The Stacks project

Lemma 39.15.5. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that

  1. $U$, $R$ are affine,

  2. there exist $e_ i \in \mathcal{O}_ R(R)$ such that every element $g \in \mathcal{O}_ R(R)$ can be uniquely written as $\sum s^*(f_ i)e_ i$ for some $f_ i \in \mathcal{O}_ U(U)$.

Then every quasi-coherent module $(\mathcal{F}, \alpha )$ on $(U, R, s, t, c)$ is a filtered colimit of finite type quasi-coherent modules.

Proof. The assumption means that $\mathcal{O}_ R(R)$ is a free $\mathcal{O}_ U(U)$-module via $s$ with basis $e_ i$. Hence for any quasi-coherent $\mathcal{O}_ U$-module $\mathcal{G}$ we see that $s^*\mathcal{G}(R) = \bigoplus _ i \mathcal{G}(U)e_ i$. We will write $s(-)$ to indicate pullback of sections by $s$ and similarly for other morphisms. Let $(\mathcal{F}, \alpha )$ be a quasi-coherent module on $(U, R, s, t, c)$. Let $\sigma \in \mathcal{F}(U)$. By the above we can write

\[ \alpha (t(\sigma )) = \sum s(\sigma _ i) e_ i \]

for some unique $\sigma _ i \in \mathcal{F}(U)$ (all but finitely many are zero of course). We can also write

\[ c(e_ i) = \sum \text{pr}_1(f_{ij}) \text{pr}_0(e_ j) \]

as functions on $R \times _{s, U, t}R$. Then the commutativity of the diagram in Definition 39.14.1 means that

\[ \sum \text{pr}_1(\alpha (t(\sigma _ i))) \text{pr}_0(e_ i) = \sum \text{pr}_1(s(\sigma _ i)f_{ij}) \text{pr}_0(e_ j) \]

(calculation omitted). Picking off the coefficients of $\text{pr}_0(e_ l)$ we see that $\alpha (t(\sigma _ l)) = \sum s(\sigma _ i)f_{il}$. Hence the submodule $\mathcal{G} \subset \mathcal{F}$ generated by the elements $\sigma _ i$ defines a finite type quasi-coherent module preserved by $\alpha $. Hence it is a subobject of $\mathcal{F}$ in $\mathit{QCoh}(U, R, s, t, c)$. This submodule contains $\sigma $ (as one sees by pulling back the first relation by $e$). Hence we win. $\square$

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