Gorenstein Global Dimensions and Cotorsion Dimension of Rings

TL;DR

This paper establishes an upper bound on the Gorenstein global dimension of commutative rings based on the global cotorsion dimension, extending classical results and applying to specific ring extensions.

Contribution

It generalizes classical homological dimension bounds to Gorenstein dimensions and computes these for trivial and group ring extensions.

Findings

Upper bound on Gorenstein global dimension using cotorsion dimension

Computed Gorenstein global dimension for trivial extensions

Computed Gorenstein global dimension for group rings

Abstract

In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings.