Lemma 21.10.3. Let \mathcal{C} be a site. Let \mathcal{U} = \{ U_ i \to U\} _{i \in I} be a covering of \mathcal{C}. There is a transformation
of functors \textit{Ab}(\mathcal{C}) \to D^{+}(\mathbf{Z}). In particular this gives a transformation of functors \check{H}^ p(U, \mathcal{F}) \to H^ p(U, \mathcal{F}) for \mathcal{F} ranging over \textit{Ab}(\mathcal{C}).
Proof. Let \mathcal{F} be an abelian sheaf. Choose an injective resolution \mathcal{F} \to \mathcal{I}^\bullet . Consider the double complex \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet ) with terms \check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{I}^ q). Next, consider the associated total complex \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet )), see Homology, Definition 12.18.3. There is a map of complexes
coming from the maps \mathcal{I}^ q(U) \to \check{H}^0(\mathcal{U}, \mathcal{I}^ q) and a map of complexes
coming from the map \mathcal{F} \to \mathcal{I}^0. We can apply Homology, Lemma 12.25.4 to see that \alpha is a quasi-isomorphism. Namely, Lemma 21.10.2 implies that the qth row of the double complex \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet ) is a resolution of \Gamma (U, \mathcal{I}^ q). Hence \alpha becomes invertible in D^{+}(\mathbf{Z}) and the transformation of the lemma is the composition of \beta followed by the inverse of \alpha . We omit the verification that this is functorial. \square
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