Lemma 21.10.5. Let \mathcal{C} be a site. Consider the functor i : \textit{Ab}(\mathcal{C}) \to \textit{PAb}(\mathcal{C}). It is a left exact functor with right derived functors given by
R^ pi(\mathcal{F}) = \underline{H}^ p(\mathcal{F}) : U \longmapsto H^ p(U, \mathcal{F})
see discussion in Section 21.7.
Proof.
It is clear that i is left exact. Choose an injective resolution \mathcal{F} \to \mathcal{I}^\bullet . By definition R^ pi is the pth cohomology presheaf of the complex \mathcal{I}^\bullet . In other words, the sections of R^ pi(\mathcal{F}) over an object U of \mathcal{C} are given by
\frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ n(U) \to \mathcal{I}^{n + 1}(U))}{\mathop{\mathrm{Im}}(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^ n(U))}.
which is the definition of H^ p(U, \mathcal{F}).
\square
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