Gorenstein projective modules and recollements over triangular matrix rings

TL;DR

This paper characterizes Gorenstein projective modules over triangular matrix rings and explores how recollements of derived categories restrict to subcategories with finite Gorenstein projective dimension, with applications to stable and defect categories.

Contribution

It refines existing results on Gorenstein projective modules over triangular matrix rings and studies the restriction of recollements to subcategories with finite Gorenstein projective dimension.

Findings

Explicit description of Gorenstein projective modules over T

Conditions for recollement restriction to subcategories

Recollements of stable and Gorenstein defect categories

Abstract

Let $T=\left( \begin{array}{cc} R & M 0 & S \end{array} \right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of Enochs, Cort\'{e}s-Izurdiaga and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of $\mathbb{D}^b(T{\text-} Mod)$ restricts to a recollement of its subcategory $\mathbb{D}^b(T{\text-} Mod)_{fgp}$ consisting of complexes with finite Gorenstein projective dimension. As applications, we obtain recollements of the stable category $\underline{T{\text-} GProj}$ and recollements of the Gorenstein defect category $\mathbb{D}_{def}(T{\text-} Mod)$.