] = L$ ; the one-relation form of the Jacobian presentation [F2] gives $Ω_{L / k} ≅ L / (f^{'} (α))$ . Since $f^{'} (α) \neq 0$ in the field $L$ , the ideal $(f^{'} (α))$ is all of $L$ and $Ω_{L / k} = 0$ . This proves claim 2 for every finite separable extension.

step 1.1F2F18

2.2

Forward, localising at the zero prime. The set $S : = B ∖ {0}$ is a multiplicative subset of the domain $B$ with $S^{- 1} B = Frac (B) = L$ [step 1.2], so [F3] gives $S^{- 1} Ω_{B / k} ≅ Ω_{L / k}$ ; under the hypothesis of claim 1 this is $0$ .

givenstep 1.2F3

3.1

Clearing denominators. The $B$ -module $Ω_{B / k}$ is finitely generated [step 1.2] and vanishes at the prime ideal $(0)$ of the domain $B$ [step 2.2], so [F4] provides $s \in B ∖ {0}$ with $(Ω_{B / k})_{s} = 0$ . Fix such an $s$ and put $C : = B_{s} \otimes_{k} K$ , the principal localisation $B_{s}$ being as in [F7].

step 1.2step 2.2F4F7

4.1

The differentials of $C$ vanish. By [F3] applied to the multiplicative set ${1, s, s^{2}, \dots}$ we have $Ω_{B_{s} / k} ≅ (Ω_{B / k})_{s} = 0$ , and [F6] gives $Ω_{C / K} = Ω_{B_{s} \otimes_{k} K / K} ≅ Ω_{B_{s} / k} \otimes_{B_{s}} C = 0$ .

step 3.1F3F6

4.2

$C$ is nonzero. The localisation map $B \to B_{s}$ is injective, since $B$ is a domain with $s \neq 0$ . The canonical map $B_{s} \to C$ , $b \mapsto b \otimes 1$ , is obtained by tensoring the injection $k ↪ K$ with the $k$ -module $B_{s}$ , which is flat by [F8]; hence it is injective by [F8], and $C \neq 0$ because $B_{s} \neq 0$ .

step 1.3step 3.1F8

4.3

$C$ is a finitely generated $K$ -algebra. The $k$ -algebra $B = k [y_{1}, \dots, y_{m}]$ is generated by $y_{1}, \dots, y_{m}$ , so $B_{s}$ is generated by the images of $y_{1}, \dots, y_{m}$ and of $s^{- 1}$ [F7]; hence $C = B_{s} \otimes_{k} K$ is generated as a $K$ -algebra by the images of these same elements, using the presentation $B_{s} ≅ k [x_{1}, \dots, x_{m}, z] / (I, z σ - 1)$ , where $σ$ represents $s$ in $k [x_{1}, \dots, x_{m}]$ and $z$ represents $s^{- 1}$ . Its base change is $K [x_{1}, \dots, x_{m}, z] / (I, z σ - 1) K [x_{1}, \dots, x_{m}, z]$ [F7]. So $C$ is of finite type over $K$ .

step 1.2step 3.1F7

5.1

$C$ is Noetherian. The field $K$ is a Noetherian ring [F2], and $C$ is a finitely generated $K$ -algebra [step 4.3], so $C$ is Noetherian by [F15]; in particular every ideal of $C$ is a finitely generated $C$ -module.

step 4.3F2F15

5.2

Maximal ideals are rational and have vanishing cotangent space. Let $m \subseteq C$ be a maximal ideal; one exists because $C \neq 0$ [step 4.2] and every proper ideal lies in a maximal ideal [F14]. By [F9] the field $C / m$ is a finite extension of $K$ , and since $K$ is algebraically closed [step 1.3] it has no nontrivial finite extension [F9], so $C / m = K$ : thus $m$ is a $K$ -rational point of $Spec C$ . By [F10], together with $Ω_{C / K} = 0$ [step 4.1], $m / m^{2} ≅ Ω_{C / K} \otimes_{C} (C / m) = 0.$

step 1.3step 4.1step 4.2F9F10F14

6.1

The local ring at each maximal ideal is a field. Let $m$ be a maximal ideal and $n : = m C_{m}$ , the maximal ideal of the local ring $C_{m}$ [F11]. Localising the short exact sequence $0 \to m^{2} \to m \to m / m^{2} \to 0$ at $C ∖ m$ is exact [F11] and gives $n / n^{2} ≅ (m / m^{2})_{m} = 0$ [step 5.2], so $n = n^{2}$ . The ideal $m$ is finitely generated [step 5.1], hence so is the $C_{m}$ -module $n$ ; since $n = J (C_{m})$ is the Jacobson radical of the local ring $C_{m}$ [F12], Nakayama's lemma [F12] with $I = n$ and $M = n$ gives $n = 0$ . Therefore the maximal ideal of the nonzero ring $C_{m}$ is zero, so $C_{m}$ is a field, and its residue field is, by [F11], $C_{m} / n = C_{m} ≅ Frac (C / m) = C / m = K$ [step 5.2]; in particular $C_{m} ≅ K$ .

