Gorenstein projective modules and recollements over triangular matrix rings

Abstract

Let $T=\left(\begin{array}{ccc}R& M0& S\end{array}\right)$ 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 ${\mathbb{D}}^b(T\text{-}Mod{)}_{fgp}$ 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 ${\mathbb{D}}_{def}(T\text{-}Mod)$.

To appear in Comm. Algebra