Trace ideals and maximal Cohen-Macaulay modules over a Gorenstein local ring

Authors
Mohammad Bagherpoor
Abdoljavad Taherizadeh
Kharazmi University, Tehran. Iran
Abstract
All rings throughout this paper are commutative and Noetherian. Semidualizing modules were studied independently by Foxby [4], Golod [5], and Vasconcelos [11]. A finite R-module C is called semidualizing if the natural homothety map is an isomorphism and for all . If a semidualizing R-module has finite injective dimension, it is called dualizing and is denoted by D. The ring itself is an example of a semidualizing R-module. Many researchers, in particular Sather-Wagstaff [12], have studied the semidualizing modules.

Let M be an R-module. The trace ideal of M, denoted by , is the sum of images of all homomorphisms from M to R. Trace ideals have attracted the attention of many researchers in recent years. In particular, Herzog et al. [6] and Dao et al. [3] studied the trace ideals of canonical modules. Also, the trace ideals of semidualizing modules were studied in [1].

In this paper, we study the trace ideals of tensor product of two arbitrary modules. We prove some known facts with a different approach via trace ideals. For example, let C and be two semidualizing R-modules. We show that is projective if and only if C and are projective R-modules of rank 1. Also, we study the trace ideals of maximal Cohen-Macaulay modules over a Gorenstein local ring.

Keywords
commutative Noetherian ring
semidualizing module
trace ideal
Gorenstein ring
maximal Cohen-Macaulay module
projective module
free module
tensor product
canonical module
Bass class
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Mathematical Research
Volume 9, Issue 4 - Serial Number 27
Autumn 2023
Pages 111-121