Stability of Gorenstein categories

By Sean Sather-Wagstaff, Tirdad Sharif, D. White. 2007

155 citations recorded by the source corpus.

We show that an iteration of the procedure used to define the Gorenstein projective modules over a commutative ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective R‐modules G=⋯→∂2GG1→∂1GG0→∂0G⋯ such that the complexes HomR(G, H) and HomR(H, G) are exact for each Gorenstein projective R‐module H, the module Coker (∂1G) is Gorenstein projective. The proof of this result hinges upon our analysis of Gorenstein subcategories of abelian categories.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3d0f2d4d633dc962bbe50a01f186e3531830180f

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Beyond totally reflexive modules and back (2008, background): “Sharif, Sather-Wagstaff, and White [94] study totally acyclic complexes of Gorenstein projective modules.”
Proper Resolutions and Gorenstein Categories (2012, background): “Dually, we define cores C̃ = {M is an object in A | there exists a HomA (−,C )-exact exact sequence 0 → M → C0 → C1 → ·· · → C i → ·· · in A with all C i objects in C } (see [18]).”
Comparison of relative cohomology theories with respect to semidualizing modules (2007, background): “The next definition was first introduced by Auslander and Bridger [ 2 ,3] in the case C = R, and in this generality by Golod [19] and Vasconcelos [28].”
AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules (2008, methodology): “In Section 6 we apply these techniques to continue our study of stability properties of Gorenstein categories, initiated in [23].”