The Stacks project

Lemma 39.15.4. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume that

  1. $U$, $R$ are Noetherian,

  2. $s, t$ are flat, quasi-compact, and quasi-separated.

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

Proof. We will use the characterization of Cohomology of Schemes, Lemma 30.9.1 of coherent modules on locally Noetherian scheme without further mention. We can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i$ as the filtered colimit of coherent submodules $\mathcal{H}_ i \subset \mathcal{F}$, see Cohomology of Schemes, Lemma 30.10.4. Given a quasi-coherent sheaf $\mathcal{H}$ on $U$ we denote $(s_*t^*\mathcal{H}, \alpha )$ the quasi-coherent sheaf on $(U, R, s, t, c)$ of Lemma 39.15.1. Consider the adjunction map $(\mathcal{F}, \beta ) \to (s_*t^*\mathcal{F}, \alpha )$ in $\mathit{QCoh}(U, R, s, t, c)$, see Remark 39.15.2. Set

\[ (\mathcal{F}_ i, \beta _ i) = (\mathcal{F}, \beta ) \times _{(s_*t^*\mathcal{F}, \alpha )} (s_*t^*\mathcal{H}_ i, \alpha ) \]

in $\mathit{QCoh}(U, R, s, t, c)$. Since restriction to $U$ is an exact functor on $\mathit{QCoh}(U, R, s, t, c)$ by the proof of Lemma 39.14.6 we obtain a pullback diagram

\[ \xymatrix{ \mathcal{F} \ar[r] & s_*t^*\mathcal{F} \\ \mathcal{F}_ i \ar[r] \ar[u] & s_*t^*\mathcal{H}_ i \ar[u] } \]

in other words $\mathcal{F}_ i = \mathcal{F} \cap s_*t^*\mathcal{H}_ i$. By the description of the adjunction map in Remark 39.15.2 this diagram is isomorphic to the diagram

\[ \xymatrix{ \mathcal{F} \ar[r] & s_*s^*\mathcal{F} \\ \mathcal{F}_ i \ar[r] \ar[u] & s_*t^*\mathcal{H}_ i \ar[u] } \]

where the right vertical arrow is the result of appplying $s_*$ to the map

\[ t^*\mathcal{H}_ i \to t^*\mathcal{F} \xrightarrow {\beta } s^*\mathcal{F} \]

This arrow is injective as $t$ is a flat morphism. It follows that $\mathcal{F}_ i$ is coherent by Lemma 39.15.3. Finally, because $s$ is quasi-compact and quasi-separated we see that $s_*$ commutes with colimits (see Cohomology of Schemes, Lemma 30.6.1). Hence $s_*t^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits s_*t^*\mathcal{H}_ i$ and hence $(\mathcal{F}, \beta ) = \mathop{\mathrm{colim}}\nolimits (\mathcal{F}_ i, \beta _ i)$ as desired. $\square$

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