# Linkage of modules with respect to a semidualizing module

## TL;DR

This paper introduces the concept of linkage with respect to a semidualizing module and explores its implications for Cohen-Macaulay modules and Serre conditions over Cohen-Macaulay local rings.

## Contribution

It defines linkage relative to semidualizing modules and establishes new relationships between linked modules, Cohen-Macaulay properties, and local cohomology.

## Key findings

- Cohen-Macaulay modules of finite Gorenstein injective dimension are linked via the canonical module.
- Connections between Serre conditions and local cohomology vanishings are established.
- The notion of linkage is extended to modules with respect to semidualizing modules.

## Abstract

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect to the canonical module. For a linked module $M$ with respect to a semidualizing module, the connection between the Serre condition $(S_n)$ on $M$ with the vanishing of certain local cohomology modules of its linked module is discussed.