The Stacks project

Lemma 29.32.10. Let $X \to S$ be a morphism of schemes. Let $g : S' \to S$ be a morphism of schemes. Let $X' = X_{S'}$ be the base change of $X$. Denote $g' : X' \to X$ the projection. Then the map

\[ (g')^*\Omega _{X/S} \to \Omega _{X'/S'} \]

of Lemma 29.32.8 is an isomorphism.

Proof. This is the sheafified version of Algebra, Lemma 10.131.12. $\square$


Comments (3)

Comment #1063 by Charles Rezk on

Suggested slogan: The sheaf of differentials is compatible with base change.

Comment #3275 by Kevin Carlson on

Suggested slogan: Pullback commutes with differentials

Comment #3367 by on

Still not a really catchy slogan... Can't we do better?

There are also:

  • 2 comment(s) on Section 29.32: Sheaf of differentials of a morphism

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Your email address will not be published.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 01V0. Beware of the difference between the letter 'O' and the digit '0'.