step 5.1step 5.2F11F12

7.1

Primes inside a maximal ideal. Let $p \subseteq m$ be a prime ideal of $C$ with $m$ maximal. Taking $S = C ∖ m$ in [F13], the primes of $C_{m}$ correspond bijectively to the primes of $C$ contained in $m$ ; the field $C_{m}$ [step 6.1] has only the prime ideal $(0)$ , so exactly one prime of $C$ is contained in $m$ . Since $m$ itself is a prime ideal contained in $m$ and $p$ is another, $p = m$ .

step 6.1F13

8.1

Every prime of $C$ is maximal, and the minimal primes are the maximal ideals. Let $p$ be a prime ideal of $C$ . Since $C \neq 0$ [step 4.2] and $p \neq C$ , the quotient $C / p$ is a nonzero ring, so it has a maximal ideal; by [F14] its preimage $m$ in $C$ is a maximal ideal with $p \subseteq m$ , and step 7.1 gives $p = m$ . So every prime is maximal, and conversely every maximal ideal is prime [F14]. Hence the primes of $C$ are exactly the maximal ideals; no prime is strictly contained in another, so each prime is a minimal prime ideal.

step 4.2step 7.1F14

9.1

$C$ is reduced. By [F15] the nilradical of $C$ is the intersection of the prime ideals, which by step 8.1 is the intersection of all maximal ideals. Let $x \in Nil (C)$ and let $m$ be any maximal ideal. The image $x / 1 \in C_{m}$ is nilpotent and $C_{m}$ is a field [step 6.1], so $x / 1 = 0$ ; by [F11] there is $t \notin m$ with $t x = 0$ , so the annihilator of $x$ is not contained in $m$ . As this holds for every maximal ideal and every proper ideal lies in a maximal ideal [F14], the annihilator of $x$ is $C$ and $x = 0$ . Hence $Nil (C) = 0$ and $C$ is reduced [F15].

step 6.1step 8.1F11F14F15

10.1

$C$ is a finite product of copies of $K$ . By [F15] the Noetherian ring $C$ [step 5.1] has only finitely many minimal primes, which by step 8.1 are exactly its maximal ideals $m_{1}, \dots, m_{r}$ ; here $r \geq 1$ because $C \neq 0$ has a maximal ideal [step 4.2, F14]. Distinct maximal ideals are comaximal, and the intersection $⋂_{i = 1}^{r} m_{i}$ of all primes is the nilradical of $C$ [F15], which is zero [step 9.1]. The Chinese remainder theorem [F16] therefore gives $C ≅ C / ⋂_{i = 1}^{r} m_{i} ≅ \prod_{i = 1}^{r} C / m_{i} = K^{r},$ using $C / m_{i} = K$ from step 5.2.

step 4.2step 5.1step 5.2step 9.1F14F15F16

11.1

$B_{s}$ is finite-dimensional over $k$ . Choose a $k$ -basis $(v_{i})_{i \in I}$ of $B_{s}$ [F8]. For any finitely many basis elements $v_{i_{1}}, \dots, v_{i_{N}}$ , put $V : = ⨁_{j = 1}^{N} k v_{i_{j}}$ , so that $V ↪ B_{s}$ is injective; tensoring with the flat $k$ -module $K$ [F8] gives an injection $V \otimes_{k} K ↪ B_{s} \otimes_{k} K = C$ [step 3.1], and [F8] gives $V \otimes_{k} K ≅ ⨁_{j = 1}^{N} K (v_{i_{j}} \otimes 1),$ using $k \otimes_{k} K ≅ K$ . Hence the elements $v_{i_{j}} \otimes 1$ are $K$ -linearly independent in $C$ . Therefore ${v_{i} \otimes 1 : i \in I}$ is a $K$ -linearly independent subset of $C$ , and since $C ≅ K^{r}$ has a spanning set with $r$ elements [step 10.1], [F21] shows that it is finite with at most $r$ elements. The map $B_{s} \to C$ is injective [step 4.2], so the image of the basis also has $∣ I ∣$ elements and $∣ I ∣ \leq r$ : the $k$ -vector space $B_{s}$ is finite-dimensional.

