n-abelian and n-exact categories

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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 algebra, commutative and non-commutative algebraic geometry.
Original languageEnglish
Pages (from-to)703-759
Number of pages57
JournalMathematische Zeitschrift
Issue number3-4
Publication statusPublished - 2016 Jan
Externally publishedYes

Subject classification (UKÄ)

  • Algebra and Logic


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