The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

27.23 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 27.23.1. 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 27.23.2. 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 27.23.3. 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 quasi-coherent subsheaves.

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 25.7.3. 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

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

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 27.23.4. 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 25.24 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 27.23.3. Pick a collection $(\mathcal{F}_ t)_{t \in T}$ of $\kappa $-generated quasi-coherent sheaves as in Lemma 27.23.2. 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.

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