Lemma 77.12.2. Let $B \to S$ as in Section 77.3. 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 77.12.1 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 $\alpha$. $\square$

Comment #6701 by on

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

There are also:

• 2 comment(s) on Section 77.12: Quasi-coherent sheaves on groupoids

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).