The stable module category of a general ring

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.

References in corpus (1)

• Deconstructibility and the Hill lemma in Grothendieck categories

Cited by in corpus (23)

• Finiteness conditions and cotorsion pairs
• Frobenius pairs in abelian categories: correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Right Bousfield Localization and Operadic Algebras
• Relative FP-injective and FP-flat complexes and their model structures
• Homological properties of modules with finite weak injective and weak flat dimensions
• Locally type ${\text{FP}}_n$ and $n$-coherent categories
• Cut cotorsion pairs
• Model structures, n-Gorenstein flat modules and PGF dimensions
• The derived category with respect to a generator
• Special precovering classes in comma categories
• Frobenius functors and Gorenstein projective precovers
• Remarks on Gorenstein weak injective and weak flat modules
• Foxby equivalence relative to $C$-weak injective and $C$-weak flat modules
• Model structures and relative Gorenstein flat modules and chain complexes
• Exact model structures and recollements
• When the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module
• Category of n-weak injective and n-weak flat modules with respect to special super presented modules
• Quillen equivalence of singular model categories
• Absolutely Clean, Level, and Gorenstein AC-Injective Complexes
• Gorenstein flat modules with respect to duality pairs
• Tate-Vogel and relative cohomologies of complexes with respect to cotorsion pairs
• Hereditary abelian model categegories