Semi-dualizing modules and related Gorenstein homological dimensions

By Henrik Holm, Peter Jørgensen. 2004

175 citations recorded by the source corpus.

Abstract A semi-dualizing module over a commutative noetherian ring A is a finitely generated module C with RHom A ( C , C ) ≃ A in the derived category D ( A ) . We show how each such module gives rise to three new homological dimensions which we call C -Gorenstein projective, C -Gorenstein injective, and C -Gorenstein flat dimension, and investigate the properties of these dimensions.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/93c42cb109aca762da266f682b6976563e4474b9

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Citations in the corpus

Tate cohomology with respect to semidualizing modules (2009, methodology): “Based on the work of Enochs and Jenda [7], the following notions were introduced and studied in this generality by Holm and Jørgensen [14] and White [23].”
Comparison of relative cohomology theories with respect to semidualizing modules (2007, methodology): “Based on the work of Enochs and Jenda [13], the following notions were introduced and studied in this generality by Holm and Jørgensen [21] and White [29].”
Proper Resolutions and Gorenstein Categories (2012, background): “Gorenstein projective and injective modules and some related generalized versions have been studied by many authors, see [1,4,6–8,10–21], and the literatures listed in them.”
Beyond totally reflexive modules and back (2008, background): “An unpublished result of Auslander states that every finitely generated module over a Gorenstein local ring has a totally reflexive cover; see Enochs, Jenda, and Xu [50] for a generalization.”
Gorenstein Projective Dimension Relative to a Semidualizing Bimodule (2013, background): “(1) It was proved by Holm and Jørgensen in [11] for commutative rings, see Step 1 in the proof of [11, Theorem 4.”
Foxby equivalence over associative rings (2006, background): “ctive dimension over Sand ﬁnite IC-injective dimension over R; see 1.6. We now prove some additional properties of the classes FC, PC, and IC. When Ris commutative and noetherian, part (a) appears in [21][(2.14)]. Proposition 5.5. Let SCR be a semidualizing bimodule. (a) The class FC(S) is closed un”
AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules (2008, methodology): “Recently, Holm and Jørgensen [17] have unified these approaches with the GC -projective modules and the GC -flat modules.”
Stability of Gorenstein categories (2007, methodology): “H. Holm and P. Jørgensen, ‘Semi-dualizing modules and related Gorenstein homological dimensions’, J. Pure Appl.”