Lemma 66.26.1 (03LV)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.26: Sheaves of modules on algebraic spaces
Lemma 66.26.1 (cite)

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Lemma 66.26.1. Let $S$ be a scheme. Let $f : X \to Y$ be an étale morphism of algebraic spaces over $S$ . Then $f^{-1}\mathcal{O}_ Y = \mathcal{O}_ X$ , and $f^*\mathcal{G} = f_{small}^{-1}\mathcal{G}$ for any sheaf of $\mathcal{O}_ Y$ -modules $\mathcal{G}$ . In particular, $f^* : \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X)$ is exact.

Proof. By the description of inverse image in Lemma 66.18.11 and the definition of the structure sheaves it is clear that $f_{small}^{-1}\mathcal{O}_ Y = \mathcal{O}_ X$ . Since the pullback

$f^*\mathcal{G} = f_{small}^{-1}\mathcal{G} \otimes _{f_{small}^{-1}\mathcal{O}_ Y} \mathcal{O}_ X$

by definition we conclude that $f^*\mathcal{G} = f_{small}^{-1}\mathcal{G}$ . The exactness is clear because $f_{small}^{-1}$ is exact, as $f_{small}$ is a morphism of topoi. $\square$

Comments (2)

Comment #7746 by Mingchen on September 10, 2022 at 06:51

Typo: f^* maps Mod(O_Y) to Mod(O_X), not the other direction.

Comment #7991 by Stacks Project on January 14, 2023 at 12:17

Thanks and fixed here.

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