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Paper IPM / M / 17650  


Abstract:  
Let A be a finitedimensional algebra, and M be a dcluster tilting subcategory
of modA. From the viewpoint of higher homological algebra, a natural question
to ask is when M induces a dcluster tilting subcategory in ModA. In this paper, we
investigate this question in a more general form. We considerMas a small dabelian category,
known to be equivalent to a dcluster 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 dexact functors. Notably, while
Ind(M) as a subcategory of ModM
Eff(M) , satisfies all properties of a dcluster tilting subcategory
except drigidity, it falls short of being a dcluster tilting category. For a dcluster
tilting subcategory M of modA,
ÃÂ¢??ÃÂ¢??
M, consists of all filtered colimits of objects from M,
is a generatingcogenerating, functorially finite subcategory of ModA. The question of
whether M is a drigid subcategory remains unanswered. However, if it is indeed drigid,
it qualifies as a dcluster tilting subcategory. In the case d = 2, employing cotorsion
theory, we establish that
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M is a 2cluster tilting subcategory if and only if M is of finite
type. Thus, the question regarding whether
ÃÂ¢??ÃÂ¢??
Mis a dcluster tilting subcategory of ModA
appears to be equivalent to the IyamaÃÂ¢??s qestion about the finiteness of M. Furthermore,
for general d, we address the problem and present several equivalent conditions for the
IyamaÃÂ¢??s question.
2010 Mathematics Subject Classification. 18E10, 18E20, 18E99.
Key words and phrases. dabelian category, dcluster tilting subcategory, Completion.
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