The stable module category of a general ring

By Daniel Bravo, James Gillespie, Mark Hovey. 2014

134 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bfe922ccb87d6fde0329630be48702d2e6ef6bfd

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In Algebra and Number Theory

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Citations in the corpus

On Ding injective, Ding projective, and Ding flat modules and complexes (2015, background): “It admittedly seems strange to require that HomR(P,F) remain exact for all ﬂat modules F rather than just all projectives. After all, it feels like we are requiring too much. However, it was shown in [BGH13] that the Ding projectives are the coﬁbrant objects of an especially nice Quillen model struc”
Hereditary abelian model categories (2015, methodology): “This is a summary of the paper [11], which grew out of the thesis of Bravo [10].”
Singular compactness and definability for $$\Sigma $$-cotorsion and Gorenstein modules (2018, background): “ral statement which yields PGF⊆⊥RR. Consequently, each module from PGFis Gorenstein projective. On the other hand, each Gorenstein AC-projective module in the sense of Bravo, Gillespie and Hovey (cf. [14]) is a PGF-module. Moreover, if R is left coherent, then PGFis precisely the class of all Gorens”
Finiteness conditions and cotorsion pairs (2015, background): “.e. for all M ∈ FP0. Similarly, Stenstr¨om [Ste70] deﬁnes the class of FP-injective as those modules M such that Ext1 R(F,M) = 0 for all ﬁnitely presented modules F, i.e. for all F ∈ FP1. Recently in [BGH14], the class of absolutely clean modules where deﬁned as those modules M such that Ext1 R(F,M)”