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$(\mathcal{F},\mathcal{A})$-Gorenstein flat homological dimensions (2203.09012v2)

Published 17 Mar 2022 in math.RA

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.

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