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Paper IPM / M / 17917  


Abstract:  
In this paper, we investigate locally finitely presented pure semisimple (hereditary) Grothendieck categories. We show that every locally finitely presented pure semisimple (resp., hereditary) Grothendieck category $\mathscr{A}$ is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when ${\rm Mod}({\rm fp}(\mathscr{A}))$ is a QF3 category and every representable functor in ${\rm Mod}({\rm fp}(\mathscr{A}))$ has finitely generated essential socle. In fact, we show that there exists a bijection between Morita equivalence classes of left pure semisimple (resp., left hereditary) rings $\Lambda$ and equivalence classes of locally finitely presented pure semisimple (resp., hereditary) Grothendieck categories $\mathscr{A}$ that ${\rm Mod}({\rm fp}(\mathscr{A}))$ is a QF3 category and every representable functor in ${\rm Mod}({\rm fp}(\mathscr{A}))$ has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using AuslanderÃÂ¢??s ideas. We show that there exists, up to equivalence, a bijection between the class of left pure semisimple rings and the class of rings with nice homological properties. These results extend the Auslander and RingelTachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity
conjecture.
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