$C$-CANONICAL MODULES

Mohammad Bagheri,Abdol-Javad Taherizadeh

doi:10.24330/ieja.969917

Mohammad Bagheri, Abdol-Javad Taherizadeh

Open Access

https://doi.org/10.24330/ieja.969917

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Abstract

Let $C$ be a semidualizing module over a commutative Noetherian local ring $R$. In this paper we introduce a new class of modules, namely $C$-canonical modules which are a generalization of canonical modules. It is shown that if the canonical module exists then the $C$-canonical module exists and the converse holds under special conditions. Also, a new characterization of Gorenstein local rings is given via $C$-canonical modules.

Highlights

Throughout this introduction (R, m) is a commutative Noetherian local ring of dimension n, ER(R/m) denotes the injective envelope of R/m and Hnm(R) is the n-th local cohomology module of M with respect to m.Grothendieck [11] defined a canonical module over a complete local ring and called it a module of dualizing differentials; see [11, page 94]
Herzog and Kunz defined a canonical module for R as a finitely generated R-module K for which K ⊗R R ∼= HomR(Hnm(R), ER(R/m)) [12, Definition 5.6]
Canonical modules play an important role in studying Cohen-Macaulay local rings

Summary

Introduction

Throughout this introduction (R, m) is a commutative Noetherian local ring of dimension n, ER(R/m) denotes the injective envelope of R/m and Hnm(R) is the n-th local cohomology module of M with respect to m. A canonical module of a Cohen-Macaulay local ring (if it exists) is the dualizing module. It is known that if R is a homomorphic image of a Gorenstein local ring, it has the canonical module, and the converse holds when R is Cohen-Macaulay. We prove some new results concerning the existence of the canonical module over Cohen-Macaulay rings via C-canonical modules. Sharp [17] showed that over a Cohen-Macaulay local ring with the canonical module ωR, any maximal Cohen-Macaulay R-module with finite injective dimension, is equal to a finite direct sum of copies of ωR (see [17, Theorem 2.1 (v)]). By the following result (Theorem 4.9), we obtain a similar representation for some subclasses of maximal Cohen-Macaulay R-modules, via C-canonical modules. Theorem: Let R be a Cohen-Macaulay ring, C be a semidualizing module, and suppose ωC exists.

Preliminaries

Main results

The Cohen-Macaulay case