step 1.3step 4.2step 10.1F8F21

12.1

$L = B_{s}$ is finite over $k$ . The ring $B_{s}$ is a domain, being a subring of the field $L$ , and finite-dimensional over $k$ [step 11.1]. For $0 \neq b \in B_{s}$ the multiplication map $b \cdot ⁣ : B_{s} \to B_{s}$ is $k$ -linear with kernel zero; by rank-nullity [F17] it is surjective, so some $b^{'}$ satisfies $b b^{'} = 1$ and $b$ is a unit. Hence $B_{s}$ is a field. Since $B \subseteq B_{s} \subseteq L$ and $Frac (B) = L$ [step 1.2], we get $L = Frac (B) \subseteq Frac (B_{s}) = B_{s} \subseteq L$ , so $B_{s} = L$ ; in particular $L$ is finite-dimensional over $k$ , that is, $L / k$ is finite.

step 1.2step 11.1F17

13.1

An element with non-separable minimal polynomial. Suppose now that $L / k$ is not separable. Since it is finite [step 12.1], it is algebraic [F18], and by [F18] some $α \in L$ fails to be separable over $k$ , which for an algebraic element means that its minimal polynomial $f \in k [x]$ is not separable. If $char k = 0$ then $k$ would be perfect [F5], so every irreducible polynomial over $k$ would be separable, a contradiction; hence $char k = p > 0$ . By [F19] there are a unique $e \geq 0$ and an irreducible separable $g \in k [x]$ with $f (x) = g (x^{p^{e}})$ , and since $f$ is not separable the case $e = 0$ does not occur, so $e \geq 1$ . Then $g$ is nonconstant of some degree $d \geq 1$ , and $\deg f = p^{e} d$ .

step 12.1F5F18F19

14.1

A nonzero nilpotent in $B^{'} \otimes_{k} K$ . The field $K$ has characteristic $p$ and is perfect, so by [F5] its $p^{e}$ -th power map is surjective. Write $g = \sum_{i} a_{i} x^{i}$ and choose $b_{i} \in K$ with $b_{i}^{p^{e}} = a_{i}$ , and set $g_{1} : = \sum_{i} b_{i} x^{i} \in K [x]$ . Since $K [x]$ , like $K$ , has characteristic $p$ , the binomial theorem gives $(u + v)^{p^{e}} = u^{p^{e}} + v^{p^{e}}$ in $K [x]$ [F5], whence $g_{1} (x)^{p^{e}} = \sum_{i} (b_{i} x^{i})^{p^{e}} = \sum_{i} b_{i}^{p^{e}} x^{i p^{e}} = \sum_{i} a_{i} x^{i p^{e}} = g (x^{p^{e}}) = f (x)$ in $K [x]$ ; also $1 \leq d = \deg g_{1} < \deg f = p^{e} d$ . Let $B^{'} : = k [α] \subseteq L$ , which by [F18] satisfies $B^{'} ≅ k [x] / (f)$ and is a field; by [F7] the $K$ -algebra $M : = B^{'} \otimes_{k} K ≅ K [x] / (f) = K [x] / (g_{1}^{p^{e}})$ contains the class $z$ of $g_{1}$ , which is nonzero because $\deg g_{1} < \deg g_{1}^{p^{e}} = \deg f$ , while $z^{p^{e}} = 0$ because $g_{1}^{p^{e}} = f \equiv 0$ .

step 13.1F5F7F18F19

15.1

The canonical map $M \to C$ is injective. The field $B^{'} = k [α]$ is a subfield of $L$ , so $B^{'} ↪ L$ is injective, and $M = B^{'} \otimes_{k} K$ is flat over the field $B^{'}$ because it is the extension of scalars of the flat $k$ -module $K$ [F8, F20]. Hence $M ≅ M \otimes_{B^{'}} B^{'} ↪ M \otimes_{B^{'}} L$ is injective [F8], and by the canonical identification $(B^{'} \otimes_{k} K) \otimes_{B^{'}} L ≅ L \otimes_{k} K = B_{s} \otimes_{k} K = C$ [F20, step 12.1] this map is the canonical map $M \to C$ , $b \otimes c \mapsto b \otimes c$ .

step 3.1step 14.1F8F20

16.1

Conclusion. The image of the nonzero nilpotent $z$ of step 14.1 under the injective map of step 15.1 is a nonzero element of $C$ whose $p^{e}$ -th power is $0$ ; this contradicts step 10.1, since in the product of fields $C ≅ K^{r}$ the only nilpotent element is $0$ . Hence $L / k$ is separable, and together with step 12.1 it is finite and separable, so claim 1 holds; claim 2 is step 2.1.

