$n$-abelian and $n$-exact categories

Abstract

We introduce $n$-abelian and $n$-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that $n$-cluster-tilting subcategories of abelian (resp. exact) categories are $n$-abelian (resp. $n$-exact). These results allow to construct several examples of $n$-abelian and $n$-exact categories. Conversely, we prove that $n$-abelian categories satisfying certain mild assumptions can be realized as $n$-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius $n$-exact category has a natural $(n+2)$-angulated structure in the sense of Geiß-Keller-Oppermann. We give several examples of $n$-abelian and $n$-exact categories which have appeared in representation theory, commutative ring theory, commutative and non-commutative algebraic geometry.

58 pages. Corrected an error in Thm 3.20 and Lemma 3.22 and several typos. Accepted for publication in Mathematische Zeitschrift

References in corpus (1)

• ${\tau }^2$-stable tilting complexes over weighted projective lines

Cited by in corpus (27)

• Higher Nakayama algebras I: Construction
• Transport of structure in higher homological algebra
• An introduction to higher Auslander-Reiten theory
• d-Representation-finite self-injective algebras
• Simplicial structures in higher Auslander-Reiten theory
• Classifying exact categories via Wakamatsu tilting
• $n$-cluster tilting subcategories of representation-directed algebras
• Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Brüning
• $d$-abelian quotients of $(d+2)$-angulated categories
• Higher Auslander's formula
• The symplectic geometry of higher Auslander algebras: Symmetric products of disks
• The category of extensions and a characterisation of $n$-exangulated functors
• Maximal ${\tau }_d$-rigid pairs
• $d$-Auslander-Reiten sequences in subcategories
• Gluing of $n$-cluster tilting subcategories for representation-directed algebras
• Recollements for dualizing $k$-varieties and Auslander's formulas
• Tensor products of n-complete algebras
• On higher torsion classes
• Grothendieck groups of $d$-exangulated categories and a modified Caldero-Chapoton map
• Torsion and torsion-free classes from objects of finite type in Grothendieck categories
• Idempotent completions of $n$-exangulated categories
• A resolution theorem for extriangulated categories with applications to the index
• A functorial approach to $n$-abelian categories
• The index in $d$-exact categories
• A characterisation of higher torsion classes
• On the equivalence between the existence of $n$-kernels and $n$-cokernels
• The rigidity of filtered colimits of n-cluster tilting subcategories