2010/11/24 by Riccardo Colpi, Colpi, Riccardo, Francesca Mantese +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1011.5345
openalex publication_date 2010/11/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category \citeMit. 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 \citeCGM. It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By \citeCGM 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 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.