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.

References in corpus (10)

• The Auslander-Reiten Translation in Submodule Categories
• Submodule Categories of Wild Representation Type
• An introduction to higher Auslander-Reiten theory
• Quotients of exact categories by cluster tilting subcategories as module categories
• From triangulated categories to module categories via localisation II: Calculus of fractions
• From triangulated categories to module categories via localisation
• Projectively generated $d$-abelian categories are $d$-cluster tilting
• Auslander-Reiten duality for subcategories
• From submodule categories to preprojective algebras
• From triangulated categories to abelian categories--cluster tilting in a general framework

Cited by in corpus (1)

• Recollements, comma categories and morphic enhancements