Lemma 18.14.1. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. The category $\textit{Mod}(\mathcal{O})$ is an abelian category. The forgetful functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$ is exact, hence kernels, cokernels and exactness of $\mathcal{O}$-modules, correspond to the corresponding notions for abelian sheaves.
18.14 The abelian category of sheaves of modules
Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}$-modules, see Sheaves, Definition 6.10.1. Let $\varphi , \psi : \mathcal{F} \to \mathcal{G}$ be morphisms of sheaves of $\mathcal{O}$-modules. We define $\varphi + \psi : \mathcal{F} \to \mathcal{G}$ to be the sum of $\varphi $ and $\psi $ as morphisms of abelian sheaves. This is clearly again a map of $\mathcal{O}$-modules. It is also clear that composition of maps of $\mathcal{O}$-modules is bilinear with respect to this addition. Thus $\textit{Mod}(\mathcal{O})$ is a pre-additive category, see Homology, Definition 12.3.1.
We will denote $0$ the sheaf of $\mathcal{O}$-modules which has constant value $\{ 0\} $ for all objects $U$ of $\mathcal{C}$. Clearly this is both a final and an initial object of $\textit{Mod}(\mathcal{O})$. Given a morphism of $\mathcal{O}$-modules $\varphi : \mathcal{F} \to \mathcal{G}$ the following are equivalent: (a) $\varphi $ is zero, (b) $\varphi $ factors through $0$, (c) $\varphi $ is zero on sections over each object $U$.
Moreover, given a pair $\mathcal{F}$, $\mathcal{G}$ of sheaves of $\mathcal{O}$-modules we may define the direct sum as
\[ \mathcal{F} \oplus \mathcal{G} = \mathcal{F} \times \mathcal{G} \]
with obvious maps $(i, j, p, q)$ as in Homology, Definition 12.3.5. Thus $\textit{Mod}(\mathcal{O})$ is an additive category, see Homology, Definition 12.3.8.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of $\mathcal{O}$-modules. We may define $\mathop{\mathrm{Ker}}(\varphi )$ to be the kernel of $\varphi $ as a map of abelian sheaves. By Section 18.3 this is the subsheaf of $\mathcal{F}$ with sections
\[ \mathop{\mathrm{Ker}}(\varphi )(U) = \{ s \in \mathcal{F}(U) \mid \varphi (s) = 0 \text{ in } \mathcal{G}(U)\} \]
for all objects $U$ of $\mathcal{C}$. It is easy to see that this is indeed a kernel in the category of $\mathcal{O}$-modules. In other words, a morphism $\alpha : \mathcal{H} \to \mathcal{F}$ factors through $\mathop{\mathrm{Ker}}(\varphi )$ if and only if $\varphi \circ \alpha = 0$.
Similarly, we define $\mathop{\mathrm{Coker}}(\varphi )$ as the cokernel of $\varphi $ as a map of abelian sheaves. There is a unique multiplication map
\[ \mathcal{O} \times \mathop{\mathrm{Coker}}(\varphi ) \longrightarrow \mathop{\mathrm{Coker}}(\varphi ) \]
such that the map $\mathcal{G} \to \mathop{\mathrm{Coker}}(\varphi )$ becomes a morphism of $\mathcal{O}$-modules (verification omitted). The map $\mathcal{G} \to \mathop{\mathrm{Coker}}(\varphi )$ is surjective (as a map of sheaves of sets, see Section 18.3). To show that $\mathop{\mathrm{Coker}}(\varphi )$ is a cokernel in $\textit{Mod}(\mathcal{O})$, note that if $\beta : \mathcal{G} \to \mathcal{H}$ is a morphism of $\mathcal{O}$-modules such that $\beta \circ \varphi $ is zero, then you get for every object $U$ of $\mathcal{C}$ a map induced by $\beta $ from $\mathcal{G}(U)/\varphi (\mathcal{F}(U))$ into $\mathcal{H}(U)$. By the universal property of sheafification (see Sheaves, Lemma 6.20.1) we obtain a canonical map $\mathop{\mathrm{Coker}}(\varphi ) \to \mathcal{H}$ such that the original $\beta $ is equal to the composition $\mathcal{G} \to \mathop{\mathrm{Coker}}(\varphi ) \to \mathcal{H}$. The morphism $\mathop{\mathrm{Coker}}(\varphi ) \to \mathcal{H}$ is unique because of the surjectivity mentioned above.
Proof. Above we have seen that $\textit{Mod}(\mathcal{O})$ is an additive category, with kernels and cokernels and that $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$ preserves kernels and cokernels. By Homology, Definition 12.5.1 we have to show that image and coimage agree. This is clear because it is true in $\textit{Ab}(\mathcal{C})$. The lemma follows. $\square$
Lemma 18.14.2. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. All limits and colimits exist in $\textit{Mod}(\mathcal{O})$ and the forgetful functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$ commutes with them. Moreover, filtered colimits are exact.
Proof. The final statement follows from the first as filtered colimits are exact in $\textit{Ab}(\mathcal{C})$ by Lemma 18.3.2. Let $\mathcal{I} \to \textit{Mod}(\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Let $\mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$ be the limit of the diagram in $\textit{Ab}(\mathcal{C})$. By the description of this limit in Lemma 18.3.2 we see immediately that there exists a multiplication
\[ \mathcal{O} \times \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i \longrightarrow \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i \]
which turns $\mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$ into a sheaf of $\mathcal{O}$-modules. It is easy to see that this is the limit of the diagram in $\textit{Mod}(\mathcal{C})$. Let $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ be the colimit of the diagram in $\textit{PAb}(\mathcal{C})$. By the description of this colimit in the proof of Lemma 18.2.1 we see immediately that there exists a multiplication
\[ \mathcal{O} \times \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i \]
which turns $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ into a presheaf of $\mathcal{O}$-modules. Applying sheafification we get a sheaf of $\mathcal{O}$-modules $(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i)^\# $, see Lemma 18.11.1. It is easy to see that $(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i)^\# $ is the colimit of the diagram in $\textit{Mod}(\mathcal{O})$, and by Lemma 18.3.2 forgetting the $\mathcal{O}$-module structure is the colimit in $\textit{Ab}(\mathcal{C})$. $\square$
The existence of limits and colimits allows us to consider exactness properties of functors defined on the category of $\mathcal{O}$-modules in terms of limits and colimits, as in Categories, Section 4.23. See Homology, Lemma 12.7.2 for a description of exactness properties in terms of short exact sequences.
Lemma 18.14.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi.
The functor $f_*$ is left exact. In fact it commutes with all limits.
The functor $f^*$ is right exact. In fact it commutes with all colimits.
Proof. This is true because $(f^*, f_*)$ is an adjoint pair of functors, see Lemma 18.13.2. See Categories, Section 4.24. $\square$
Lemma 18.14.4. Let $\mathcal{C}$ be a site. If $\{ p_ i\} _{i \in I}$ is a conservative family of points, then we may check exactness of a sequence of abelian sheaves on the stalks at the points $p_ i$, $i \in I$. If $\mathcal{C}$ has enough points, then exactness of a sequence of abelian sheaves may be checked on stalks.
Proof. This is immediate from Sites, Lemma 7.38.2. $\square$
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