The Stacks project

Lemma 37.9.9. Let

\[ \xymatrix{ X_1 \ar[d] & X_2 \ar[l]^ f \ar[d] \\ S_1 & S_2 \ar[l] } \]

be a commutative diagram of schemes with $X_2 \to X_1$ and $S_2 \to S_1$ étale. Then the map $c_ f : f^*\Omega _{X_1/S_1} \to \Omega _{X_2/S_2}$ of Morphisms, Lemma 29.32.8 is an isomorphism.

Proof. We recall that an étale morphism $U \to V$ is a smooth morphism with $\Omega _{U/V} = 0$. Using this we see that Morphisms, Lemma 29.32.9 implies $\Omega _{X_2/S_2} = \Omega _{X_2/S_1}$ and Morphisms, Lemma 29.34.16 implies that the map $f^*\Omega _{X_1/S_1} \to \Omega _{X_2/S_1}$ (for the morphism $f$ seen as a morphism over $S_1$) is an isomorphism. Hence the lemma follows. $\square$


Comments (0)


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Your email address will not be published.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 04BX. Beware of the difference between the letter 'O' and the digit '0'.