Locally finitely presented Grothendieck categories and the pure semisimplicity conjecture

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 A is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when Mod(fp(A)) is a QF-3 category and every representable functor in Mod(fp(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 A that Mod(fp(A)) is a QF-3 category and every representable functor in Mod(fp(A)) has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using Auslander'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 Ringel-Tachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity conjecture.