On a Conjecture on the weak global dimension of Gaussian rings

TL;DR

This paper proves Bazzoni and Glaz's conjecture that Gaussian rings have weak global dimension 0, 1, or infinity, except for a specific non-reduced local case.

Contribution

The paper confirms the conjecture for all Gaussian rings except a particular class of non-reduced local rings with nilradical squared to zero.

Findings

Confirmed the conjecture for all Gaussian rings except a specific case.

Identified the remaining case where the conjecture is unresolved.

Extended understanding of the weak global dimension in Gaussian rings.

Abstract

Bazzoni and Glaz conjecture that the weak global dimension of a Gaussian ring is 0,1 or \infty. In this paper, we prove their conjecture in all cases except when R is a non-reduced local Gaussian ring with nilradical $\mathcal{N} satisfying $\mathcal{N}^2=0$.