When an abelian category with a tilting object is equivalent to a module category

Abstract

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category \cite{Mit}. A tilting object in a abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts \cite{CGM}. It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By \cite{CGM} 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 \emph{heart of the $t$-structure} $\H(\X,\Y)$ associated to $(\X,\Y)$ is equivalent to a category of modules. In this paper we give a complete answer to this question, proving necessary and sufficient condition 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.