step 2.1step 10.1step 12.1step 14.1step 15.1∎

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

Unramified residue extensions are finite separable

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $f ⁣ : X \to S$ be a morphism of schemes, let $x \in X$ and put $s = f (x)$ . Suppose that $f$ is locally of finite type at $x$ (Locally finite type and finite type morphisms): there are affine opens $Spec B \subseteq X$ containing $x$ and $Spec A \subseteq S$ containing $s$ with $f (Spec B) \subseteq Spec A$ and $B$ a finitely generated $A$ -algebra. Suppose further that the stalk at $x$ of $Ω_{X / S}$ vanishes (Sheaf of relative Kähler differentials). Write $m_{x}$ for the maximal ideal of $O_{X, x}$ and $m_{s}$ for the maximal ideal of $O_{S, s}$ , and let $κ (x)$ and $κ (s)$ be the residue fields (The residue field at a point of an affine scheme). Then

$κ (x) / κ (s)$ is a finite separable extension (Separable algebraic elements and separable extensions), and
$m_{s} O_{X, x} = m_{x}$ .

In particular both conclusions hold at every point of an unramified morphism (Unramified morphism), and at every point of a morphism which is locally of finite type and formally etale (Formally etale morphism). The Axiom of Choice is used only through the finite-type field lemma Finite-type field extensions with zero Ω and Nakayama's lemma; the separable-residue cotangent input Separable residue and the cotangent sequence of a local algebra is choice-free. No flatness, finite presentation or separatedness hypothesis is imposed.

Facts & Assumptions

Given: A morphism $f ⁣ : X \to S$ of schemes, a point $x \in X$ with $s = f (x)$ , affine opens $Spec B \subseteq X$ and $Spec A \subseteq S$ with $x \in Spec B$ , $f (Spec B) \subseteq Spec A$ and $B$ a finitely generated $A$ -algebra, and $Ω_{X / S, x} = 0$ .

[F1]

Locally finite type and finite type morphisms: $f$ is locally of finite type at $x$ exactly when $x$ has an affine open neighbourhood $U = Spec B$ whose image lies in an affine open $V = Spec A$ of $S$ with $A \to B$ of finite type, that is, $B$ generated as an $A$ -algebra by finitely many elements $b_{1}, \dots, b_{N}$ .

[F2]

Affine charts recover the algebraic module of differentials, Kähler differentials commute with localization, Localisation at a prime ideal: $R_{p} = (R ∖ p)^{- 1} R$ , $R_{p}$ is local with unique maximal ideal $p R_{p}$ , The residue field at a point of an affine scheme and $R_{p} / p R_{p} ≅ Frac (R / p)$ is the residue field at $p$ : on the affine chart $Spec B$ the sheaf $Ω_{X / S}$ is the sheaf attached to $Ω_{B / A}$ , so for the prime $p \subseteq B$ with $x = p$ and $q = p \cap A$ the stalk is $Ω_{X / S, x} ≅ (Ω_{B / A})_{p} ≅ Ω_{B_{p} / A_{q}}, B_{p} = O_{X, x}, A_{q} = O_{S, s} .$ Moreover $B_{p}$ is a local ring with maximal ideal $m : = p B_{p}$ , $m_{s} : = q A_{q}$ is the maximal ideal of the local ring $A_{q}$ , one has $m_{s} B_{p} \subseteq m$ , and $κ (x) ≅ B_{p} / p B_{p} ≅ Frac (B / p), κ (s) ≅ A_{q} / q A_{q} ≅ Frac (A / q) .$

[F3]

Finitely generated field extensions $F (a_{1}, \dots, a_{r})$ : a field extension $K = k (α_{1}, \dots, α_{n})$ generated by finitely many elements is finitely generated; an algebraic finitely generated extension inside a fixed finitely generated one is finite by An extension generated by finitely many algebraic elements is finite.

[F4]

The Axiom of Choice: the Axiom of Choice is assumed in this item; it is consumed by Finite-type field extensions with zero Ω and Assuming the Axiom of Choice, Nakayama's lemma.

[F5]

Conormal exact sequence for an algebra quotient, Transitivity sequence for differential modules and Derivations are maps out of Ω: for a ring map $A^{'} \to P$ and an ideal $I \subseteq P$ with $B^{'} = P / I$ the sequence $I / I^{2} \to B^{'} \otimes_{P} Ω_{P / A^{'}} \to Ω_{B^{'} / A^{'}} \to 0$ is exact; for ring maps $A^{'} \to B^{'} \to C^{'}$ the sequence $C^{'} \otimes_{B^{'}} Ω_{B^{'} / A^{'}} \to Ω_{C^{'} / A^{'}} \to Ω_{C^{'} / B^{'}} \to 0$ is exact; and $Ω_{A^{'} / A^{'}} = 0$ because $Hom (Ω_{A^{'} / A^{'}}, M) ≅ {Der}_{A^{'}} (A^{'}, M) = 0$ for every $A^{'}$ -module $M$ . In particular, if $A^{'} \to B^{'}$ is surjective then $Ω_{B^{'} / A^{'}} = 0$ : apply the conormal sequence to $P = A^{'}$ , $I = \ker (A^{'} \to B^{'})$ .

