Enough injective sheaves of modules

A morphism $\phi :\mathcal{F}\to \mathcal{G}$ of ${\mathcal{O}}_X$-modules is a family of ${\mathcal{O}}_X(U)$-linear maps ${\phi }_U:\mathcal{F}(U)\to \mathcal{G}(U)$ over the open sets of $X$, compatible with restrictions, and it is determined by that family [F1]. Hence $\mathrm{Hom}(\mathcal{F},\mathcal{G})$ is a subset of ${\prod }_U{\mathrm{Hom}}_{{\mathcal{O}}_X(U)}(\mathcal{F}(U),\mathcal{G}(U))$, a product of sets indexed by the set of open subsets of $X$, so it is a set and $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ is locally small.

Let $({\mathcal{F}}_i{)}_{i\in I}$ be a set-indexed family of ${\mathcal{O}}_X$-modules and let $\mathcal{P}$ be the presheaf $U\mapsto {⨁}_{i\in I}{\mathcal{F}}_i(U)$ with componentwise restriction maps and the diagonal action $a\cdot (s_i{)}_i=(a\cdot s_i{)}_i$ of $a\in {\mathcal{O}}_X(U)$; this is a presheaf of ${\mathcal{O}}_X$-modules, so in particular a presheaf of sets, with sheafification $\eta :\mathcal{P}\to a\mathcal{P}$ [F1, F4].

Because $R\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ is a Grothendieck category for every ring $R$, every small filtered colimit of $R$-modules is exact and the filtered colimit functor preserves finite limits; in particular a filtered colimit of short exact sequences of modules over one ring is short exact.

Fix an open $U\subseteq X$, write ${\mathcal{O}}_U={\mathcal{O}}_X{\mathrm{\mid }}_U$ and $G_U:=j_{U!}{\mathcal{O}}_U$, the extension by zero of the structure sheaf restricted to $U$ [F5]. Its sections are $G_U(V)=\{a\in {\mathcal{O}}_X(V\cap U):\mathrm{Supp}(a)\text{ closed in }V\}$, and multiplication by $b\in {\mathcal{O}}_X(V)$ is $(b{\mathrm{\mid }}_{V\cap U})a$; its support remains inside $\mathrm{Supp}(a)$, so this makes $G_U$ an ${\mathcal{O}}_X$-module. For $s\in \mathcal{F}(U)$ and $a\in G_U(V)$, the product $as{\mathrm{\mid }}_{V\cap U}$ is a section of $\mathcal{F}$ on $V\cap U$. On the open complement $V\setminus \mathrm{Supp}(a)$ take the zero section. The two sections agree on their overlap because $a$ vanishes there, so they glue uniquely to a section ${\phi }_V(a)\in \mathcal{F}(V)$. This construction commutes with restriction and is ${\mathcal{O}}_X$-linear, hence defines $\phi :G_U\to \mathcal{F}$ with ${\phi }_U(1_U)=s$. Conversely, on $V\cap U$ any ${\mathcal{O}}_X$-linear morphism with this value sends $a$ to $as{\mathrm{\mid }}_{V\cap U}$, while on $V\setminus \mathrm{Supp}(a)$ it sends the zero section to zero; the same open cover therefore makes it unique. Thus ${\mathrm{Hom}}_{\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)}(G_U,\mathcal{F})\cong \mathcal{F}(U)$.

A Grothendieck category is an abelian category satisfying AB5 and possessing a generator; for a cocomplete abelian category AB5 holds if and only if every small filtered colimit functor is exact.

The Axiom of Choice is available as a hypothesis and will be applied below, where the Grothendieck injective-embedding theorem is invoked for a locally small Grothendieck category.

The presheaf ${\mathcal{P}}^{+}$ of the plus construction carries an ${\mathcal{O}}_X$-module structure for which $\eta$ is linear, and the same holds for $a\mathcal{P}=({\mathcal{P}}^{+}{)}^{+}$: for a germ-compatible presentation $(U_i,s_i)$ over $U$ and $a\in {\mathcal{O}}_X(U)$, the class of $(U_i,a{\mathrm{\mid }}_{U_i}\cdot s_i)$ is independent of the chosen representative because multiplying representatives by $a$ preserves equality of germs, and the class of $(U_i\cap V_j,s_i{\mathrm{\mid }}_{U_i\cap V_j}+t_j{\mathrm{\mid }}_{U_i\cap V_j})$ defines the sum of two presentations; the module axioms hold because they hold sectionwise in the modules $\mathcal{P}(U_i)$ and are compatible with the equivalence relation on germ-compatible presentations. Since a morphism of presheaves of modules $\mathcal{P}\to \mathcal{G}$ into the sheaf $\mathcal{G}$ is in particular a presheaf of sets morphism, [F4] factors it uniquely through $\eta$; the factorisation $a\mathcal{P}\to \mathcal{G}$ is ${\mathcal{O}}_X$-linear because linearity can be checked on the local presentations, which generate every section of $a\mathcal{P}$ over a cover, and on those it is just the linearity of $\mathcal{P}\to \mathcal{G}$. Hence ${\mathrm{Hom}}_{\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)}(a\mathcal{P},\mathcal{G})\cong {\mathrm{Hom}}_{\text{presheaves of modules}}(\mathcal{P},\mathcal{G})\cong {\prod }_{i\in I}{\mathrm{Hom}}_{\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)}({\mathcal{F}}_i,\mathcal{G})$, the second bijection because a compatible family of ${\mathcal{O}}_X(U)$-linear maps out of the direct sums ${⨁}_i{\mathcal{F}}_i(U)$ is the same thing as a family of morphisms ${\mathcal{F}}_i\to \mathcal{G}$. Thus $a\mathcal{P}$ is a coproduct of the family $({\mathcal{F}}_i)$ in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$, so small coproducts exist.

