Lemma 75.5.6. Let $S$ be a scheme. Let $Z \to Y \to X$ be immersions of algebraic spaces. Then there is a canonical exact sequence

where the maps come from Lemma 75.5.3 and $i : Z \to Y$ is the first morphism.

Lemma 75.5.6. Let $S$ be a scheme. Let $Z \to Y \to X$ be immersions of algebraic spaces. Then there is a canonical exact sequence

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

where the maps come from Lemma 75.5.3 and $i : Z \to Y$ is the first morphism.

**Proof.**
Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. Via Lemmas 75.5.2 and 75.5.4 the exactness of the sequence translates immediately into the exactness of the corresponding sequence for the immersions of schemes $Z \times _ X U \to Y \times _ X U \to U$. Hence the lemma follows from Morphisms, Lemma 29.31.5.
$\square$

Your email address will not be published.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.

## Comments (0)