Lemma 78.12.2 (077W)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.12: Quasi-coherent sheaves on groupoids
Lemma 78.12.2 (cite)

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Lemma 78.12.2. Let $B \to S$ as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$ . If $(\mathcal{F}, \alpha )$ is a quasi-coherent module on $(U, R, s, t, c)$ then $\alpha$ is an isomorphism.

Proof. Pull back the commutative diagram of Definition 78.12.1 by the morphism $(i, 1) : R \to R \times _{s, U, t} R$ . Then we see that $i^*\alpha \circ \alpha = s^*e^*\alpha$ . Pulling back by the morphism $(1, i)$ we obtain the relation $\alpha \circ i^*\alpha = t^*e^*\alpha$ . By the second assumption these morphisms are the identity. Hence $i^*\alpha$ is an inverse of $\alpha$ . $\square$

Comments (2)

Comment #6701 by Raymond on November 06, 2021 at 11:24

Perhaps, in this section, $(U,R,s,t,c)$ wants to be a groupoid in algebraic spaces over some $B$ ?

Comment #6906 by Johan on January 19, 2022 at 10:07

Thanks and fixed here.

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