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When the heart of a faithful torsion pair is a module category
An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits a...
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Published in: | Journal of pure and applied algebra 2011-12, Vol.215 (12), p.2923-2936 |
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Main Authors: | , , |
Format: | Article |
Language: | English |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
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Summary: | An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964)
[17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. (2007)
[8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007)
[8], the problem simplifies in understanding when, given an associative ring
R
and a faithful torsion pair
(
X
,
Y
)
in the category of right
R
-modules, the
heart
H
(
X
,
Y
)
of the t-structure associated with
(
X
,
Y
)
is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions on
(
X
,
Y
)
for
H
(
X
,
Y
)
to be equivalent to a module category. We analyze in detail the case when
R
is right artinian. |
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ISSN: | 0022-4049 1873-1376 |
DOI: | 10.1016/j.jpaa.2011.04.013 |