Locally finitely presented Grothendieck categories and the pure semisimplicity conjecture
Ziba Fazelpour, Alireza Nasr-Isfahani
TL;DR
The paper addresses the pure semisimplicity conjecture by establishing Morita-type correspondences between left pure semisimple (and hereditary) rings and locally finitely presented pure semisimple (and hereditary) Grothendieck categories under QF-3 and finite socle hypotheses. It extends Auslander's and Ringel– Tachikawa's classical correspondences to the nonunital, locally unitary setting via functor rings and Hom–tensor adjunctions, proving bijections that classify these rings up to Morita equivalence and relate them to their associated categories of flat modules. A central theme is that the functor rings of these categories satisfy a fixed set of structural properties, which in turn characterize left pure semisimple rings; conversely, rings with these properties yield pure semisimple Grothendieck categories, notably Flat$(R)$, with property (*). The work further shows that the pure semisimplicity conjecture is equivalent to finiteness statements (finite representation type) for the corresponding Grothendieck categories, and it discusses implications for representations of quivers and Dynkin-type graphs, offering a new operational framework for tackling the conjecture. Overall, the results provide a unifying, ring–category perspective that connects homological and representation-theoretic aspects through Morita theory and functor rings.
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 $Λ$ 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.