The Stacks project

Lemma 28.23.3. Let $X$ be a scheme. There exists a cardinal $\kappa $ such that every quasi-coherent module $\mathcal{F}$ is the directed colimit of its quasi-coherent $\kappa $-generated submodules.

Proof. Choose an affine open covering $X = \bigcup _{i \in I} U_ i$. For each pair $i, j$ choose an affine open covering $U_ i \cap U_ j = \bigcup _{k \in I_{ij}} U_{ijk}$. Write $U_ i = \mathop{\mathrm{Spec}}(A_ i)$ and $U_{ijk} = \mathop{\mathrm{Spec}}(A_{ijk})$. Let $\kappa $ be any infinite cardinal $\geq $ than the cardinality of any of the sets $I$, $I_{ij}$.

Let $\mathcal{F}$ be a quasi-coherent sheaf. Set $M_ i = \mathcal{F}(U_ i)$ and $M_{ijk} = \mathcal{F}(U_{ijk})$. Note that

\[ M_ i \otimes _{A_ i} A_{ijk} = M_{ijk} = M_ j \otimes _{A_ j} A_{ijk}. \]

see Schemes, Lemma 26.7.3. Using the axiom of choice we choose a map

\[ (i, j, k, m) \mapsto S(i, j, k, m) \]

which associates to every $i, j \in I$, $k \in I_{ij}$ and $m \in M_ i$ a finite subset $S(i, j, k, m) \subset M_ j$ such that we have

\[ m \otimes 1 = \sum \nolimits _{m' \in S(i, j, k, m)} m' \otimes a_{m'} \]

in $M_{ijk}$ for some $a_{m'} \in A_{ijk}$. Moreover, let's agree that $S(i, i, k, m) = \{ m\} $ for all $i, j = i, k, m$ as above. Fix such a map.

Given a family $\mathcal{S} = (S_ i)_{i \in I}$ of subsets $S_ i \subset M_ i$ of cardinality at most $\kappa $ we set $\mathcal{S}' = (S'_ i)$ where

\[ S'_ j = \bigcup \nolimits _{(i, k, m)\text{ such that }m \in S_ i} S(i, j, k, m) \]

Note that $S_ i \subset S'_ i$. Note that $S'_ i$ has cardinality at most $\kappa $ because it is a union over a set of cardinality at most $\kappa $ of finite sets. Set $\mathcal{S}^{(0)} = \mathcal{S}$, $\mathcal{S}^{(1)} = \mathcal{S}'$ and by induction $\mathcal{S}^{(n + 1)} = (\mathcal{S}^{(n)})'$. Then set $\mathcal{S}^{(\infty )} = \bigcup _{n \geq 0} \mathcal{S}^{(n)}$. Writing $\mathcal{S}^{(\infty )} = (S^{(\infty )}_ i)$ we see that for any element $m \in S^{(\infty )}_ i$ the image of $m$ in $M_{ijk}$ can be written as a finite sum $\sum m' \otimes a_{m'}$ with $m' \in S_ j^{(\infty )}$. In this way we see that setting

\[ N_ i = A_ i\text{-submodule of }M_ i\text{ generated by }S^{(\infty )}_ i \]

we have

\[ N_ i \otimes _{A_ i} A_{ijk} = N_ j \otimes _{A_ j} A_{ijk}. \]

as submodules of $M_{ijk}$. Thus there exists a quasi-coherent subsheaf $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{G}(U_ i) = N_ i$. Moreover, by construction the sheaf $\mathcal{G}$ is $\kappa $-generated.

Let $\{ \mathcal{G}_ t\} _{t \in T}$ be the set of $\kappa $-generated quasi-coherent subsheaves. If $t, t' \in T$ then $\mathcal{G}_ t + \mathcal{G}_{t'}$ is also a $\kappa $-generated quasi-coherent subsheaf as it is the image of the map $\mathcal{G}_ t \oplus \mathcal{G}_{t'} \to \mathcal{F}$. Hence the system (ordered by inclusion) is directed. The arguments above show that every section of $\mathcal{F}$ over $U_ i$ is in one of the $\mathcal{G}_ t$ (because we can start with $\mathcal{S}$ such that the given section is an element of $S_ i$). Hence $\mathop{\mathrm{colim}}\nolimits _ t \mathcal{G}_ t \to \mathcal{F}$ is both injective and surjective as desired. $\square$

Comments (2)

Comment #5884 by Laurent Moret-Bailly on

In the statement, delete the second (or the third) occurrence of "quasi-coherent". Also, I had some trouble understanding the definition of . For more clarity, I think the union should run over triples .

There are also:

  • 4 comment(s) on Section 28.23: Gabber's result

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