The Stacks project

Remark 25.11.2. Let us repackage this information in yet another way. Namely, suppose that $(I, \{ U_ i\} )$ is a hypercovering of the topological space $X$. Given this data we can construct a simplicial topological space $U_\bullet $ by setting

\[ U_ n = \coprod \nolimits _{i \in I_ n} U_ i, \]

and where for given $\varphi : [n] \to [m]$ we let morphisms $U(\varphi ) : U_ n \to U_ m$ be the morphism coming from the inclusions $U_ i \subset U_{\varphi (i)}$ for $i \in I_ n$. This simplicial topological space comes with an augmentation $\epsilon : U_\bullet \to X$. With this morphism the simplicial space $U_\bullet $ becomes a hypercovering of $X$ along which one has cohomological descent in the sense of [Exposé Vbis, SGA4]. In other words, $H^ n(U_\bullet , \epsilon ^*\mathcal{F}) = H^ n(X, \mathcal{F})$. (Insert future reference here to cohomology over simplicial spaces and cohomological descent formulated in those terms.) Suppose that $\mathcal{F}$ is an abelian sheaf on $X$. In this case the spectral sequence of Lemma 25.5.3 becomes the spectral sequence with $E_1$-term

\[ E_1^{p, q} = H^ q(U_ p, \epsilon _ q^*\mathcal{F}) \Rightarrow H^{p + q}(U_\bullet , \epsilon ^*\mathcal{F}) = H^{p + q}(X, \mathcal{F}) \]

comparing the total cohomology of $\epsilon ^*\mathcal{F}$ to the cohomology groups of $\mathcal{F}$ over the pieces of $U_\bullet $. (Insert future reference to this spectral sequence here.)

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