## 28.23 Gabber's result

In this section we prove a result of Gabber which guarantees that on every scheme there exists a cardinal $\kappa$ such that every quasi-coherent module $\mathcal{F}$ is the union of its quasi-coherent $\kappa$-generated subsheaves. It follows that the category of quasi-coherent sheaves on a scheme is a Grothendieck abelian category having limits and enough injectives1.

Definition 28.23.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\kappa$ be an infinite cardinal. We say a sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ is $\kappa$-generated if there exists an open covering $X = \bigcup U_ i$ such that $\mathcal{F}|_{U_ i}$ is generated by a subset $R_ i \subset \mathcal{F}(U_ i)$ whose cardinality is at most $\kappa$.

Note that a direct sum of at most $\kappa$ $\kappa$-generated modules is again $\kappa$-generated because $\kappa \otimes \kappa = \kappa$, see Sets, Section 3.6. In particular this holds for the direct sum of two $\kappa$-generated modules. Moreover, a quotient of a $\kappa$-generated sheaf is $\kappa$-generated. (But the same needn't be true for submodules.)

Lemma 28.23.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\kappa$ be a cardinal. There exists a set $T$ and a family $(\mathcal{F}_ t)_{t \in T}$ of $\kappa$-generated $\mathcal{O}_ X$-modules such that every $\kappa$-generated $\mathcal{O}_ X$-module is isomorphic to one of the $\mathcal{F}_ t$.

Proof. There is a set of coverings of $X$ (provided we disallow repeats). Suppose $X = \bigcup U_ i$ is a covering and suppose $\mathcal{F}_ i$ is an $\mathcal{O}_{U_ i}$-module. Then there is a set of isomorphism classes of $\mathcal{O}_ X$-modules $\mathcal{F}$ with the property that $\mathcal{F}|_{U_ i} \cong \mathcal{F}_ i$ since there is a set of glueing maps. This reduces us to proving there is a set of (isomorphism classes of) quotients $\oplus _{k \in \kappa } \mathcal{O}_ X \to \mathcal{F}$ for any ringed space $X$. This is clear. $\square$

Here is the result the title of this section refers to.

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. $\square$

Proposition 28.23.4. Let $X$ be a scheme.

1. The category $\mathit{QCoh}(\mathcal{O}_ X)$ is a Grothendieck abelian category. Consequently, $\mathit{QCoh}(\mathcal{O}_ X)$ has enough injectives and all limits.

2. The inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ has a right adjoint2

$Q : \textit{Mod}(\mathcal{O}_ X) \longrightarrow \mathit{QCoh}(\mathcal{O}_ X)$

such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism.

Proof. Part (1) means $\mathit{QCoh}(\mathcal{O}_ X)$ (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section 19.10. By Schemes, Section 26.24 colimits in $\mathit{QCoh}(\mathcal{O}_ X)$ exist and agree with colimits in $\textit{Mod}(\mathcal{O}_ X)$. By Modules, Lemma 17.3.2 filtered colimits are exact. Hence (a) and (b) hold. To construct a generator $U$, pick a cardinal $\kappa$ as in Lemma 28.23.3. Pick a collection $(\mathcal{F}_ t)_{t \in T}$ of $\kappa$-generated quasi-coherent sheaves as in Lemma 28.23.2. Set $U = \bigoplus _{t \in T} \mathcal{F}_ t$. Since every object of $\mathit{QCoh}(\mathcal{O}_ X)$ is a filtered colimit of $\kappa$-generated quasi-coherent modules, i.e., of objects isomorphic to $\mathcal{F}_ t$, it is clear that $U$ is a generator. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem 19.11.7 and Lemma 19.13.2.

Proof of (2). To construct $Q$ we use the following general procedure. Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_ X)$ we consider the functor

$\mathit{QCoh}(\mathcal{O}_ X)^{opp} \longrightarrow \textit{Sets},\quad \mathcal{G} \longmapsto \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F})$

This functor transforms colimits into limits, hence is representable, see Injectives, Lemma 19.13.1. Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$ and a functorial isomorphism $\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, Q(\mathcal{F}))$ for $\mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_ X)$. By the Yoneda lemma (Categories, Lemma 4.3.5) the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is functorial in $\mathcal{F}$. By construction $Q$ is a right adjoint to the inclusion functor. The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact that the inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ is fully faithful. $\square$

[1] Nicely explained in a blog post by Akhil Mathew.
[2] This functor is sometimes called the coherator.

Comment #4725 by Herr on

In Proposition 28.23.4. , shouldn't the canonical map be $\mathcal{F}\rightarrow Q(\mathcal{F})$?

Comment #4813 by on

No, because $Q$ is a right adjoint.

Comment #7080 by Matthieu Romagny on

In a blog post here https://www.math.columbia.edu/~dejong/wordpress/?p=1982 you write that "it occurred to me that the exact same results hold for algebraic stacks, with the exact same proof", does this appear presentlyin the Stacks Project?

Comment #7257 by on

Yes, you can find it as Proposition 96.15.2. For some reason this doesn't show up when you search for "coherator" in the search bar. I'm not sure why. Anyway, I have added the keyword in this commit so it will show up in the future (as soon as I update the website again).

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