---
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 $n_0, n_1, \ldots , n_r  {\geqslant}0$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ and ${\mathrm{G}\text{-}\!\dim}_R(N) {\leqslant}r$.

2.  \[thm110527a2\] If ${\operatorname{pd}}_R(M) = r < \infty$, then there exist an $R$-regular sequence $\underline z = z_1, \ldots, z_r$, integers $n_0, n_1, \ldots , n_r  {\geqslant}0$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ and ${\operatorname{pd}}_R(N) {\leqslant}r$.

The point of Theorem \[thm110527a\], as Hochster and Yao indicate, is that the given module of finite G-dimension or projective dimension embeds in a module that *obviously* has finite projective dimension, with some extra control on the G-dimension or projective dimension of the cokernel.

The first of our results, stated next, is for modules of finite ${\text{G}_C}$-dimension and modules of finite ${{\mathcal{P}}_C}$-projective dimension, where $C$ is a semidualizing $R$-module, after Golod [@golod:gdagpi], Holm and Jørgensen [@holm:smarghd] and White [@white:gpdrsm].[^3] (See Section \[sec110530a\] for definitions and background material.) On the one hand, this result is a corollary to Theorem \[thm110527a\]. On the other hand, Theorem \[thm110527a\] is the special case $C=R$, so our result generalizes this one. This result is proved in Proofs \[proof110530a\] and \[proof110530b\], which are similar to the proof of [@hochster:etmf Theorem 4.2].

\[thm110527b\] Let $M$ be a non-zero finitely generated $R$-module, and let $C$ be a semidualizing $R$-module. Then

1.  \[thm110527b1\] If ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(M) = r < \infty$ and $M$ is in the Bass class ${{\mathcal{B}}_C}(R)$, then there exist an $R$-regular sequence $\underline z = z_1, \ldots, z_r$, integers $n_0, n_1, \ldots , n_r  {\geqslant}0$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(C/(\underline z_{[i]})C)^{n_i}$ and ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(N) {\leqslant}r$ and $N\in{{\mathcal{B}}_C}(R)$.

2.  \[thm110527b2\] If ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M) = r < \infty$, then there exist an $R$-regular sequence $\underline z = z_1, \ldots, z_r$, integers $n_0, n_1, \ldots , n_r  {\geqslant}0$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(C/(\underline z_{[i]})C)^{n_i}$ and ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(N) {\leqslant}r$.

The next result we extend is the following:

\[thm110530a\] Let $R$ be a Cohen-Macaulay ring with a pointwise dualizing module $\omega$. Then, for any non-zero finitely generated $R$-module $M$ with locally finite injective dimension, there exist an integer $r = {\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,M)})$, an $R$-regular sequence $\underline z = z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$, such that $N$ has locally finite injective dimension and ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,N)}){\leqslant}r$.

We extend this in two directions in Theorems \[thm110608a\] and \[thm110530b\]. First, we have the version for G-injective dimension. Second, we give versions relative to a semidualizing module. These are proved in Proofs \[proof110609a\], \[proof110609b\], and \[proof110609c\].

\[thm110608a\] Let $R$ be a Cohen-Macaulay ring with a pointwise dualizing module $\omega$. For any non-zero finitely generated $R$-module $M$ with locally finite G-injective dimension, there exist an integer $r = {\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,M)})$, an $R$-regular sequence $\underline z = z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ with $Z = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$ such that $N$ has locally finite G-injective dimension and ${\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,N)}){\leqslant}r$.

\[thm110530b\] Let $R$ be a Cohen-Macaulay ring with a pointwise dualizing module $\omega$. Let $M$ a non-zero finitely generated $R$-module, and let $C$ be a semidualizing $R$-module. Set ${C^{\dagger}}={\operatorname{Hom}_{R}(C,\omega)}$. Then

1.  \[thm110530b1\] If $M$ has locally finite ${{\mathcal{GI}}_C}\text{-}{\operatorname{id}}$ and $M$ is in the Auslander class ${{\mathcal{A}}_C}(R)$, then there exist an integer $r = {\mathrm{G}\text{-}\!\dim}({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}M})})$, an $R$-regular sequence $\underline z = z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ such that $N$ has locally finite ${{\mathcal{GI}}_C}\text{-}{\operatorname{id}}$, and $Z = \oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}$, $N \in {{\mathcal{A}}_C}(R)$ and ${\mathrm{G}\text{-}\!\dim}({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})}){\leqslant}r$.

