Definition 21.13.4 (072Y)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.13: Totally acyclic sheaves
Definition 21.13.4 (cite)

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Definition 21.13.4. Let $\mathcal{C}$ be a site. We say an abelian sheaf $\mathcal{F}$ is totally acyclic¹ if for every sheaf of sets $K$ we have $H^ p(K, \mathcal{F}) = 0$ for all $p \geq 1$ .

[1] Although this terminology is is used in [Vbis, Proposition 1.3.10, SGA4] this is probably nonstandard notation. In [V, Definition 4.1, SGA4] this property is dubbed “flasque”, but we cannot use this because it would clash with our definition of flasque sheaves on topological spaces. Please email stacks.project@gmail.com if you have a better suggestion.

Comments (2)

Comment #5065 by Remy on April 30, 2020 at 18:58

In SGA IV $_2$ , Exp. V $^{\text{bis}}$ , Prop. 1.3.10, this is called totally acyclic.

Comment #5281 by Johan on June 16, 2020 at 15:46

Thanks this is much better. Changes are here.

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