Perfect linkage of Cohen--Macaulay modules over Cohen--Macaulay rings

TL;DR

This paper introduces the concept of perfect linkage for Cohen-Macaulay modules over Cohen-Macaulay rings, connecting it with syzygies, approximations, and classical linkage theory, and provides structural and criterion results.

Contribution

It develops the theory of perfect linkage, linking it to existing concepts, and offers new criteria and examples for Cohen-Macaulay modules and ideals.

Findings

Recovered a theorem of Yoshino and Isogawa.

Established a criterion for codimension one Cohen-Macaulay modules to be perfectly linked.

Constructed various examples of module and ideal linkage.

Abstract

In this paper, we introduce and study the notion of linkage by perfect modules, which we call perfect linkage, for Cohen-Macaulay modules over Cohen--Macaulay local rings. We explore perfect linkage in connection with syzygies, maximal Cohen-Macaulay approximations and Yoshino-Isogawa linkage. We recover a theorem of Yoshino and Isogawa, and analyze the structure of double perfect linkage. Moreover, we establish a criterion for two Cohen-Macaulay modules of codimension one to be perfectly linked, and apply it to the classical linkage theory for ideals. We also construct various examples of linkage of modules and ideals.