---
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 ${\text{G}_C}$-dimension, finite ${{\mathcal{P}}_C}$-projective dimension, locally finite ${{\mathcal{GI}}_C}$-injective dimension, or locally finite ${{\mathcal{I}}_C}$-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.'
address: 'Mathematics Department, NDSU Dept \# 2750, PO Box 6050, Fargo, ND 58108-6050 USA'
author:
- 'Sean Sather-Wagstaff'
title: Embedding modules of finite homological dimension
---

[^1]

Introduction {#sec0}
============

Throughout this note, $R$ is a commutative noetherian ring with identity. The purpose of this note is to extend some results of Hochster and Yao, beginning with the following, wherein $\underline z_{[i]}=z_1,\ldots,z_i$ and ${\mathrm{G}\text{-}\!\dim}_R(M)$ is the G-dimension of Auslander and Bridger [@auslander:adgeteac; @auslander:smt].[^2]

\[thm110527a\] Let $M$ be a non-zero finitely generated $R$-module. Then

1.  \[thm110527a1\] If ${\mathrm{G}\text{-}\!\dim}_R(M) = r < \infty$, then there are an $R$-regular sequence $\underline z = z_1, \ldots, z_r$, integers $n_0, n_1, \ldots , 