The Stacks project

Lemma 76.7.11. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $i : Z \to X$ be an immersion of algebraic spaces over $B$, and assume $i$ (étale locally) has a left inverse. Then the canonical sequence

\[ 0 \to \mathcal{C}_{Z/X} \to i^*\Omega _{X/B} \to \Omega _{Z/B} \to 0 \]

of Lemma 76.7.10 is (étale locally) split exact.

Proof. Clarification: we claim that if $g : X \to Z$ is a left inverse of $i$ over $B$, then $i^*c_ g$ is a right inverse of the map $i^*\Omega _{X/B} \to \Omega _{Z/B}$. Having said this, the result follows from the corresponding result for morphisms of schemes by étale localization, see Lemmas 76.7.3 and 76.5.2. $\square$


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Your email address will not be published.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 05ZB. Beware of the difference between the letter 'O' and the digit '0'.