Homological Dimensions with Respect to a Semidualizing Complex

TL;DR

This paper extends the concepts of projective and injective dimensions to semidualizing complexes, establishing foundational properties and exploring their interactions with other invariants in homological algebra.

Contribution

It introduces new homological dimensions for semidualizing complexes, generalizing previous notions and providing a framework for their analysis.

Findings

Established base change results for the new dimensions

Proved local-global principles for these dimensions

Explored interactions with other algebraic invariants

Abstract

In this paper we build off of Takahashi and White's $\mathcal{P}_C$-projective dimension and $\mathcal{I}_C$-injective dimension to define these dimensions for when $C$ is a semidaulizing complex. We develop the framework for these homological dimensions by establishing base change results and local-global behavior. Furthermore, we investigate how these dimensions interact with other invariants.