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

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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.

Autor: Riccardo Colpi; Francesca Mantese; Alberto Tonolo

Fuente: https://archive.org/

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