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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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