## 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.

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