Ascent and descent of Gorenstein homological properties

TL;DR

This paper establishes criteria for when Gorenstein homological properties are preserved or reflected through ring homomorphisms, aiding in understanding Gorenstein conditions across related rings and their module categories.

Contribution

It provides new criteria for the ascent and descent of Gorenstein properties along ring homomorphisms, and characterizes when these induce equivalences of stable categories.

Findings

Criteria for ascent of Gorenstein properties

Criteria for descent of Gorenstein properties

Conditions for triangle equivalences of stable categories

Abstract

Let $\varphi\colon R\rightarrow A$ be a ring homomorphism, where $R$ is a commutative noetherian ring and $A$ is a finite $R$-algebra. We provide criteria for detecting the ascent and descent of Gorenstein homological properties. %As an application, one can deduce a result that supports a question of Avramov and Foxby. We observe that the ascent and descent of Gorenstein homological property can detect the Gorenstein properties of rings along $\varphi$. Finally, we describe when $\varphi$ induces a triangle equivalence between the stable categories of finitely generated Gorenstein projective modules over $R$ and $A$.