Transfer of Gorenstein dimensions along ring homomorphisms

TL;DR

This paper advances the understanding of Gorenstein dimensions by providing a resolution-free characterization of Gorenstein injective dimension over local rings and extending classical formulas relating depth and injective dimension.

Contribution

It offers the first resolution-free characterization of Gorenstein injective dimension and generalizes classical depth formulas to this setting.

Findings

Resolution-free characterization of Gorenstein injective dimension.

Formulas relating Gorenstein injective dimension to depth invariant.

Extension of Bass and Chouinard formulas to Gorenstein context.

Abstract

A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard.