The Stacks project

Lemma 20.11.6. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be a covering. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Assume that $H^ i(U_{i_0 \ldots i_ p}, \mathcal{F}) = 0$ for all $i > 0$, all $p \geq 0$ and all $i_0, \ldots , i_ p \in I$. Then $\check{H}^ p(\mathcal{U}, \mathcal{F}) = H^ p(U, \mathcal{F})$ as $\mathcal{O}_ X(U)$-modules.

Proof. We will use the spectral sequence of Lemma 20.11.5. The assumptions mean that $E_2^{p, q} = 0$ for all $(p, q)$ with $q \not= 0$. Hence the spectral sequence degenerates at $E_2$ and the result follows. $\square$

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