The Stacks project

Lemma 77.13.4. Let $B \to S$ be as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Assume that

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

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

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

Proof. We will use the characterization of Cohomology of Spaces, Lemma 68.12.2 of coherent modules on locally Noetherian algebraic spaces 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 Spaces, Lemma 68.15.1. 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 77.13.1. Consider the adjunction map $(\mathcal{F}, \beta ) \to (s_*t^*\mathcal{F}, \alpha )$ in $\mathit{QCoh}(U, R, s, t, c)$, see Remark 77.13.2. 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 77.12.6 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 77.13.2 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 77.13.3. 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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