Gorenstein projective dimension with respect to a semidualizing module

By D. White. 2006

130 citations recorded by the source corpus.

We introduce and investigate the notion of $\gc$-projective modules over (possibly non-noetherian) commutative rings, where $C$ is a semidualizing module. This extends Holm and J{\o}rgensen's notion of $C$-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite $\gc$-projective dimension, showing in particular that they admit $\gc$-projective approximations, a generalization of the maximal Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local (noetherian) ring, we provide necessary and sufficient conditions for a $G_C$-approximation to be minimal.

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

View Paper: https://www.semanticscholar.org/paper/b0a34c200e6474821674c9c3daabe021a655cbdd

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Citations in the corpus

Tate cohomology with respect to semidualizing modules (2009, methodology): “H. Holm and D. White, Foxby equivalence over associative rings, J. Math.”
Foxby equivalence over associative rings (2006, background): “Recently, Araya-Takahashi-Yoshino [1] extended this definition to a pair of non-commutative, but noetherian rings, while White [28] extended the definition to the non-noetherian, but commutative, setting.”
Gorenstein Projective Dimension Relative to a Semidualizing Bimodule (2013, background): “The following notions were introduced by Holm and Jørgensen in [11] and White in [17] for commutative rings.”
Comparison of relative cohomology theories with respect to semidualizing modules (2007, methodology): “Via the next sequence, the previous display works with [29] to show that M is in P B”
Homological aspects of semidualizing modules (2007, unknown): “These form the building blocks of the so-called GC -projectives, which are studied in depth in [13].”