Gorensteinness from duality pairs induced via Foxby equivalences

TL;DR

This paper explores how duality pairs induced by Foxby equivalences relate to Gorenstein modules and properties, extending duality concepts in homological algebra.

Contribution

It introduces a framework for transferring duality pair properties via Foxby equivalences and studies their impact on Gorenstein modules.

Findings

Duality pairs are preserved under Foxby equivalences.

Relations between Gorenstein injective and flat modules are established.

Properties of duality pairs are transferred through the induced equivalences.

Abstract

We define and study induced duality pairs under Foxby equivalences. Given a semidualizing $(S,R)$-bimodule ${}_S C_R$, if $(\mathcal{A}_C(R),\mathcal{B}_C(R^{\rm op}))$ and $(\mathcal{A}_C(S^{\rm op}),\mathcal{B}_C(S))$ denote the duality pairs formed by the corresponding classes of Auslander and Bass modules, and if $(\mathcal{M,N})$ is a duality pair over $R$, we study the duality pair formed by the essential images of the restricted Foxby equivalences $(C \otimes_R \sim)|_{\mathcal{A}_C(R) \cap \mathcal{M}}$ and $\mathrm{Hom}_{R^{\rm op}}(C,\sim) |_{\mathcal{B}_C(R^{\rm op}) \cap \mathcal{N}}$, denoted by $\mathcal{M}^C(S)$ and $\mathcal{N}^C(S^{\rm op})$. We investigate which additional properties of the duality pair $(\mathcal{M,N})$ are transferred to $(\mathcal{M}^C(S),\mathcal{N}^C(S^{\rm op}))$. We also study several versions of Gorenstein injective and Gorenstein flat modules relative to the pairs $(\mathcal{A}_C(R) \cap \mathcal{M},\mathcal{B}_C(R^{\rm op}) \cap \mathcal{N})$ and $(\mathcal{M}^C(S),\mathcal{N}^C(S^{\rm op}))$. For instance, we explore the relation between these classes of modules under Foxby equivalences and under Pontryagin duality.