Gorenstein and duality pair over triangular matrix rings

Abstract

Let $A$, $B$ be two rings and $T=\left(\begin{array}{cc}A& M\\ 0& B\end{array}\right)$ 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.