The Stacks project

Lemma 85.25.6. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $\mathcal{F}_ n$ be the pullback to $U_ n$. If $U$ is a proper hypercovering of $X$, then there exists a canonical spectral sequence

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

converging to $H^{p + q}(X, \mathcal{F})$.

Proof. Immediate consequence of Lemmas 85.25.4 and 85.2.10. $\square$

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