The Stacks project

59.14 Cohomology

The following is the basic result that makes it possible to define cohomology for abelian sheaves on sites.

Theorem 59.14.1. The category of abelian sheaves on a site is an abelian category which has enough injectives.

Proof. See Modules on Sites, Lemma 18.3.1 and Injectives, Theorem 19.7.4. $\square$

So we can define cohomology as the right-derived functors of the sections functor: if $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $\mathcal{F} \in \textit{Ab}(\mathcal{C})$,

\[ H^ p(U, \mathcal{F}) := R^ p\Gamma (U, \mathcal{F}) = H^ p(\Gamma (U, \mathcal{I}^\bullet )) \]

where $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution. To do this, we should check that the functor $\Gamma (U, -)$ is left exact. This is true and is part of why the category $\textit{Ab}(\mathcal{C})$ is abelian, see Modules on Sites, Lemma 18.3.1. For more general discussion of cohomology on sites (including the global sections functor and its right derived functors), see Cohomology on Sites, Section 21.2.

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Your email address will not be published.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 03NT. Beware of the difference between the letter 'O' and the digit '0'.