Lemma 21.44.5 (08FN)—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 21.44.5 (cite)

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Lemma 21.44.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$ . Given a solid diagram of $\mathcal{O}_ U$ -modules

$\xymatrix{ \mathcal{E} \ar@{..>}[dr] \ar[r] & \mathcal{F} \\ & \mathcal{G} \ar[u]_ p }$

with $\mathcal{E}$ a direct summand of a finite free $\mathcal{O}_ U$ -module and $p$ surjective, then there exists a covering $\{ U_ i \to U\}$ such that a dotted arrow making the diagram commute exists over each $U_ i$ .

Proof. We may assume $\mathcal{E} = \mathcal{O}_ U^{\oplus n}$ for some $n$ . In this case finding the dotted arrow is equivalent to lifting the images of the basis elements in $\Gamma (U, \mathcal{F})$ . This is locally possible by the characterization of surjective maps of sheaves (Sites, Section 7.11). $\square$

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