Gorenstein weak dimension of a coherent power series rings

TL;DR

This paper calculates the Gorenstein weak dimension of coherent power series rings over commutative rings, revealing that a low Gorenstein weak dimension does not necessarily mean the ring is arithmetical.

Contribution

It provides the first computation of Gorenstein weak dimension for coherent power series rings and clarifies its relationship with arithmetical rings.

Findings

Gorenstein weak dimension of coherent power series rings can be explicitly computed.

Having Gorenstein weak dimension ≤ 1 does not imply the ring is arithmetical.

The paper establishes new bounds and properties of Gorenstein dimensions in power series contexts.

Abstract

We compute the Gorenstein weak dimension of a coherent power series rings over a commutative rings and we show that, in general, $\gwd(R) \leq 1$ does not imply that $R$ is an arithmetical ring.