The Stacks project

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

\[ E_1^{p, q} = H^ q(X_ p, \mathcal{F}_ p) \]

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

\[ A^{p, q} = \mathcal{I}^ q(X_ p) \]

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

\[ A^{p, q} = \Gamma (X_ p, \mathcal{I}_ p^ q) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}}(h_{X_ p}, \mathcal{I}^ q) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}}(\mathbf{Z}_{X_ p}, \mathcal{I}^ q) \]

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

\[ \mathcal{F}_ p \to \mathcal{I}_ p^\bullet \]

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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