The Stacks project

Lemma 29.32.12. Let $f : X \to S$ be a morphism of schemes. Assume $f$ is smooth. Then the module of differentials $\Omega _{X/S}$ of $X$ over $S$ is finite locally free and

\[ \text{rank}_ x(\Omega _{X/S}) = \dim _ x(X_{f(x)}) \]

for every $x \in X$.

Proof. The statement is local on $X$ and $S$. By Lemma 29.32.11 above we may assume that $f$ is a standard smooth morphism of affines. In this case the result follows from Algebra, Lemma 10.136.7 (and the definition of a relative global complete intersection, see Algebra, Definition 10.135.5). $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 29.32: Smooth morphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02G1. Beware of the difference between the letter 'O' and the digit '0'.