Beyond totally reflexive modules and back

Abstract

Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein rings in terms of total acyclicity of complexes.

Gorenstein flat dimension of complexes

“…If P → N is a DG-projective resolution, then C g (P ) is Gorenstein flat and H j (P ) = 0 for all j ≥ g + 1. By [6], Theorem 29, GpdC g (P ) = l < ∞. Then C g+l (P ) is Gorenstein projective.…”

A study of quasi-Gorenstein rings

“…Definition 4.1. [9] A finitely generated R-module N is said to be a totally reflexive R-module if it satisfies the following conditions.…”

Pure-injectivity in the category of Gorenstein projective modules

“…As a nice generalization of finitely generated projective modules over commutative noetherian local rings, Auslander and Bridger introduced in [AB] finitely generated modules of Gorenstein dimension zero; and then Enochs and Jenda generalized it in [EJ1] to Gorenstein projective modules (not necessarily finitely generated) and introduced the dual notion-Gorenstein injective modules over general rings. Since then, Gorenstein projective and injective modules and related modules have become very important research objects in Gorenstein homological algebra and representation theory of algebras; see [B1,B2,BK,C1,C2,Ch,ChFH,EJ1,EJ2,Ho1,J,LY,X] and references therein.…”