Lemma 21.10.7. Let $\mathcal{C}$ be a site. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a covering. Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Ab}(\mathcal{C}))$. Assume that $H^ i(U_{i_0} \times _ U \ldots \times _ U U_{i_ p}, \mathcal{F}) = 0$ for all $i > 0$, all $p \geq 0$ and all $i_0, \ldots , i_ p \in I$. Then $\check{H}^ p(\mathcal{U}, \mathcal{F}) = H^ p(U, \mathcal{F})$.
Proof. We will use the spectral sequence of Lemma 21.10.6. 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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