[F6]

Finite-type field extensions with zero Ω: assuming Choice, a finitely generated field extension with vanishing module of differentials is finite and separable.

[F7]

Separable residue and the cotangent sequence of a local algebra: let $k$ be a field and $R$ 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 exact. If in addition $κ / k$ is finite separable, then $Ω_{κ / k} = 0$ and the first map is an isomorphism $m / m^{2} ≅ Ω_{R / k} \otimes_{R} κ$ .

[F8]

Separating transcendence basis and separably generated extensions: a finitely generated extension admitting a separating transcendence basis is separably generated, and the empty tuple is a separating transcendence basis exactly when the extension is finite separable; so every finite separable extension is separably generated.

[F9]

Kähler differentials commute with scalar base change, A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring and Every quotient and every localisation of a Noetherian ring is Noetherian: for ring maps $A \to B$ , $A \to A^{'}$ there is an isomorphism $Ω_{B / A} \otimes_{B} (B \otimes_{A} A^{'}) ≅ Ω_{(B \otimes_{A} A^{'}) / A^{'}}$ ; a field is a Noetherian ring, every finitely generated algebra over a Noetherian ring is Noetherian, and quotients and localisations of Noetherian rings are Noetherian.

[F10]

Localisation of modules is extension of scalars and $M \otimes_{R} R / I ≅ M / I M$ naturally: for a ring $R$ , multiplicative $S \subseteq R$ and $R$ -module $M$ one has $S^{- 1} M ≅ S^{- 1} R \otimes_{R} M$ , and for an ideal $I \subseteq R$ one has $M \otimes_{R} (R / I) ≅ M / I M$ .

[F11]

Assuming the Axiom of Choice, Nakayama's lemma, The Jacobson radical of a ring and A local ring is a nonzero commutative ring with a unique maximal ideal: assuming Choice, if $I \subseteq J (R)$ and $M$ is a finitely generated $R$ -module with $I M = M$ , then $M = 0$ ; in a local ring $J (R)$ is the unique maximal ideal and the maximal ideal of a nonzero local ring is finitely generated as soon as the ring is Noetherian.

[F12]

Unramified morphism, Formal unramifiedness iff Omega vanishes and Formally etale morphism: $f$ is unramified exactly when it is locally of finite type and $Ω_{X / S} = 0$ ; a morphism is formally unramified exactly when $Ω_{X / S} = 0$ ; and $f$ is formally etale when it is formally smooth and formally unramified, so a formally etale morphism satisfies $Ω_{X / S} = 0$ .

Proof

technique · direct

1.1

The local picture. Let $p \subseteq B$ be the prime with $x = p$ and $q = p \cap A$ , so that $s = f (x)$ corresponds to $q$ . Put $R : = B_{p} = O_{X, x}$ and $A^{'} : = A_{q} = O_{S, s}$ , with maximal ideals $m = p B_{p}$ and $m_{s} = q A_{q}$ . By [F2], $κ (x) = Frac (B / p)$ , $κ (s) = Frac (A / q)$ , $m_{s} R \subseteq m$ , and the hypothesis reads $Ω_{R / A^{'}} = (Ω_{B / A})_{p} = Ω_{X / S, x} = 0$ .

givenF1F2

1.2

Choice. Assume the Axiom of Choice [F4]; it is consumed below only by the two Choice-dependent results [F6] and [F11], while the separable-residue supplier [F7] is choice-free.

givenF4

2.1

The residue extension is finitely generated. By [F1] the $A$ -algebra $B$ is generated by finitely many elements $b_{1}, \dots, b_{N}$ , so $B / p$ is generated as an $A / q$ -algebra, hence as a $κ (s)$ -algebra, by the images of the $b_{i}$ ; therefore $κ (x) = Frac (B / p)$ is a finitely generated field extension of $κ (s)$ in the sense of [F3].

