Gorenstein and duality pair over triangular matrix rings

TL;DR

This paper constructs duality pairs for modules over triangular matrix rings and characterizes Gorenstein modules, establishing model structures and recollements that extend previous results in the area.

Contribution

It introduces a semi-complete duality pair for modules over triangular matrix rings and characterizes Gorenstein modules, leading to new model structures and recollement results.

Findings

Constructed a semi-complete duality pair for T-modules.

Characterized Gorenstein D_T-projective, injective, and flat modules.

Established model structures and recollements for the homotopy categories.

Abstract

Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M \\ 0 & B \\\end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. We first construct a semi-complete duality pair $\mathcal{D}_{T}$ of $T$-modules using duality pairs in $A$-Mod and $B$-Mod respectively. Then we characterize when a left $T$-module is Gorenstein $D_{T}$-projective, Gorenstein $D_{T}$-injective or Gorenstein $D_{T}$-flat. These three class of $T$-modules will induce model structures on $T$-Mod. Finally we show that the homotopy category of each of model structures above admits a recollement relative to corresponding stable categories. Our results give new characterizations to earlier results in this direction.