Homological aspects of semidualizing modules

TL;DR

This paper explores the properties of C-projective and C-injective dimensions in modules relative to semidualizing modules, establishing their equivalences and connections to classical homological dimensions.

Contribution

It introduces and compares three definitions of finite C-projective dimension and develops the dual theory for C-injective dimension, revealing deep homological relationships.

Findings

Three definitions of finite C-projective dimension are equivalent

Connections between relative and absolute cohomology modules are established

Results demonstrate links between modules of finite projective and C-projective dimensions

Abstract

We investigate the notion of the C-projective dimension of a module, where C is a semidualizing module. When C=R, this recovers the standard projective dimension. We show that three natural definitions of finite C-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite C-projective dimension. In parallel, we develop the dual theory for injective dimension and C-injective dimension.