step 1.1F1F3

2.2

The differentials of the residue extension vanish. Apply the conormal sequence [F5] to the ring map $A^{'} \to R$ and the ideal $m \subseteq R$ with $R / m = κ (x)$ : the sequence $m / m^{2} \to κ (x) \otimes_{R} Ω_{R / A^{'}} \to Ω_{κ (x) / A^{'}} \to 0$ is exact, and $Ω_{R / A^{'}} = 0$ by step 1.1, so $Ω_{κ (x) / A^{'}} = 0$ . The structure map $A^{'} \to κ (x)$ factors as $A^{'} \to κ (s) \to κ (x)$ with $A^{'} \to κ (s)$ surjective, and $Ω_{κ (s) / A^{'}} = 0$ by [F5]; the transitivity sequence [F5] for $A^{'} \to κ (s) \to κ (x)$ has first term $Ω_{κ (s) / A^{'}} \otimes_{κ (s)} κ (x) = 0$ and is exact at $Ω_{κ (x) / A^{'}}$ , so the natural map $Ω_{κ (x) / A^{'}} \to Ω_{κ (x) / κ (s)}$ is an isomorphism. Hence $Ω_{κ (x) / κ (s)} = 0$ .

step 1.1F5

2.3

The fibre ring. Put $\overset{ˉ}{R} : = R / m_{s} R$ , $\overset{ˉ}{m} : = m / m_{s} R$ . By [F10], $\overset{ˉ}{R} ≅ R \otimes_{A^{'}} κ (s) = B_{p} \otimes_{A_{q}} κ (s) ≅ (B \otimes_{A} κ (s))_{p}$ , the last isomorphism because localisation is extension of scalars and $κ (s) = A_{q} / q A_{q}$ ; hence $\overset{ˉ}{R}$ is a localisation of the finitely generated $κ (s)$ -algebra $B \otimes_{A} κ (s)$ [F9], so $\overset{ˉ}{R}$ is a Noetherian local $κ (s)$ -algebra with maximal ideal $\overset{ˉ}{m}$ and residue field $\overset{ˉ}{R} / \overset{ˉ}{m} ≅ R / m = κ (x)$ .

step 1.1F9F10

3.1

$κ (x) / κ (s)$ is finite separable. By step 2.1 the extension $κ (x) / κ (s)$ is finitely generated and by step 2.2 it has vanishing module of differentials, so [F6], applied under the Axiom of Choice of step 1.2, shows that $κ (x) / κ (s)$ is finite and separable.

step 1.2step 2.1step 2.2F6

3.2

The differentials of the fibre ring vanish. By [F9], $Ω_{(B \otimes_{A} κ (s)) / κ (s)} ≅ Ω_{B / A} \otimes_{B} (B \otimes_{A} κ (s))$ ; localising at $p$ and using $Ω_{R / A^{'}} = (Ω_{B / A})_{p} = 0$ from step 1.1 together with $\overset{ˉ}{R} ≅ (B \otimes_{A} κ (s))_{p}$ from step 2.3 gives $Ω_{\overset{ˉ}{R} / κ (s)} ≅ (Ω_{B / A})_{p} \otimes_{B_{p}} \overset{ˉ}{R} = 0$ .

step 1.1step 2.3F9

4.1

The cotangent space of the fibre ring vanishes. The field $κ (x)$ is a finite separable extension of $κ (s)$ by step 3.1, hence separably generated over $κ (s)$ by [F8]; the ring $\overset{ˉ}{R}$ is a Noetherian local $κ (s)$ -algebra with residue field $κ (x)$ by step 2.3, so the supplier [F7] applies and the injective cotangent map is an isomorphism $\overset{ˉ}{m} / {\overset{ˉ}{m}}^{2} ≅ Ω_{\overset{ˉ}{R} / κ (s)} \otimes_{\overset{ˉ}{R}} κ (x) = 0$ , the vanishing being step 3.2.

step 2.3step 3.1step 3.2F7F8

5.1

The maximal ideal of the fibre ring is zero. Since $\overset{ˉ}{R}$ is Noetherian [step 2.3], the ideal $\overset{ˉ}{m}$ is finitely generated, and $\overset{ˉ}{m} / {\overset{ˉ}{m}}^{2} = 0$ by step 4.1 means $\overset{ˉ}{m} = {\overset{ˉ}{m}}^{2}$ . As $\overset{ˉ}{m} = J (\overset{ˉ}{R})$ is the Jacobson radical of the local ring $\overset{ˉ}{R}$ [F11], Nakayama's lemma [F11] with $I = M = \overset{ˉ}{m}$ gives $\overset{ˉ}{m} = 0$ .

step 2.3step 4.1F11

6.1

The maximal ideals match. Since $\overset{ˉ}{m} = m / m_{s} R$ is zero by step 5.1, we get $m = m_{s} R$ , that is $m_{f (x)} O_{X, x} = m_{x}$ .

step 1.1step 2.3step 5.1

7.1

Conclusion. Steps 3.1 and 6.1 prove the two assertions under the stated hypotheses. If $f$ is unramified then $Ω_{X / S} = 0$ by [F12], so the hypotheses hold at every point $x$ ; if $f$ is formally etale and locally of finite type then $Ω_{X / S} = 0$ by [F12] and again the hypotheses hold at every point. The Axiom of Choice entered only through [F6] in step 3.1 and [F11] in step 5.1.

step 3.1step 6.1F12∎

DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-09-27Open item page →

Relative differential-rank condition

Definition

Let $f ⁣ : X \to S$ be a morphism of schemes with sheaf of relative differentials $Ω_{X / S}$ (Sheaf of relative Kähler differentials), and let $n \geq 0$ be an integer.

