Recollements associated to cotorsion pairs over upper triangular matrix rings (1911.02478v1)

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