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

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.

**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

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

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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)