Lemma 20.11.6. Let X be a ringed space. Let \mathcal{U} : U = \bigcup _{i \in I} U_ i be an open 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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