Homological Systems in Triangulated Categories

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• 作者：O. Mendoza ; V. Santiago
• 关键词：Triangulated categories ; Standardly stratified algebras ; Exceptional sequences ; Homological systems ; Primary ; 18E30 ; 18E40 ; Secondary ; 18G25
• 刊名：Applied Categorical Structures
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：24
• 期：1
• 页码：1-35
• 全文大小：2,536 KB
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• 作者单位：O. Mendoza (1)
  V. Santiago (1)
  
  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, C.P. 04510, México, D.F., Mexico
  
• 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

We introduce the notion of homological systems Θ for triangulated categories. Homological systems generalize, on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional sequences in triangulated categories. We prove that, attached to the homological system Θ, there are two standardly stratified algebras A and B, which are derived equivalent. Furthermore, it is proved that the category \(\mathfrak {F}({\Theta }),\) of the Θ-filtered objects in a triangulated category \(\mathcal {T},\) admits in a very natural way a structure of an exact category, and then there are exact equivalences between the exact category \(\mathfrak {F}({\Theta })\) and the exact categories of the Δ-good modules associated to the standardly stratified algebras A and B. Some of the obtained results can be seen also under the light of the cotorsion pairs in the sense of Iyama-Nakaoka-Yoshino (see 6.6 and 6.7 ). We recall that cotorsion pairs are studied extensively in relation with cluster tilting categories, t-structures and co-t-structures. Keywords Triangulated categories Standardly stratified algebras Exceptional sequences Homological systems