The Stacks project

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

\[ \mathcal{F} \to \mathcal{G} \to \mathcal{H} \]

is exact at $\mathcal{G}$ if and only if for all $x \in X$ the complex

\[ \mathcal{F}_ x \to \mathcal{G}_ x \to \mathcal{H}_ x \]

is exact at $\mathcal{G}_ x$.

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. $\square$

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