Let $D$ be a small filtered diagram in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ and let $\mathcal{Q}(U):={\mathrm{colim}}_j{\mathcal{F}}_j(U)$ be the objectwise filtered colimit presheaf, with the induced ${\mathcal{O}}_X(U)$-actions; by the same universal-property argument as in step 2.1 applied to filtered colimits instead of coproducts, its sheafification $a\mathcal{Q}$ is the colimit of $D$ in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$. Since sheafification is stalkwise a bijection [F4] and stalks are themselves filtered colimits over the neighbourhood filter, each stalk is $({\mathrm{colim}}_j{\mathcal{F}}_j{)}_x={\mathrm{colim}}_j({\mathcal{F}}_j{)}_x$, the two sides being the filtered colimit of the same diagram of modules indexed by $j$ and a neighbourhood of $x$, with equality of representatives governed by [F9]. Now let a filtered diagram of short exact sequences $0\to {\mathcal{A}}_j\to {\mathcal{B}}_j\to {\mathcal{C}}_j\to 0$ of ${\mathcal{O}}_X$-modules be given. On stalks over $x$ one obtains the filtered colimit of the short exact sequences $0\to ({\mathcal{A}}_j{)}_x\to ({\mathcal{B}}_j{)}_x\to ({\mathcal{C}}_j{)}_x\to 0$ of ${\mathcal{O}}_{X,x}$-modules, which is short exact by step 1.3; hence the stalk sequence of the colimit sequence is exact, and by the stalkwise exactness criterion [F3] the colimit sequence $0\to {\mathrm{colim}}_j{\mathcal{A}}_j\to {\mathrm{colim}}_j{\mathcal{B}}_j\to {\mathrm{colim}}_j{\mathcal{C}}_j\to 0$ is short exact. This holds for every small filtered diagram of short exact sequences, so every small filtered colimit functor on $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ is exact.

Let $\mathcal{G}:={\coprod }_{U\subseteq X\text{ open}}{G}_U$ be the small coproduct of step 1.4, which exists by step 2.1, and let $f\mathrm{\ne }g:\mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }}$ be distinct morphisms of ${\mathcal{O}}_X$-modules. Since morphisms are determined by their components [F1], there are an open $U$ and a section $s\in \mathcal{F}(U)$ with $f_U(s)\mathrm{\ne }{g}_U(s)$. By the bijection of step 1.4 there is $\phi \in \mathrm{Hom}(G_U,\mathcal{F})$ with ${\phi }_U(1_U)=s$, and the coproduct universal property of step 2.1 extends $\phi$ to $\bar{\phi }:\mathcal{G}\to \mathcal{F}$ with $\bar{\phi }{\mathrm{\mid }}_{G_U}=\phi$. Then $(f\circ \bar{\phi }{)}_U(1_U)=f_U(s)\mathrm{\ne }{g}_U(s)=(g\circ \bar{\phi }{)}_U(1_U)$, so $f\circ \bar{\phi }\mathrm{\ne }g\circ \bar{\phi }$; hence the singleton family $\{\mathcal{G}\}$ separates morphisms and $\mathcal{G}$ is a generator of $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$.

Since $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ is abelian [F2], it has finite coproducts and coequalizers, and step 2.1 supplies set-indexed coproducts; a small colimit of a diagram $D$ is the coequalizer of the two canonical maps ${\coprod }_{f\in \mathrm{Mor}(D)}{\mathcal{F}}_{f(0)}\rightrightarrows {\coprod }_{i\in \mathrm{Ob}(D)}{\mathcal{F}}_i$, so $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ is cocomplete and satisfies AB3.

By step 3.1 every small filtered colimit functor on the cocomplete abelian category $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ is exact, so by the AB5 criterion of [F7] the category satisfies AB5.

Steps 1.1, 3.3, 4.1 and 3.2 show that $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ is locally small, abelian, cocomplete, satisfies AB5 and has the generator $\mathcal{G}$; by the definition of a Grothendieck category [F7] it is therefore a locally small Grothendieck category.

By step 5.1 and [F8], whose hypothesis is exactly the Axiom of Choice and the locally small Grothendieck structure just verified, $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ has enough injectives, every ${\mathcal{O}}_X$-module embeds into an injective ${\mathcal{O}}_X$-module, and there is a functorial monomorphism ${\eta }_{\mathcal{F}}:\mathcal{F}\rightarrowtail E(\mathcal{F})$ into an injective object.

Fix an ${\mathcal{O}}_X$-module $\mathcal{F}$. Set $I^0(\mathcal{F}):=E(\mathcal{F})$ and suppose a coaugmented complex $0\to \mathcal{F}\to I^0\to \cdots \to I^n$ of ${\mathcal{O}}_X$-modules has been constructed which is exact at every displayed term except possibly at $I^n$, with all $I^j$ injective. For $n\ge 0$ put $C^n:=\mathrm{coker}(I^{n-1}\to I^n)$, with $I^{-1}$ read as $\mathcal{F}$, and set $I^{n+1}(\mathcal{F}):=E(C^n)$ with the monomorphism $C^n\rightarrowtail I^{n+1}(\mathcal{F})$; the composite $I^n\twoheadrightarrow C^n\rightarrowtail I^{n+1}(\mathcal{F})$ extends the complex by one term and makes it exact at $I^n$, all terms remaining injective. Recursing this construction over $n=0,1,2,\dots$ uses only the fixed functorial embedding $E$ applied to the canonical cokernel of the previously constructed map, so it selects nothing further and yields one specific injective resolution $0\to \mathcal{F}\to I^0(\mathcal{F})\to I^1(\mathcal{F})\to \cdots$ of $\mathcal{F}$.