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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