The Stacks project

Lemma 78.13.4. Let $B \to S$ be as in Section 78.3. 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 69.12.2 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 69.15.1. 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 78.13.1. Consider the adjunction map $(\mathcal{F}, \beta ) \to (s_*t^*\mathcal{F}, \alpha )$ in $\mathit{QCoh}(U, R, s, t, c)$, see Remark 78.13.2. 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 78.12.6 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 78.13.2 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 applying $s_*$ 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 78.13.3. 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GPS. Beware of the difference between the letter 'O' and the digit '0'.