Lemma 6.26.4 (0098)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.26: Morphisms of ringed spaces and modules
Lemma 6.26.4 (cite)

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Lemma 6.26.4. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ Y$ -modules. Let $x \in X$ . Then

$(f^*\mathcal{G})_ x = \mathcal{G}_{f(x)} \otimes _{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x}$

as $\mathcal{O}_{X, x}$ -modules where the tensor product on the right uses $f^\sharp _ x : \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ .

Proof. This follows from Lemma 6.20.3 and the identification of the stalks of pullback sheaves at $x$ with the corresponding stalks at $f(x)$ . See the formulae in Section 6.23 for example. $\square$

Comments (2)

Comment #746 by Anfang Zhou on June 27, 2014 at 00:38

Typo. It should be $\mathcal{F}_{f(x)} \otimes_{\mathcal{O}_{Y, f(x)}} \mathcal{O}_{X, x}$

Comment #752 by Johan on June 28, 2014 at 15:48

Thanks for the comment. I have fixed this as well as the other typo you pointed out. See this commit.

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