$(\mathcal{F},\mathcal{A})$-Gorenstein flat homological dimensions

Abstract

In this paper we develop the homological properties of the $(\mathcal{L},\mathcal{A})$-Gorenstein flat $R$-modules ${\mathcal{GF}}_{(\mathcal{F}(R),\mathcal{A})}$ proposed by Gillespie. Where the class $\mathcal{A}\subseteq \mathrm{Mod}(R^{op})$ sometimes corresponds to a duality pair $(\mathcal{L},\mathcal{A})$. We study the weak global and finitistic dimensions that comes with ${\mathcal{GF}}_{(\mathcal{F}(R),\mathcal{A})}$ and show that over a $(\mathcal{L},\mathcal{A})$-Gorenstein ring, the functor $-{\otimes }_R-$ is left balanced over $\mathrm{Mod}(R^{op})\times \mathrm{Mod}(R)$ by the classes ${\mathcal{GF}}_{(\mathcal{F}(R^{op}),\mathcal{A})}\times {\mathcal{GF}}_{(\mathcal{F}(R),\mathcal{A})}$. When the duality pair is $(\mathcal{F}(R),\mathcal{F}\mathcal{P}Inj(R^{op}))$ we recover the G. Yang's result over a Ding-Chen ring, and we see that is new for $(\mathrm{Lev}(R),\mathrm{AC}(R^{op}))$ among others.

21 pages