Journal of the
Korean Mathematical Society JKMS

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

Keywords: Bi-complete duality pair, $(\mathcal{L}, \mathcal{A})$-Gorenstein ring, balanced pair, weak global dimension, finitistic dimension, relative Gorenstein flat

MSC numbers: Primary 16E10, 18G25; Secondary 18G10, 16E65

Supported by: The author was fully supported by a CONACyT Postdoctoral Fellowship, managed by the Universidad Michoacana de San Nicolas de Hidalgo.