Abelian quotients of the categories of short exact sequences

Abstract

We mainly investigate abelian quotients of the categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as modules categories. These ideas not only can be used to recover the abelian quotients produced by cluster-tilting subcategories of both exact categories and triangulated categories, but also can be used to reach our goal. Let $(\mathcal{C},\mathcal{E})$ be an exact category. We denote by $\mathcal{E}(\mathcal{C})$ the category of bounded complexes whose objects are given by short exact sequences in $\mathcal{E}$ and by $S\mathcal{E}(\mathcal{C})$ the full subcategory formed by split short exact sequences. In general, $\mathcal{E}(\mathcal{C})$ is just an exact category, but the quotient $\mathcal{E}(\mathcal{C})/[S\mathcal{E}(\mathcal{C})]$ turns out to be abelian. In particular, if $(\mathcal{C},\mathcal{E})$ is Frobenius, we present three equivalent abelian quotients of $\mathcal{E}(\mathcal{C})$ and point out that the equivalences are actually given by left and right rotations. The abelian quotient $\mathcal{E}(\mathcal{C})/[S\mathcal{E}(\mathcal{C})]$ admits some nice properties. We explicitly describe the abelian structure, projective objects, injective objects and simple objects, which provide a new viewpoint to understanding Hilton-Rees Theorem and Auslander-Reiten theory. Furthermore, we present some analogous results both for $n$-exact versions and for triangulated versions.