Mathematische Zeitschrift

, Volume 283, Issue 3–4, pp 703–759 | Cite as

n-Abelian and n-exact categories

  • Gustavo JassoEmail author


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.


Abelian category Exact category Triangulated category n-Angulated category Homological algebra Cluster-tilting 

Mathematics Subject Classification

Primary 18E99 Secondary 18E10 18E30 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  2. 2.Mathematik ZentrumUniversität BonnBonnGermany

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