Lemma 21.44.4 (08H3)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.44: Strictly perfect complexes
Lemma 21.44.4 (cite)

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Lemma 21.44.4. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. If $\mathcal{F}^\bullet$ is a strictly perfect complex of $\mathcal{O}_\mathcal {D}$ -modules, then $f^*\mathcal{F}^\bullet$ is a strictly perfect complex of $\mathcal{O}_\mathcal {C}$ -modules.

Proof. We have seen in Modules on Sites, Lemma 18.17.2 that the pullback of a finite free module is finite free. The functor $f^*$ is additive functor hence preserves direct summands. The lemma follows. $\square$

Comments (2)

Comment #1259 by typo on January 13, 2015 at 16:19

By the reference, the result seems to be true for a morphism $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ of ringed topoi.

Maybe should extend the definition of a strictly perfect complex (Tag 08FL) to ringed topoi.

Comment #1270 by Johan on January 22, 2015 at 19:44

OK, thanks. Fixed here.

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