Foxby equivalence relative to $C$-$fp_n$-injective and $C$-$fp_{n}$-flat modules

TL;DR

This paper introduces and studies new classes of modules called $C$-$fp_n$-injective and $C$-$fp_n$-flat, generalizing known modules, and explores their properties, dimensions, and Foxby equivalence in the context of ring extensions.

Contribution

It defines $C$-$fp_n$-injective and $C$-$fp_n$-flat modules, investigates their dimensions, and establishes Foxby equivalence and exchange properties related to these classes.

Findings

Established Foxby equivalence for the new module classes.

Proved existence of preenvelopes and covers for these modules.

Analyzed exchange properties under ring extensions.

Abstract

Let $R$ and $S$ be rings, $C= {}_SC_R$ a (faithfully) semidualizing bimodule, and $n$ a positive integer or $n=\infty$. In this paper, we introduce the concepts of $C$-$fp_n$-injective $R$-modules and $C$-$fp_n$-flat $S$-modules as a common generalization of some known modules such as $C$-$FP_{n}$-injective (resp. $C$-weak injective) $R$-modules and $C$-$FP_{n}$-flat (resp. $C$-weak flat) $S$-modules. Then we investigate $C$-$fp_{n}$-injective and $C$-$fp_{n}$-flat dimensions of modules, where the classes of these modules, namely $Cfp_nI(R)_{\leq k}$ and $Cfp_nF(S)_{\leq k}$, respectively. We study Foxby equivalence relative to these classes, and also the existence of $Cfp_nI(R)_{\leq k}$ and $Cfp_nF(S)_{\leq k}$ preenvelopes and covers. Finally, we study the exchange properties of these classes, as well as preenvelopes (resp. precovers) and Foxby equivalence, under almost excellent extensions of rings.