Torsion Pairs and Simple-Minded Systems in Triangulated Categories

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• 作者：Alex Dugas
• 关键词：Simple ; minded system ; Mutation ; Torsion pair ; Derived equivalence ; Stable equivalence
• 刊名：Applied Categorical Structures
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：23
• 期：3
• 页码：507-526
• 全文大小：466 KB
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• 作者单位：Alex Dugas (1)
  
  1. Department of Mathematics, University of the Pacific, 3601 Pacific Ave, Stockton, CA, 95211, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Logic and Foundations
  Theory of Computation
  Convex and Discrete Geometry
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9095

文摘

Let e a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of We construct torsion pairs in ssociated to any subset f a simple-minded system and use these to define left and right mutations of elative to When as a Serre functor ν and nd re invariant under ν 1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in D b (mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to ields the images of the simple Γ-modules under a stable equivalence mod-Γ od-Λ, where Γ is the tilting mutation of Λ relative to