Lemma 68.15.2. Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$ with $Y$ Noetherian. Then every quasi-coherent $\mathcal{O}_ X$-module is a filtered colimit of finitely presented $\mathcal{O}_ X$-modules.

Proof. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Write $f_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i$ with $\mathcal{H}_ i$ a coherent $\mathcal{O}_ Y$-module, see Lemma 68.15.1. By Lemma 68.12.2 the modules $\mathcal{H}_ i$ are $\mathcal{O}_ Y$-modules of finite presentation. Hence $f^*\mathcal{H}_ i$ is an $\mathcal{O}_ X$-module of finite presentation, see Properties of Spaces, Section 65.30. We claim the map

$\mathop{\mathrm{colim}}\nolimits f^*\mathcal{H}_ i = f^*f_*\mathcal{F} \to \mathcal{F}$

is surjective as $f$ is assumed affine, Namely, choose a scheme $V$ and a surjective étale morphism $V \to Y$. Set $U = X \times _ Y V$. Then $U$ is a scheme, $f' : U \to V$ is affine, and $U \to X$ is surjective étale. By Properties of Spaces, Lemma 65.26.2 we see that $f'_*(\mathcal{F}|_ U) = f_*\mathcal{F}|_ V$ and similarly for pullbacks. Thus the restriction of $f^*f_*\mathcal{F} \to \mathcal{F}$ to $U$ is the map

$f^*f_*\mathcal{F}|_ U = (f')^*(f_*\mathcal{F})|_ V) = (f')^*f'_*(\mathcal{F}|_ U) \to \mathcal{F}|_ U$

which is surjective as $f'$ is an affine morphism of schemes. Hence the claim holds.

We conclude that every quasi-coherent module on $X$ is a quotient of a filtered colimit of finitely presented modules. In particular, we see that $\mathcal{F}$ is a cokernel of a map

$\mathop{\mathrm{colim}}\nolimits _{j \in J} \mathcal{G}_ j \longrightarrow \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{H}_ i$

with $\mathcal{G}_ j$ and $\mathcal{H}_ i$ finitely presented. Note that for every $j \in I$ there exist $i \in I$ and a morphism $\alpha : \mathcal{G}_ j \to \mathcal{H}_ i$ such that

$\xymatrix{ \mathcal{G}_ j \ar[r]_\alpha \ar[d] & \mathcal{H}_ i \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{j \in J} \mathcal{G}_ j \ar[r] & \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{H}_ i }$

commutes, see Lemma 68.5.3. In this situation $\mathop{\mathrm{Coker}}(\alpha )$ is a finitely presented $\mathcal{O}_ X$-module which comes endowed with a map $\mathop{\mathrm{Coker}}(\alpha ) \to \mathcal{F}$. Consider the set $K$ of triples $(i, j, \alpha )$ as above. We say that $(i, j, \alpha ) \leq (i', j', \alpha ')$ if and only if $i \leq i'$, $j \leq j'$, and the diagram

$\xymatrix{ \mathcal{G}_ j \ar[r]_\alpha \ar[d] & \mathcal{H}_ i \ar[d] \\ \mathcal{G}_{j'} \ar[r]^{\alpha '} & \mathcal{H}_{i'} }$

commutes. It follows from the above that $K$ is a directed partially ordered set,

$\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{(i, j, \alpha ) \in K} \mathop{\mathrm{Coker}}(\alpha ),$

and we win. $\square$

## 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 07UW. Beware of the difference between the letter 'O' and the digit '0'.