## Tag `01AG`

Chapter 17: Sheaves of Modules > Section 17.3: The abelian category of sheaves of modules

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. $\square$

The code snippet corresponding to this tag is a part of the file `modules.tex` and is located in lines 148–162 (see updates for more information).

```
\begin{lemma}
\label{lemma-abelian}
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 $\mathcal{G}_x$.
\end{lemma}
\begin{proof}
By Homology, Definition \ref{homology-definition-abelian-category}
we have to show that image and coimage agree. By Sheaves,
Lemma \ref{sheaves-lemma-points-exactness} 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.
\end{proof}
```

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