Embedding modules of finite homological dimension

TL;DR

This paper extends embedding results for modules of finite homological dimensions to a relative setting involving semidualizing modules, broadening the scope of such embeddings in homological algebra.

Contribution

It introduces new embeddings for modules of finite G_C-dimension, P_C-projective dimension, and locally finite I_C-injective dimension in the relative context.

Findings

Extended embedding results to modules with finite G_C-dimension.

Generalized results to modules with locally finite homological dimensions.

Broadened the applicability of homological dimension embeddings in the relative setting.

Abstract

This paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension, or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite G_C-dimension, finite P_C-projective dimension, locally finite GI_C-injective dimension, or locally finite I_C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homological dimension in the relative setting.