2.  \[thm110530b2\] If $M$ has locally finite ${{\mathcal{I}}_C}\text{-}{\operatorname{id}}$, then there exist an integer $$r = {\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}M})})$$ an $R$-regular sequence $\underline z = z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$0 \to M \to Z \to N \to 0$$ such that $N$ has locally finite ${{\mathcal{I}}_C}\text{-}{\operatorname{id}}$, and $Z = \oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}$ and ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})}){\leqslant}r$.

We summarize the organization of this paper. Section \[sec110530a\] contains the proof of Theorem \[thm110527b\], along with the necessary background material. Section \[sec110608a\] similarly treats Theorems \[thm110608a\] and \[thm110530b\]; this section also contains several lemmas about modules of locally finite G-injective dimension, locally finite ${\text{G}_C}$-injective dimension, and locally finite ${{\mathcal{I}}_C}$-injective dimension where $C$ is a semidualizing module, as these notions have not been developed in the literature as best we know.[^4] Note that we do not develop the background material in the most optimal manner, focusing instead on accessibility. To be specific, we present background material as it is needed in Sections \[sec110530a\] and \[sec110608a\] instead of putting it all in its own section. Also, we feel that the definition of locally finite ${{\mathcal{GI}}_C}\text{-}{\operatorname{id}}$ is a bit much to swallow on the first bite, so we first discuss the special case of ${{{\mathcal{G}}\text{-}{\operatorname{id}}}}$ and work our way up.

Proof of Theorem \[thm110527b\] {#sec110530a}
===============================

\[notn1xx\] We let ${\operatorname{pd}}_R(-)$ and ${\operatorname{id}}_R(-)$ denote the classical projective dimension and the classical injective dimension, respectively.

The study of semidualizing modules was initiated independently (with different names) by Foxby [@foxby:gmarm], Golod [@golod:gdagpi], and Vasconcelos [@vasconcelos:dtmc]. For example, a finitely generated projective $R$-module of rank 1 is semidualizing.

\[notation08a\] A finitely generated $R$-module $C$ is *semidualizing* if the natural map $R\to{\operatorname{Hom}_{R}(C,C)}$ is an isomorphism, and ${\operatorname{Ext}}_R^{{\geqslant}1}(C,C)=0$. An $R$-module $C$ is *pointwise dualizing* if it is semidualizing and ${\operatorname{id}}_{R_{{{\mathfrak{m}}}}}(C_{{{\mathfrak{m}}}})<\infty$ for all maximal ideals ${{\mathfrak{m}}}\subset R$.[^5] An $R$-module $C$ is *dualizing* if it is semidualizing and ${\operatorname{id}}_R(C)<\infty$.

\[fact110527a\] (a) It is straightforward to show that the semidualizing property is local. That is, if $C$ is a semidualizing $R$-module, then for each multiplicatively closed subset $U\subseteq R$ the localization $U^{-1}C$ is a semidualizing $U^{-1}R$-module; and conversely, if $M$ is a finitely generated $R$-module such that $M_{{{\mathfrak{m}}}}$ is a semidualizing $R_{{{\mathfrak{m}}}}$-module for each maximal ideal ${{\mathfrak{m}}}\subset R$, then $M$ is a semidualizing $R$-module. Similarly, if $C$ is a (pointwise) dualizing module for $R$, then for each multiplicatively closed subset $U\subseteq R$ the localization $U^{-1}C$ is a (pointwise) dualizing module for $U^{-1}R$; and conversely, if $M$ is a finitely generated $R$-module such that $M_{{{\mathfrak{m}}}}$ is a dualizing module for $R_{{{\mathfrak{m}}}}$ for each maximal ideal ${{\mathfrak{m}}}\subset R$, then $M$ is a pointwise dualizing module for $R$. Moreover, an $R$-module $M$ is a dualizing module for $R$ if and only if it is a pointwise dualizing module for $R$ and the Krull dimension of $R$ is finite.

