The Stacks project

Lemma 36.6.13. Let $X$ be a scheme. Let $Z \to X$ be a closed immersion of finite presentation whose conormal sheaf $\mathcal{C}_{Z/X}$ is locally free of rank $c$. Then there is a canonical map

\[ c : \wedge ^ c(\mathcal{C}_{Z/X})^\vee \otimes _{\mathcal{O}_ Z} i^*\mathcal{F} \longrightarrow \mathcal{H}_ Z^ c(\mathcal{F}) \]

functorial in the quasi-coherent module $\mathcal{F}$.

Proof. Follows from the construction in Remark 36.6.10 and the independence of the choice of generators of the ideal sheaf shown in Lemma 36.6.12. Some details omitted. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 36.6: Cohomology with support in a closed subset

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Your email address will not be published.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 0G7T. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 36.6.13 as pdf