Lemma 39.12.2 (03LG)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 39: Groupoid Schemes
Section 39.12: Equivariant quasi-coherent sheaves
Lemma 39.12.2 (cite)

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Lemma 39.12.2. Let $S$ be a scheme. Let $G$ be a group scheme over $S$. Let $f : Y \to X$ be a $G$-equivariant morphism between $S$-schemes endowed with $G$-actions. Then pullback $f^*$ given by $(\mathcal{F}, \alpha ) \mapsto (f^*\mathcal{F}, (1_ G \times f)^*\alpha )$ defines a functor from the category of $G$-equivariant quasi-coherent $\mathcal{O}_ X$-modules to the category of $G$-equivariant quasi-coherent $\mathcal{O}_ Y$-modules.

Proof. Omitted. $\square$

Comments (2)

Comment #4284 by Laurent Moret-Bailly on July 02, 2019 at 08:41

Statement of lemma: why is "quasi-coherent" omitted on the $X$ side?

Comment #4448 by Johan on September 01, 2019 at 10:15

Indeed, weird. Thanks and fixed here.

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