Lemma 25.6.4. Let \mathcal{C} be a site with equalizers and fibre products. Let \mathcal{G} be a presheaf on \mathcal{C}. Let K be a hypercovering of \mathcal{G}. Let \mathcal{F} be a sheaf of abelian groups on \mathcal{C}. There is a map
s(\mathcal{F}(K)) \longrightarrow R\Gamma (\mathcal{G}, \mathcal{F})
in D^{+}(\textit{Ab}) functorial in \mathcal{F}, which induces a natural transformation
\check{H}^ i(K, -) \longrightarrow H^ i(\mathcal{G}, -)
of functors \textit{Ab}(\mathcal{C}) \to \textit{Ab}. Moreover, there is a spectral sequence (E_ r, d_ r)_{r \geq 0} with
E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F}))
converging to H^{p + q}(\mathcal{G}, \mathcal{F}). This spectral sequence is functorial in \mathcal{F} and in the hypercovering K.
Proof.
Choose an injective resolution \mathcal{F} \to \mathcal{I}^\bullet in the category of abelian sheaves on \mathcal{C}. Consider the double complex A^{\bullet , \bullet } with terms
A^{p, q} = \mathcal{I}^ q(K_ p)
where the differential d_1^{p, q} : A^{p, q} \to A^{p + 1, q} is the one coming from the differential \mathcal{I}^ p \to \mathcal{I}^{p + 1} and the differential d_2^{p, q} : A^{p, q} \to A^{p, q + 1} is the one coming from the differential on the complex s(\mathcal{I}^ p(K)) associated to the cosimplicial abelian group \mathcal{I}^ p(K) as explained above. We will use the two spectral sequences ({}'E_ r, {}'d_ r) and ({}''E_ r, {}''d_ r) associated to this double complex, see Homology, Section 12.25.
By Lemma 25.6.3 the complexes s(\mathcal{I}^ p(K)) are acyclic in positive degrees and have H^0 equal to H^0(\mathcal{G}, \mathcal{I}^ p). Hence by Homology, Lemma 12.25.4 and its proof the spectral sequence ({}'E_ r, {}'d_ r) degenerates, and the natural map
H^0(\mathcal{G}, \mathcal{I}^\bullet ) \longrightarrow \text{Tot}(A^{\bullet , \bullet })
is a quasi-isomorphism of complexes of abelian groups. The map s(\mathcal{F}(K)) \longrightarrow R\Gamma (\mathcal{G}, \mathcal{F}) of the lemma is the composition of the natural map s(\mathcal{F}(K)) \to \text{Tot}(A^{\bullet , \bullet }) followed by the inverse of the displayed quasi-isomorphism above. This works because H^0(\mathcal{G}, \mathcal{I}^\bullet ) is a representative of R\Gamma (\mathcal{G}, \mathcal{F}).
Consider the spectral sequence ({}''E_ r, {}''d_ r)_{r \geq 0}. By Homology, Lemma 12.25.1 we see that
{}''E_2^{p, q} = H^ p_{II}(H^ q_ I(A^{\bullet , \bullet }))
In other words, we first take cohomology with respect to d_1 which gives the groups {}''E_1^{p, q} = \underline{H}^ p(\mathcal{F})(K_ q). Hence it is indeed the case (by the description of the differential {}''d_1) that {}''E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F})). Since this spectral sequence converges to the cohomology of \text{Tot}(A^{\bullet , \bullet }) the proof is finished.
\square
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