Relative Gorenstein dimensions over triangular matrix rings

TL;DR

This paper explores the structure of Gorenstein modules over triangular matrix rings, linking properties of the base rings and bimodule to the global and Gorenstein dimensions of the combined ring.

Contribution

It introduces methods to construct and analyze Gorenstein modules over triangular matrix rings using properties of the component rings and bimodules, extending existing theory.

Findings

Describes construction of tilting and semidualizing modules over T

Characterizes G_C-projective modules over T under compatibility conditions

Provides formulas for relative global dimension of T based on A and B

Abstract

Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\begin{pmatrix} A&0\\U&B \end{pmatrix}$ the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over $T$ using the corresponding ones over $A$ and $B$. We show that when $U$ is relative (weakly) compatible we are able to describe the structure of $G_C$-projective modules over $T$. As an application, we study when a morphism in $T$-Mod has a special $G_CP(T)$-precover and when the class $G_CP(T)$ is a special precovering class. In addition, we study the relative global dimension of $T$. In some cases, we show that it can be computed from the relative global dimensions of $A$ and $B$. We end the paper with a counterexample to a result that characterizes when a $T$-module has a finite projective dimension.