Modules of finite homological dimension with respect to a semidualizing module

TL;DR

This paper extends classical results on modules of finite homological dimensions by incorporating semidualizing modules, providing new insights into Gorenstein dimensions and addressing open questions in the field.

Contribution

It generalizes Foxby and Holm's results to modules relative to semidualizing modules and verifies specific cases of Takahashi and White's question.

Findings

Extended results on Gorenstein injective and projective dimensions

Verified special cases of Takahashi and White's question

Provided new characterizations of modules with finite homological dimensions

Abstract

We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of Takahashi and White.