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| Paper IPM / M / 17650 |
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| Abstract: | |||||
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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.
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