The Stacks project

Lemma 50.2.2. Consider a commutative diagram of schemes

\[ \xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S } \]

If $X' \to X$ and $S' \to S$ are étale, then the maps discussed above induce isomorphisms $f^*\Omega ^ p_{X/S} \to \Omega ^ p_{X'/S'}$.

Proof. We have $\Omega _{S'/S} = 0$ and $\Omega _{X'/X} = 0$, see for example Morphisms, Lemma 29.36.15. Then by the short exact sequences of Morphisms, Lemmas 29.32.9 and 29.34.16 we see that $\Omega _{X'/S'} = \Omega _{X'/S} = f^*\Omega _{X/S}$. Taking exterior powers we conclude. $\square$


Comments (0)

There are also:

  • 3 comment(s) on Section 50.2: The de Rham complex

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Your email address will not be published.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