\(b) Let $C$ be a semidualizing $R$-module. The isomorphism $R\cong{\operatorname{Hom}_{R}(C,C)}$ implies that ${\operatorname{Supp}}_R(C)={\operatorname{Spec}}(R)$ and ${\operatorname{Ass}}_R(C)={\operatorname{Ass}}(R)$. Thus, an element $x\in R$ is $C$-regular if and only if it is $R$-regular. When $x$ is $R$-regular, the quotient $C/xC$ is a semidualizing $R/xR$-module. Thus, by induction, a sequence $\underline z=z_1,\ldots,z_r\in R$ is $R$-regular if and only if it is $C$-regular; see [@frankild:rrhffd Theorem 4.5]. Also, from [@holm:fear Proposition 3.1] we know that, for all $R$-modules $M\neq 0$, we have ${C\otimes_{R}M}\neq 0\neq{\operatorname{Hom}_{R}(C,M)}$.

The next categories were introduced by Foxby [@foxby:gdcmr] when $C$ is dualizing, and by Vasconcelos [@vasconcelos:dtmc §4.4] for arbitrary $C$, with different notation.

\[notation08d\] Let $C$ be a semidualizing $R$-module. The *Auslander class* of $C$ is the class ${{\mathcal{A}}_C}(R)$ of $R$-modules $M$ such that ${\operatorname{Tor}}^R_{{\geqslant}1}(C,M)=0={\operatorname{Ext}}_R^{{\geqslant}1}(C,C\otimes_R M)$, and the natural map $M\to{\operatorname{Hom}_{R}(C,C\otimes_R M)}$ is an isomorphism. The *Bass class* of $C$ is the class ${{\mathcal{B}}_C}(R)$ of $R$-modules $N$ such that ${\operatorname{Ext}}_R^{{\geqslant}1}(C,M)=0={\operatorname{Tor}}^R_{{\geqslant}1}(C,{\operatorname{Hom}_{R}(C,M)})$, and the natural evaluation map $C\otimes_R{\operatorname{Hom}_{R}(C,N)}\to N$ is an isomorphism.

\[disc110530a\] It is straightforward to show that the Auslander and Bass classes for $C=R$ are trivial: ${{\mathcal{A}}}_{R}(R)$ and ${{\mathcal{B}}}_R(R)$ both contain all $R$-modules.

The following notion was introduced and studied by Holm and Jørgensen [@holm:smarghd] and White [@white:gpdrsm]. Note that in the special case $C=R$, we recover the class of projective $R$-modules and the classical projective dimension.

\[notation08bxy\] Let $C$ be a semidualizing $R$-module. Let ${{\mathcal{P}}_C}(R)$ denote the class of modules $M\cong P\otimes_R C$ with $P$ projective. Modules in ${{\mathcal{P}}_C}(R)$ are called *$C$-projective*. We let ${{\mathcal{P}}_C}\text{-}{\operatorname{pd}}_R(-)$ denote the homological dimension obtained from resolutions in ${{\mathcal{P}}_C}(R)$, with the convention that ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(0)=-\infty$.[^6]

\[disc01xyz\] Let $C$ be a semidualizing $R$-module. The Bass class ${{\mathcal{B}}_C}(R)$ contains all $R$-modules of finite ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}$; see [@holm:fear Lemmas 4.1 and 5.1, and Corollary 6.3]. Given an $R$-module $M$, one has ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M)={\operatorname{pd}}_R({\operatorname{Hom}_{R}(C,M)})$ and ${\operatorname{pd}}_R(M)={{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R({C\otimes_{R}M})$ by [@takahashi:hasm Theorem 2.11]. In particular, one has ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M)<\infty$ if and only if ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(C,M)})<\infty$, and one has ${\operatorname{pd}}_R(M)<\infty$ if and only if ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R({C\otimes_{R}M})<\infty$.

