n-Abelian and n-exact categories

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• 作者：Gustavo Jasso
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：August 2016
• 年：2016
• 卷：283
• 期：3-4
• 页码：703-759
• 全文大小：5,765 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823
• 卷排序：283

文摘

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.KeywordsAbelian categoryExact categoryTriangulated categoryn-Angulated categoryHomological algebraCluster-tilting