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

Bagherpoor, Mohammad; Taherizadeh, Abdoljavad

Volume 9, Issue 4 (12-2023) mmr 2023, 9(4): 111-121 | Back to browse issues page

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Bagherpoor M, Taherizadeh A. Trace ideals and maximal Cohen-Macaulay modules over a Gorenstein local ring. mmr 2023; 9 (4) :111-121
URL: http://mmr.khu.ac.ir/article-1-3304-en.html

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

Mohammad Bagherpoor

¹, Abdoljavad Taherizadeh²

1- Kharazmi University, Tehran. Iran , bagherpour.mohammad.2019@gmail.com
2- Kharazmi University, Tehran. Iran

Abstract: (896 Views)

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.