Codualizing modules

TL;DR

This paper introduces the concept of codualizing modules as a dual notion to dualizing modules in Noetherian local rings, exploring their properties and applications, including characterizations of Gorenstein rings.

Contribution

It defines codualizing modules, establishes an equivalence between certain module categories, and provides conditions characterizing Gorenstein rings.

Findings

Established an equivalence between noetherian modules of finite projective dimension and artinian modules of finite projective dimension.

Provided a mixed identity involving quasidualizing modules that characterizes codualizing modules.

Derived necessary and sufficient conditions for a ring to be Gorenstein.

Abstract

Let $(R, \mathfrak{m})$ be a Noetherian local ring. In this paper, we introduce a dual notion for dualizing modules, namely codualizing modules. We study the basic properties of codualizing modules and use them to establish an equivalence between the category of noetherian modules of finite projective dimension and the category of artinian modules of finite projective dimension. Next, we give some applications of codualizing modules. Finally, we present a mixed identity involving quasidualizing module that characterize the codualizing module. As an application, we obtain a necessary and sufficient condition for $R$ to be Gorenstein.