The Stacks project

Lemma 39.15.6. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $\kappa $ be a cardinal. There exists a set $T$ and a family $(\mathcal{F}_ t, \alpha _ t)_{t \in T}$ of $\kappa $-generated quasi-coherent modules on $(U, R, s, t, c)$ such that every $\kappa $-generated quasi-coherent module on $(U, R, s, t, c)$ is isomorphic to one of the $(\mathcal{F}_ t, \alpha _ t)$.

Proof. For each quasi-coherent module $\mathcal{F}$ on $U$ there is a (possibly empty) set of maps $\alpha : t^*\mathcal{F} \to s^*\mathcal{F}$ such that $(\mathcal{F}, \alpha )$ is a quasi-coherent modules on $(U, R, s, t, c)$. By Properties, Lemma 28.23.2 there exists a set of isomorphism classes of $\kappa $-generated quasi-coherent $\mathcal{O}_ U$-modules. $\square$

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