Duality Pairs Induced by Auslander and Bass Classes

TL;DR

This paper explores duality pairs formed by Auslander and Bass classes induced by a semidualizing bimodule over rings, establishing their properties and implications for module dimensions.

Contribution

It introduces and analyzes duality pairs involving Auslander and Bass classes, proving their closure properties and conditions for being covering and preenveloping.

Findings

Both pairs are coproduct-closed and product-closed duality pairs.

The pairs are covering and preenveloping, with one being perfect.

Applications to Auslander projective dimension are provided.

Abstract

Let $R$ and $S$ be any rings and $_RC_S$ a semidualizing bimodule, and let $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ be the Auslander and Bass classes respectively. Then both the pairs $$(\mathcal{A}_C(R^{op}),\mathcal{B}_C(R))\ {\rm and}\ (\mathcal{B}_C(R),\mathcal{A}_C(R^{op}))$$ are coproduct-closed and product-closed duality pairs and both $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if $\mathcal{B}_C(R)$ is enveloping in $\Mod R$, then $\mathcal{A}_C(S)$ is enveloping in $\Mod S$. Then some applications to the Auslander projective dimension of modules are given.