When the heart of a faithful torsion pair is a module category |
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Authors: | Riccardo Colpi |
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Affiliation: | a Dip. Matematica Pura ed Applicata, Università degli studi di Padova, via Trieste 63, I-35121 Padova, Italyb Dipartimento di Informatica, Università degli Studi di Verona, strada Le Grazie 15, I-37134 Verona, Italy |
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Abstract: | 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 heartH(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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Keywords: | 18E10 18E40 16D90 |
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