Locally free of constant rank $n$ . An $O_{X}$ -module $F$ is locally free of constant rank $n$ on an open subscheme $U \subseteq X$ when every point $x \in U$ has an open neighbourhood $W \subseteq U$ together with an isomorphism of $O_{W}$ -modules $F ∣_{W} ≅ O_{W}^{\oplus n} .$ Equivalently, $F ∣_{U}$ is a locally free $O_{U}$ -module whose rank function $x \mapsto {rk}_{O_{X, x}} F_{x}$ is constant equal to $n$ on $U$ ; the locally free rank is locally constant, so if $U$ is nonempty and $F ∣_{U}$ is locally free of constant rank $n$ , then $n$ is determined by $U$ and $F$ . On $U = \emptyset$ the condition holds for every $n$ and determines no rank. No quasi-coherence, finiteness or flatness hypothesis on $f$ is built into this definition; the hypothesis is placed on the module $F = Ω_{X / S}$ alone.

Differential rank. The morphism $f$ has differential rank $n$ on the open subscheme $U \subseteq X$ when the restriction $Ω_{X / S} ∣_{U}$ is locally free of constant rank $n$ on $U$ in the sense above. Thus differential rank $0$ on $U$ means that $Ω_{X / S}$ vanishes locally on $U$ , and differential rank $n$ for $n > 0$ means that the module of relative differentials is locally standard of rank $n$ over $U$ .

This condition alone does not define smoothness. Differential rank $n$ is a statement about the first-order infinitesimal structure of $f$ ; it is not a smoothness criterion. In the source treatment the relative-dimension notion smooth of relative dimension $n$ is defined as smoothness together with finiteness and local freeness of constant rank $n$ of $Ω_{X / S}$ , and it is equivalently described by the four hypotheses: locally of finite presentation, flat, all nonempty fibres equidimensional of dimension $n$ , and $Ω_{X / S}$ finite locally free of rank $n$ . None of these four hypotheses beyond the last is built into the definition above, and the comparison of the rank condition with flatness and fibre conditions belongs to the smooth-morphism development of the library rather than to this definition. In particular, no item on this page may conclude smoothness from differential rank alone.

Consistency with standard smooth presentations. The condition is not empty: if $A$ is a commutative ring and $B$ is an $A$ -algebra admitting a standard smooth presentation of relative dimension $n$ (Standard smooth presentations and locally standard smooth maps), that is $B ≅ (A [x_{1}, \dots, x_{N}] / (f_{1}, \dots, f_{c}))_{g}$ with $N - c = n$ and with a $c \times c$ minor of the Jacobian matrix $(\partial f_{j} / \partial x_{i})$ invertible in $B$ , then $Ω_{B / A}$ is a free $B$ -module of rank $n$ , as follows. By Jacobian presentation of Ω applied to the presentation before inverting $g$ , the module $Ω_{(P / I) / A}$ for $P = A [x_{1}, \dots, x_{N}]$ and $I = (f_{1}, \dots, f_{c})$ is the cokernel of the $B^{'}$ -linear map $B^{' c} \to B^{' N}$ (with $B^{'} = P / I$ ) given by the transpose of the row-oriented $c \times N$ Jacobian matrix; localising at $g$ , which commutes with $Ω$ and with forming the cokernel (Kähler differentials commute with localization), presents $Ω_{B / A}$ as the cokernel of this transposed Jacobian over $B$ . Reordering the variables so that the invertible minor occupies the first $c$ columns of the row-oriented Jacobian, write its transpose in vertical blocks $(\begin{array}{c} C \\ D \end{array})$ , with $C \in {G L}_{c} (B)$ and $D \in {Mat}_{N - c, c} (B)$ . Then the map $B^{c} \to B^{N}$ , $u \mapsto (C u, D u)$ has image ${(u^{'}, v^{'}) : u^{'} \in B^{c}, v^{'} = D C^{- 1} u^{'}}$ , and the $B$ -linear map $B^{N} ⟶ B^{N - c}, (u^{'}, v^{'}) ⟼ v^{'} - D C^{- 1} u^{'},$ vanishes on this image and restricts to the identity on the complementary coordinates; hence it induces an isomorphism from the cokernel to $B^{N - c}$ . So $Ω_{B / A}$ is free of rank $N - c = n$ , and the morphism $Spec B \to Spec A$ has differential rank $n$ on its whole chart, with no smoothness hypothesis needed for this computation.

