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Paper   IPM / M / 17650
School of Mathematics
  Title:   The completion of d-abelian categories
1.  Ramin Ebrahimi
2.  Alireza Nasr-Isfahani
  Status:   Published
  Journal: J. Algebra
  Vol.:  645
  Year:  2024
  Pages:   143-163
  Supported by:  IPM
Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of modA. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d-cluster tilting subcategory in ModA. In this paper, we investigate this question in a more general form. We considerMas a small d-abelian category, known to be equivalent to a d-cluster tilting subcategory of an abelian category A. The completion of M, denoted by Ind(M), is defined as the universal completion of M with respect to filtered colimits. We explore Ind(M) and demonstrate its equivalence to the full subcategory Ld(M) of ModM, comprising left d-exact functors. Notably, while Ind(M) as a subcategory of ModM Eff(M) , satisfies all properties of a d-cluster tilting subcategory except d-rigidity, it falls short of being a d-cluster tilting category. For a d-cluster tilting subcategory M of modA, â??â?? M, consists of all filtered colimits of objects from M, is a generating-cogenerating, functorially finite subcategory of ModA. The question of whether M is a d-rigid subcategory remains unanswered. However, if it is indeed d-rigid, it qualifies as a d-cluster tilting subcategory. In the case d = 2, employing cotorsion theory, we establish that â??â?? M is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether â??â?? Mis a d-cluster tilting subcategory of ModA appears to be equivalent to the Iyamaâ??s qestion about the finiteness of M. Furthermore, for general d, we address the problem and present several equivalent conditions for the Iyamaâ??s question. 2010 Mathematics Subject Classification. 18E10, 18E20, 18E99. Key words and phrases. d-abelian category, d-cluster tilting subcategory, Completion. 1

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