A generalization of a theorem of Foxby

TL;DR

This paper extends Foxby's theorem by demonstrating that certain conditions on a finitely generated module over a commutative noetherian local ring imply the ring is Gorenstein, broadening the understanding of ring properties.

Contribution

It generalizes Foxby's theorem by establishing Gorenstein conditions under broader module assumptions involving semidualizing modules.

Findings

The ring is Gorenstein if it admits a finitely generated module with finite projective and injective dimensions relative to a semidualizing module.

The result recovers and extends a well-known theorem of Foxby.

Provides new criteria for Gorenstein rings based on module properties.

Abstract

In this paper, it is proved that a commutative noetherian local ring admitting a finitely generated module of finite projective and injective dimensions with respect to a semidualizing module is Gorenstein. This result recovers a celebrated theorem of Foxby.