The stable module category of a general ring

TL;DR

This paper constructs new triangulated categories for any ring R that generalize the stable module category, especially for Gorenstein and quasi-Frobenius rings, using model structures involving modules of type FP-infinity.

Contribution

It introduces stable module categories for general rings via homotopy categories of model structures, extending Gorenstein concepts beyond Noetherian rings.

Findings

Stable categories coincide for Gorenstein rings.

Model structures involve modules of type FP-infinity.

Extended duality between injective and flat modules.

Abstract

For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each other and with the usual stable module category. Our stable module categories are homotopy categories of Quillen model structures on the category of R-modules. These model categories involve generalizations of Gorenstein projective and injective modules that we derive by replacing finitely presented modules by modules of type FP-infinity. Along the way, we extend the perfect duality between injective left modules and flat right modules that holds over Noetherian rings to general rings by considering weaker notions of injectivity and flatness.