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

TL;DR

This paper develops the theory of $(rak{F},rak{A})$-Gorenstein flat modules, exploring their homological properties, dimensions, and balanced functors, extending known results to new classes of rings and duality pairs.

Contribution

It introduces and analyzes $(rak{F},rak{A})$-Gorenstein flat modules, generalizing existing homological frameworks and establishing new balanced functor properties for various duality pairs.

Findings

Over a $(rak{L},rak{A})$-Gorenstein ring, the tensor functor is left balanced with specific Gorenstein flat classes.

Recovers G. Yang's results over Ding-Chen rings when the duality pair is $(rak{F}(R), ext{Inj}(R^{op}))$.

Extends balanced functor properties to classes like $( ext{Lev}(R), ext{AC}(R^{op}))$.

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 $(\mathcal{F} (R), \mathcal{FP}Inj (R^{op}))$ 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.