Theorem 59.14.1. The category of abelian sheaves on a site is an abelian category which has enough injectives.
59.14 Cohomology
The following is the basic result that makes it possible to define cohomology for abelian sheaves on sites.
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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