The Stacks project

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

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

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

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.

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.