When the heart of a faithful torsion pair is a module category

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 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.