The Stacks project

Lemma 39.14.2. Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$. 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 39.14.1 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 $. $\square$

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