## 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 algebra, commutative and non-commutative algebraic geometry.

Original language | English |
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Pages (from-to) | 703-759 |

Number of pages | 57 |

Journal | Mathematische Zeitschrift |

Volume | 283 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2016 Jan |

Externally published | Yes |

## Subject classification (UKÄ)

- Algebra and Logic

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