$(n,d)$-injective and $(n,d)$-flat modules under a special semidualizing bimodule
Mostafa Amini, Alireza Vahidi, Fatemeh Ghanavati
TL;DR
The paper extends the theory of semidualizing modules to a noncommutative, $(n,d)$-relative setting by introducing a special semidualizing bimodule $_S(K_{d-1})_R$ built from the $(d-1)$st syzygy of a super finitely presented bimodule. It defines $K_{d-1}$-$(n,d)$-injective and $K_{d-1}$-$(n,d)$-flat modules as relative generalizations of known $C$-injective/weak injective and $C$-$FP_n$-injective modules, and proves fundamental structural properties: characterizations, closure under standard constructions, and dualities with Bass/Auslander classes. The work establishes that these classes are covering and preenveloping, and develops Foxby equivalence in this specialized context, linking the relative modules to Auslander and Bass classes via Hom and tensor operations with $K_{d-1}$. Over $n$-coherent rings, finite-type variants of these modules behave well with respect to extensions, kernels, and cokernels, enriching the relative homological landscape and providing tools for further investigations of noncommutative homological dimensions.
Abstract
Let $S$ and $R$ be rings, $n, d\geq 0$ be two integers or $n=\infty$. In this paper, first we introduce special (faithfully) semidualizing bimodule $_S(K_{d-1})_R$, and then introduce and study the concepts of $K_{d-1}$-$(n,d)$-injective (resp. $K_{d-1}$-$(n,d)$-flat) modules as a common generalization of some known modules such as $C$-injective, $C$-weak injective and $C$-$FP_n$-injective (resp. $C$-flat, $C$-weak flat and $C$-$FP_n$-flat) modules. Then we obtain some characterizations of two classes of these modules, namely $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$. We show that the cleasses $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$ are covering and preenveloping. Also, we investigate Foxby equivalence relative to the classes of this modules. Finally over $n$-coherent rings, we prove that the classes $\mathcal{I}_{ K_{d-1}}^{(n,d)}(R)_{<\infty}$ and $\mathcal{F}_{ K_{d-1}}^{(n,d)}(S)_{<\infty}$ are closed under extentions, kernels of epimorphisms and cokernels of monomorphisms. Keywords: $K_{d-1}$-$(n,d)$-injective module; $K_{d-1}$-$(n,d)$-flat module; Foxby equivalence; special semidualizing bimodule.