n-Abelian and n-exact categories
By Gustavo Jasso. 2014
176 citations recorded by the source corpus.
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 $$(n+2)$$(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 algebra, commutative and non-commutative algebraic geometry.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2ec3c9ac7e8cefb7376768bc8b026435ca23bf0d
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Higher Nakayama algebras I: Construction (2016, methodology): “sh category of ZA 1(resp. the tubes) admits a dZ-cluster-tilting subcategory. Finally, we remark that our constructions yield new examples of d-abelian and (d+ 2)-angulated categories in the sense of [Jas16] and [GKO13] respectively. In a forthcoming article [JK16] we establish the (almost) Koszul p”
- An introduction to higher Auslander–Reiten theory (2016, background): “The intrinsic properties of d-cluster-tilting subcategories have been investigated in [14, 25, 26].”