Let $(X, \mathcal{O}_ X)$ be a ringed space, see Sheaves, Definition 6.25.1. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_ X$-modules, see Sheaves, Definition 6.10.1. Let $\varphi , \psi : \mathcal{F} \to \mathcal{G}$ be morphisms of sheaves of $\mathcal{O}_ X$-modules. We define $\varphi + \psi : \mathcal{F} \to \mathcal{G}$ to be the map which on each open $U \subset X$ is the sum of the maps induced by $\varphi $, $\psi $. This is clearly again a map of sheaves of $\mathcal{O}_ X$-modules. It is also clear that composition of maps of $\mathcal{O}_ X$-modules is bilinear with respect to this addition. Thus $\textit{Mod}(\mathcal{O}_ X)$ is a pre-additive category, see Homology, Definition 12.3.1.
We will denote $0$ the sheaf of $\mathcal{O}_ X$-modules which has constant value $\{ 0\} $ for all open $U \subset X$. Clearly this is both a final and an initial object of $\textit{Mod}(\mathcal{O}_ X)$. Given a morphism of $\mathcal{O}_ X$-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 open $U$, and (d) $\varphi _ x = 0$ for all $x \in X$. See Sheaves, Lemma 6.16.1.
Moreover, given a pair $\mathcal{F}$, $\mathcal{G}$ of sheaves of $\mathcal{O}_ X$-modules we may define the direct sum as
with obvious maps $(i, j, p, q)$ as in Homology, Definition 12.3.5. Thus $\textit{Mod}(\mathcal{O}_ X)$ is an additive category, see Homology, Definition 12.3.8.
Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of $\mathcal{O}_ X$-modules. We may define $\mathop{\mathrm{Ker}}(\varphi )$ to be the subsheaf of $\mathcal{F}$ with sections
for all open $U \subset X$. It is easy to see that this is indeed a kernel in the category of $\mathcal{O}_ X$-modules. In other words, a morphism $\alpha : \mathcal{H} \to \mathcal{F}$ factors through $\mathop{\mathrm{Ker}}(\varphi )$ if and only if $\varphi \circ \alpha = 0$. Moreover, on the level of stalks we have $\mathop{\mathrm{Ker}}(\varphi )_ x = \mathop{\mathrm{Ker}}(\varphi _ x)$.
On the other hand, we define $\mathop{\mathrm{Coker}}(\varphi )$ as the sheaf of $\mathcal{O}_ X$-modules associated to the presheaf of $\mathcal{O}_ X$-modules defined by the rule
Since taking stalks commutes with taking sheafification, see Sheaves, Lemma 6.17.2 we see that $\mathop{\mathrm{Coker}}(\varphi )_ x = \mathop{\mathrm{Coker}}(\varphi _ x)$. Thus the map $\mathcal{G} \to \mathop{\mathrm{Coker}}(\varphi )$ is surjective (as a map of sheaves of sets), see Sheaves, Section 6.16. To show that this is a cokernel, note that if $\beta : \mathcal{G} \to \mathcal{H}$ is a morphism of $\mathcal{O}_ X$-modules such that $\beta \circ \varphi $ is zero, then you get for every open $U \subset X$ 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.
Lemma 17.3.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. The category $\textit{Mod}(\mathcal{O}_ X)$ is an abelian category. Moreover a complex is exact at $\mathcal{G}$ if and only if for all $x \in X$ the complex is exact at $\mathcal{G}_ x$.
\[ \mathcal{F} \to \mathcal{G} \to \mathcal{H} \]
\[ \mathcal{F}_ x \to \mathcal{G}_ x \to \mathcal{H}_ x \]
Proof. By Homology, Definition 12.5.1 we have to show that image and coimage agree. By Sheaves, Lemma 6.16.1 it is enough to show that image and coimage have the same stalk at every $x \in X$. By the constructions of kernels and cokernels above these stalks are the coimage and image in the categories of $\mathcal{O}_{X, x}$-modules. Thus we get the result from the fact that the category of modules over a ring is abelian. $\square$
Actually the category $\textit{Mod}(\mathcal{O}_ X)$ has many more properties. Here are two constructions we can do.
