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| Paper IPM / M / 17917 |
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| Abstract: | |||||
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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 QF-3 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 QF-3 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 Auslanderâ??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.
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