Lemma 19.8.1. The functor $\mathcal{F} \mapsto \mathcal{F}^\vee $ is exact.
19.8 Modules on a ringed site
Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. By analogy with More on Algebra, Section 15.55 let us try to prove that there are enough injective $\mathcal{O}$-modules. First of all, we pick an injective embedding
\[ \bigoplus \nolimits _{U, \mathcal{I}} j_{U!}\mathcal{O}_ U/\mathcal{I} \longrightarrow \mathcal{J} \]
where $\mathcal{J}$ is an injective abelian sheaf (which exists by the previous section). Here the direct sum is over all objects $U$ of $\mathcal{C}$ and over all $\mathcal{O}$-submodules $\mathcal{I} \subset j_{U!}\mathcal{O}_ U$. Please see Modules on Sites, Section 18.19 to read about the functors restriction and extension by $0$ for the localization functor $j_ U : \mathcal{C}/U \to \mathcal{C}$.
For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ denote
\[ \mathcal{F}^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{J}) \]
with its natural $\mathcal{O}$-module structure. Insert here future reference to internal hom. We will also need a canonical flat resolution of a sheaf of $\mathcal{O}$-modules. This we can do as follows: For any $\mathcal{O}$-module $\mathcal{F}$ we denote
\[ F(\mathcal{F}) = \bigoplus \nolimits _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}), s \in \mathcal{F}(U)} j_{U!}\mathcal{O}_ U. \]
This is a flat sheaf of $\mathcal{O}$-modules which comes equipped with a canonical surjection $F(\mathcal{F}) \to \mathcal{F}$, see Modules on Sites, Lemma 18.28.8. Moreover the construction $\mathcal{F} \mapsto F(\mathcal{F})$ is functorial in $\mathcal{F}$.
Proof. This because $\mathcal{J}$ is an injective abelian sheaf. $\square$
There is a canonical map $ev : \mathcal{F} \to (\mathcal{F}^\vee )^\vee $ given by evaluation: given $x \in \mathcal{F}(U)$ we let $ev(x) \in (\mathcal{F}^\vee )^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}^\vee , \mathcal{J})$ be the map $\varphi \mapsto \varphi (x)$.
Lemma 19.8.2. For any $\mathcal{O}$-module $\mathcal{F}$ the evaluation map $ev : \mathcal{F} \to (\mathcal{F}^\vee )^\vee $ is injective.
Proof. You can check this using the definition of $\mathcal{J}$. Namely, if $s \in \mathcal{F}(U)$ is not zero, then let $j_{U!}\mathcal{O}_ U \to \mathcal{F}$ be the map of $\mathcal{O}$-modules it corresponds to via adjunction. Let $\mathcal{I}$ be the kernel of this map. There exists a nonzero map $\mathcal{F} \supset j_{U!}\mathcal{O}_ U/\mathcal{I} \to \mathcal{J}$ which does not annihilate $s$. As $\mathcal{J}$ is an injective $\mathcal{O}$-module, this extends to a map $\varphi : \mathcal{F} \to \mathcal{J}$. Then $ev(s)(\varphi ) = \varphi (s) \not= 0$ which is what we had to prove. $\square$
The canonical surjection $F(\mathcal{F}) \to \mathcal{F}$ of $\mathcal{O}$-modules turns into a canonical injection, see above, of $\mathcal{O}$-modules
\[ (\mathcal{F}^\vee )^\vee \longrightarrow (F(\mathcal{F}^\vee ))^\vee . \]
Set $J(\mathcal{F}) = (F(\mathcal{F}^\vee ))^\vee $. The composition of $ev$ with this the displayed map gives $\mathcal{F} \to J(\mathcal{F})$ functorially in $\mathcal{F}$.
Lemma 19.8.3. Let $\mathcal{O}$ be a sheaf of rings. For every $\mathcal{O}$-module $\mathcal{F}$ the $\mathcal{O}$-module $J(\mathcal{F})$ is injective.
Proof. We have to show that the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, J(\mathcal{F}))$ is exact. Note that
\begin{eqnarray*} \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, J(\mathcal{F})) & = & \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, (F(\mathcal{F}^\vee ))^\vee ) \\ & = & \mathop{\mathrm{Hom}}\nolimits _\mathcal {O} (\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(\mathcal{F}^\vee ), \mathcal{J})) \\ & = & \mathop{\mathrm{Hom}}\nolimits (\mathcal{G} \otimes _\mathcal {O} F(\mathcal{F}^\vee ), \mathcal{J}) \end{eqnarray*}
Thus what we want follows from the fact that $F(\mathcal{F}^\vee )$ is flat and $\mathcal{J}$ is injective. $\square$
Theorem 19.8.4. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. The category of sheaves of $\mathcal{O}$-modules on a site has enough injectives. In fact there exists a functorial injective embedding, see Homology, Definition 12.27.5.
Proof. From the discussion in this section. $\square$
Proposition 19.8.5. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. The category $\textit{PMod}(\mathcal{O})$ of presheaves of $\mathcal{O}$-modules has functorial injective embeddings.
Proof. We could prove this along the lines of the discussion in Section 19.6. But instead we argue using the theorem above. Endow $\mathcal{C}$ with the structure of a site by letting the set of coverings of an object $U$ consist of all singletons $\{ f : V \to U\} $ where $f$ is an isomorphism. We omit the verification that this defines a site. A sheaf for this topology is the same as a presheaf (proof omitted). Hence the theorem applies. $\square$
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