Gorenstein projective modules and recollements over triangular matrix rings

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 ${}_R{M}_S$ an R–S-bimodule. We describe Gorenstein projective modules over T. In particular, we refine a result of Enochs, Cortés-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 ${\mathbb{D}}^b{(T‐\text{Mod})}_{\mathit{fgp}}$ consisting of complexes with finite Gorenstein projective dimension. As applications, we obtain recollements of the stable category $\underset{¯}{T‐\text{GProj}}$ and recollements of the Gorenstein defect category ${\mathbb{D}}_{\mathit{def}}(T‐\text{Mod}).$

Keywords:

• Gorenstein projective modules
• recollements
• triangulated categories
• triangle-equivalences

2010 Mathematics Subject Classification:

• 18E30
• 16E35
• 18G20

Additional information

Funding

This research was partially supported by NSFC (Grant No. 11501257, 11626179, 11671069, 11701455, 11771212), Qing Lan Project of Jiangsu Province, Jiangsu Government Scholarship for Overseas Studies (JS-2019-328), Shaanxi Province Basic Research Program of Natural Science (Grant No. 2017JQ1012), Natural Science Foundation of Zhejiang Provincial (LY18A010032) and Fundamental Research Funds for the Central Universities (Grant No. JB160703). The authors are grateful to the referee for the valuable comments.