CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-27

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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∎

Depends on

Used by

A purely inseparable field has nonzero Omega Counterexample
Zero Frobenius tangent map does not imply formal etaleness Counterexample
Relative differential-rank condition Definition
Differentials of a plane hypersurface Example
Differentials of dual numbers in both characteristics Example
Finite-type field extensions with zero Ω Lemma
Transitivity sequence for schemes Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stacks Algebra 10.131.9, 14–15 (standard reference, not scraped)
Vakil §22.2.3 and §22.2.12 (standard reference, not scraped)