The Stacks project

Lemma 39.14.5. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The category of quasi-coherent modules on $(U, R, s, t, c)$ has colimits.

Proof. Let $i \mapsto (\mathcal{F}_ i, \alpha _ i)$ be a diagram over the index category $\mathcal{I}$. We can form the colimit $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ which is a quasi-coherent sheaf on $U$, see Schemes, Section 26.24. Since colimits commute with pullback we see that $s^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits s^*\mathcal{F}_ i$ and similarly $t^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits t^*\mathcal{F}_ i$. Hence we can set $\alpha = \mathop{\mathrm{colim}}\nolimits \alpha _ i$. We omit the proof that $(\mathcal{F}, \alpha )$ is the colimit of the diagram in the category of quasi-coherent modules on $(U, R, s, t, c)$. $\square$

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