Lemma 21.51.1. Let $\mathcal{C}$ be a site. Let $U$ be a weakly contractible object of $\mathcal{C}$. Then

the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is an exact functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$,

$H^ p(U, \mathcal{F}) = 0$ for every abelian sheaf $\mathcal{F}$ and all $p \geq 1$, and

for any sheaf of groups $\mathcal{G}$ any $\mathcal{G}$-torsor has a section over $U$.

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