Lemma 85.2.10. Let $X$ be a simplicial topological space. Let $\mathcal{F}$ be an abelian sheaf on $X$. There is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with
converging to $H^{p + q}(X_{Zar}, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$.
Proof. Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. Consider the double complex with terms
and first differential given by the alternating sum along the maps $d^{p + 1}_ i$-maps $\mathcal{I}_ p^ q \to \mathcal{I}_{p + 1}^ q$, see Lemma 85.2.2. Note that
Hence it follows from Lemma 85.2.9 and Cohomology on Sites, Lemma 21.10.1 that the rows of the double complex are exact in positive degrees and evaluate to $\Gamma (X_{Zar}, \mathcal{I}^ q)$ in degree $0$. On the other hand, since restriction is exact (Lemma 85.2.5) the map
is a resolution. The sheaves $\mathcal{I}_ p^ q$ are injective abelian sheaves on $X_ p$ (Lemma 85.2.6). Hence the cohomology of the columns computes the groups $H^ q(X_ p, \mathcal{F}_ p)$. We conclude by applying Homology, Lemmas 12.25.3 and 12.25.4. $\square$
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