Given any set $I$ and for each $i \in I$ a $\mathcal{O}_ X$-module we can form the product
which is the sheaf that associates to each open $U$ the product of the modules $\mathcal{F}_ i(U)$. This is also the categorical product, as in Categories, Definition 4.14.6.
Given any set $I$ and for each $i \in I$ a $\mathcal{O}_ X$-module we can form the direct sum
which is the sheafification of the presheaf that associates to each open $U$ the direct sum of the modules $\mathcal{F}_ i(U)$. This is also the categorical coproduct, as in Categories, Definition 4.14.7. To see this you use the universal property of sheafification.
Using these we conclude that all limits and colimits exist in $\textit{Mod}(\mathcal{O}_ X)$.
Lemma 17.3.2. Let $(X, \mathcal{O}_ X)$ be a ringed space.
All limits exist in $\textit{Mod}(\mathcal{O}_ X)$. Limits are the same as the corresponding limits of presheaves of $\mathcal{O}_ X$-modules (i.e., commute with taking sections over opens).
All colimits exist in $\textit{Mod}(\mathcal{O}_ X)$. Colimits are the sheafification of the corresponding colimit in the category of presheaves. Taking colimits commutes with taking stalks.
Filtered colimits are exact.
Finite direct sums are the same as the corresponding finite direct sums of presheaves of $\mathcal{O}_ X$-modules.
Proof. As $\textit{Mod}(\mathcal{O}_ X)$ is abelian (Lemma 17.3.1) it has all finite limits and colimits (Homology, Lemma 12.5.5). Thus the existence of limits and colimits and their description follows from the existence of products and coproducts and their description (see discussion above) and Categories, Lemmas 4.14.11 and 4.14.12. Since sheafification commutes with taking stalks we see that colimits commute with taking stalks. Part (3) signifies that given a system $0 \to \mathcal{F}_ i \to \mathcal{G}_ i \to \mathcal{H}_ i \to 0$ of exact sequences of $\mathcal{O}_ X$-modules over a directed set $I$ the sequence $0 \to \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i \to 0$ is exact as well. Since we can check exactness on stalks (Lemma 17.3.1) this follows from the case of modules which is Algebra, Lemma 10.8.8. We omit the proof of (4). $\square$
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 17.3.3. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces.
The functor $f_* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ Y)$ is left exact. In fact it commutes with all limits.
The functor $f^* : \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X)$ is right exact. In fact it commutes with all colimits.
Pullback $f^{-1} : \textit{Ab}(Y) \to \textit{Ab}(X)$ on abelian sheaves is exact.
Proof. Parts (1) and (2) hold because $(f^*, f_*)$ is an adjoint pair of functors, see Sheaves, Lemma 6.26.2 and Categories, Section 4.24. Part (3) holds because exactness can be checked on stalks (Lemma 17.3.1) and the description of stalks of the pullback, see Sheaves, Lemma 6.22.1. $\square$
Lemma 17.3.4. Let $j : U \to X$ be an open immersion of topological spaces. The functor $j_! : \textit{Ab}(U) \to \textit{Ab}(X)$ is exact.
Proof. Follows from the description of stalks given in Sheaves, Lemma 6.31.6. $\square$
Lemma 17.3.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $I$ be a set. For $i \in I$, let $\mathcal{F}_ i$ be a sheaf of $\mathcal{O}_ X$-modules. For $U \subset X$ quasi-compact open the map is bijective.
\[ \bigoplus \nolimits _{i \in I} \mathcal{F}_ i(U) \longrightarrow \left(\bigoplus \nolimits _{i \in I} \mathcal{F}_ i\right)(U) \]
Proof. If $s$ is an element of the right hand side, then there exists an open covering $U = \bigcup _{j \in J} U_ j$ such that $s|_{U_ j}$ is a finite sum $\sum _{i \in I_ j} s_{ji}$ with $s_{ji} \in \mathcal{F}_ i(U_ j)$. Because $U$ is quasi-compact we may assume that the covering is finite, i.e., that $J$ is finite. Then $I' = \bigcup _{j \in J} I_ j$ is a finite subset of $I$. Clearly, $s$ is a section of the subsheaf $\bigoplus _{i \in I'} \mathcal{F}_ i$. The result follows from the fact that for a finite direct sum sheafification is not needed, see Lemma 17.3.2 above. $\square$
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