Lemma 21.14.3 (0731)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.14: The Leray spectral sequence
Lemma 21.14.3 (cite)

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Lemma 21.14.3. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be a ringed topos. A totally acyclic sheaf is right acyclic for the following functors:

the functor $H^0(U, -)$ for any object $U$ of $\mathcal{C}$ ,
the functor $\mathcal{F} \mapsto \mathcal{F}(K)$ for any presheaf of sets $K$ ,
the functor $\Gamma (\mathcal{C}, -)$ of global sections,
the functor $f_*$ for any morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ of ringed topoi.

Proof. Part (2) is the definition of a totally acyclic sheaf. Part (1) is a consequence of (2) as pointed out in the discussion following the definition of totally acyclic sheaves. Part (3) is a special case of (2) where $K = e$ is the final object of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ .

To prove (4) we may assume, by Modules on Sites, Lemma 18.7.2 that $f$ is given by a morphism of sites. In this case we see that $R^ if_*$ , $i > 0$ of a totally acyclic sheaf are zero by the description of higher direct images in Lemma 21.7.4. $\square$

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