Recollements associated to cotorsion pairs over upper triangular matrix rings

Abstract

Let $A$, $B$ be two rings and $T=\left(\begin{array}{ccc}A& M0& B\end{array}\right)$ with $M$ an $A$-$B$-bimodule. Given two complete hereditary cotorsion pairs $({\mathcal{A}}_A,{\mathcal{B}}_A)$ and $({\mathcal{C}}_B,{\mathcal{D}}_B)$ in $A$-Mod and $B$-Mod respectively. We define two cotorsion pairs $(\Phi ({\mathcal{A}}_A,{\mathcal{C}}_B),\mathrm{Rep}({\mathcal{B}}_A,{\mathcal{D}}_B))$ and $(\mathrm{Rep}({\mathcal{A}}_A,{\mathcal{C}}_B),\Psi ({\mathcal{B}}_A,{\mathcal{D}}_B))$ in $T$-Mod and show that both of these cotorsion pairs are complete and hereditary. Given two cofibrantly generated model structures ${\mathcal{M}}_A$ and ${\mathcal{M}}_B$ on $A$-Mod and $B$-Mod respectively. Using the result above, we investigate when there exist a cofibrantly generated model structure ${\mathcal{M}}_T$ on $T$-Mod and a recollement of $\mathrm{Ho}({\mathcal{M}}_T)$ relative to $\mathrm{Ho}({\mathcal{M}}_A)$ and $\mathrm{Ho}({\mathcal{M}}_B)$. Finally, some applications are given in Gorenstein homological algebra.