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

The conormal sequence is only right exact

Remark

For a homomorphism of commutative rings $A \to P$ , an ideal $I \subseteq P$ and $B = P / I$ , the conormal sequence $I / I^{2} ⟶ B \otimes_{P} Ω_{P / A} ⟶ Ω_{B / A} ⟶ 0$ of Conormal exact sequence for an algebra quotient is exact at the middle and final terms only: the left arrow is not asserted to be injective, and it need not be. The kernel of that arrow is therefore genuine information about the pair $(I, P)$ , not a defect of the construction.

The smallest witness is $P = k [x]$ with $k$ a field, $I = (x^{2})$ and $B = k [x] / (x^{2})$ . Then $I / I^{2} = (x^{2}) / (x^{4})$ , and the class $[x^{3}]$ is nonzero there, because $x^{3} \notin (x^{4})$ by degrees. Its image under the conormal map is $1 \otimes d (x^{3})$ . By Polynomial differentials are free the module $Ω_{k [x] / k}$ is free on $d x$ with $d g = g^{'} (x) d x$ , so $d (x^{3}) = 3 x^{2} d x$ ; after identifying $B \otimes_{P} Ω_{P / k} ≅ B d x$ this becomes $3 x^{2} d x = 0$ , because $x^{2} = 0$ in $B$ . The class $[x^{3}]$ thus lies in the kernel of the left arrow, in every characteristic: for $p = 3$ the coefficient $3$ is already zero in $k$ , and otherwise the coefficient dies only after passing to $B$ .

Two qualifications. First, the failure is not an artefact of a badly chosen presentation: it depends on the ideal $I$ and not on the number of generators used to describe it. Second, with additional regularity hypotheses on $I$ the left arrow can become injective, so that the sequence starts as a short exact sequence; no such hypothesis is part of the general statement, and the injectivity is never to be used on this page without an explicit regular hypothesis. The companion examples page records the witness above as a counterexample with the same computation.

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

Differential rank alone does not prove smoothness

Remark

The vanishing, or the local freeness of constant rank, of $Ω_{X / S}$ records first-order infinitesimal information about a morphism $f ⁣ : X \to S$ . The differential-rank condition of Relative differential-rank condition is therefore not a smoothness criterion. Vanishing of $Ω_{X / S}$ is equivalent to formal unramifiedness (Formal unramifiedness iff Omega vanishes), a uniqueness statement about square-zero lifts; it does not by itself imply existence of lifts or flatness. It does have further consequences under finiteness hypotheses: if $f$ is locally of finite type, vanishing of $Ω_{X / S}$ makes $f$ unramified (Unramified morphism). The warning here is that differential rank alone does not establish smoothness, not that vanishing differentials carry no geometric information.

The source treatment makes the separation explicitly. Smoothness of a ring map is defined by finite presentation together with a condition on the naive cotangent complex, not by the module of differentials alone; and after defining the relative-dimension condition the Stacks text records that it is not enough to assume that $f$ is flat, of finite presentation, and $Ω_{X / S}$ finite locally free of rank $d$ : a counterexample is given by $Spec (F_{p} [t]) ⟶ Spec (F_{p} [t^{p}]) .$ That morphism is flat of finite presentation with $Ω$ free of rank one, and it is precisely the Frobenius morphism discussed below; the rank of $Ω$ is not the fibre-dimension computation and no regularity of the fibres follows from it.

The same distinction appears at the level of tangent maps. A morphism $F ⁣ : A_{k}^{1} \to A_{k}^{1}$ over $F_{p}$ can have $d F = 0$ as a map between the absolute modules $F^{*} Ω_{A_{k}^{1} / k} \to Ω_{A_{k}^{1} / k}$ (Differential of an S-morphism), so that its dual fibre map vanishes at every point (with the target cotangent space extended to the source residue field), while the relative module $Ω_{A_{k}^{1} / A_{k}^{1}, F}$ of the morphism is nonzero, so that $F$ is not formally unramified and hence not formally etale. The companion examples page records this Frobenius witness; the moral is that a zero map on absolute differentials checks a different module from the one whose vanishing would give formal unramifiedness, and that a rank computation may not be substituted for the flatness, finiteness and fibre hypotheses that enter smoothness. In particular, no item on this page may conclude smoothness from a differential rank computation alone.

5 · Examples, counterexamples and false statements

None yet.