Lemma 21.51.1 (0946)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.51: Weakly contractible objects
Lemma 21.51.1 (cite)

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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 $U$ .

Proof. The first statement follows immediately from the definition (see also Homology, Section 12.7). The higher derived functors vanish by Derived Categories, Lemma 13.16.9. Let $\mathcal{F}$ be a $\mathcal{G}$ -torsor. Then $\mathcal{F} \to *$ is a surjective map of sheaves. Hence (3) follows from the definition as well. $\square$

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