Lemma 10.131.15 (00RY)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.131: Differentials
Lemma 10.131.15 (cite)

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Lemma 10.131.15. Suppose $R \to S$ is of finite presentation. Then $\Omega _{S/R}$ is a finitely presented $S$ -module.

Proof. Write $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ . Write $I = (f_1, \ldots , f_ m)$ . According to Lemma 10.131.9 there is an exact sequence of $S$ -modules

$I/I^2 \to \Omega _{R[x_1, \ldots , x_ n]/R} \otimes _{R[x_1, \ldots , x_ n]} S \to \Omega _{S/R} \to 0$

The result follows from the fact that $I/I^2$ is a finite $S$ -module (generated by the images of the $f_ i$ ), and that the middle term is finite free by Lemma 10.131.14. $\square$

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