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68.15 Limits of coherent modules

A colimit of coherent modules (on a locally Noetherian algebraic space) is typically not coherent. But it is quasi-coherent as any colimit of quasi-coherent modules on an algebraic space is quasi-coherent, see Properties of Spaces, Lemma 65.29.7. Conversely, if the algebraic space is Noetherian, then every quasi-coherent module is a filtered colimit of coherent modules.

Lemma 68.15.1. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Every quasi-coherent $\mathcal{O}_ X$-module is the filtered colimit of its coherent submodules.

Proof. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $\mathcal{G}, \mathcal{H} \subset \mathcal{F}$ are coherent $\mathcal{O}_ X$-submodules then the image of $\mathcal{G} \oplus \mathcal{H} \to \mathcal{F}$ is another coherent $\mathcal{O}_ X$-submodule which contains both of them (see Lemmas 68.12.3 and 68.12.4). In this way we see that the system is directed. Hence it now suffices to show that $\mathcal{F}$ can be written as a filtered colimit of coherent modules, as then we can take the images of these modules in $\mathcal{F}$ to conclude there are enough of them.

Let $U$ be an affine scheme and $U \to X$ a surjective étale morphism. Set $R = U \times _ X U$ so that $X = U/R$ as usual. By Properties of Spaces, Proposition 65.32.1 we see that $\mathit{QCoh}(\mathcal{O}_ X) = \mathit{QCoh}(U, R, s, t, c)$. Hence we reduce to showing the corresponding thing for $\mathit{QCoh}(U, R, s, t, c)$. Thus the result follows from the more general Groupoids, Lemma 39.15.4. $\square$

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. $\square$


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