\[proof110530a\] Assume that $M$ is a non-zero finitely generated $R$-module such that ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M) = r < \infty$. Then Fact \[disc01xyz\] implies that ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(C,M)})=r<\infty$. Since $M\neq 0$, we have ${\operatorname{Hom}_{R}(C,M)}\neq 0$, by Fact \[fact110527a\](b), so Theorem \[thm110527a\] provides an $R$-regular sequence $\underline z = z_1, \ldots, z_r$, integers $n_0, n_1, \ldots , n_r  {\geqslant}0$ with $n_r{\geqslant}1$, and an exact sequence $$\label{eq110530a}
0 \to {\operatorname{Hom}_{R}(C,M)} \to Z' \to N' \to 0$$ with $Z' = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ and ${\operatorname{pd}}_R(N') {\leqslant}r$. In particular, we have $N'\in{{\mathcal{A}}_C}(R)$ which implies that ${\operatorname{Tor}^{R}_{1}(C,N')}=0$. Thus, an application of ${C\otimes_{R}-}$ to the sequence  yields the next exact sequence: $$\label{eq110530b}
0 \to {C\otimes_{R}{\operatorname{Hom}_{R}(C,M)}} \to {C\otimes_{R}Z'} \to {C\otimes_{R}N'} \to 0.$$ Fact \[disc01xyz\] implies that $M\in{{\mathcal{B}}_C}(R)$, so we have ${C\otimes_{R}{\operatorname{Hom}_{R}(C,M)}}\cong M$. The equality $Z' = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ implies that $Z:={C\otimes_{R}Z'} = \oplus_{i=0}^r(C/(\underline z_{[i]})C)^{n_i}$. Because of Fact \[disc01xyz\], the condition ${\operatorname{pd}}_R(N') {\leqslant}r$ implies that ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R({C\otimes_{R}N'}){\leqslant}r$. Thus, with $N:={C\otimes_{R}N'}$, the sequence  satisfies the conclusion of Theorem \[thm110527b\].

The remainder of the proof of Theorem \[thm110527b\] is similar to the above proof, but it requires a bit more technology.

\[disc01xy\] Let $C$ be a semidualizing $R$-module.

\(a) The Auslander class ${{\mathcal{A}}_C}(R)$ contains all $R$-modules of finite projective dimension, and the Bass class ${{\mathcal{B}}_C}(R)$ contains all $R$-modules of finite injective dimension; see [@holm:fear Lemmas 4.1 and 5.1, and Corollary 6.3]. Moreover, it is straightforward to show that the Bass class satisfies the following local-global principal: For an $R$-module $M$, the following conditions are equivalent:

(i) $M\in{{\mathcal{B}}_C}(R)$;

(ii) $U^{-1}M\in{{\mathcal{B}}}_{U^{-1}C}(U^{-1}R)$ for each multiplicatively closed subset $U\subseteq R$;

(iii) $M_{{{\mathfrak{p}}}}\in{{\mathcal{B}}}_{C_{{{\mathfrak{p}}}}}(R_{{{\mathfrak{p}}}})$ for each ${{\mathfrak{p}}}\in{\operatorname{Spec}}(R)$; and

(iv) $M_{{{\mathfrak{m}}}}\in{{\mathcal{B}}}_{C_{{{\mathfrak{m}}}}}(R_{{{\mathfrak{m}}}})$ for each maximal ideal ${{\mathfrak{m}}}\in{\operatorname{Supp}}_R(M)$.

It follows that ${{\mathcal{B}}_C}(R)$ contains every $R$-module of locally finite injective dimension. The Auslander class satisfies an analogous local-global principal.

\(b) The Auslander and Bass classes also satisfy the two-of-three property by [@holm:fear Corollary 6.3]. That is, given a short exact sequence $0\to M'\to M\to M''\to 0$ of $R$-module homomorphisms, if two of the modules in the sequence are in ${{\mathcal{A}}_C}(R)$, then so is the third module, and similarly for ${{\mathcal{B}}_C}(R)$.

\(c) From [@takahashi:hasm (2.8)] we know that an $R$-module $M$ is in ${{\mathcal{B}}_C}(R)$ if and only if ${\operatorname{Hom}_{R}(C,M)}\in{{\mathcal{A}}_C}(R)$, and that $M\in{{\mathcal{A}}_C}(R)$ if and only if $C\otimes_RM\in{{\mathcal{B}}_C}(R)$. This is known as *Foxby equivalence*.

\[notation08bx\] Let $C$ be a semidualizing $R$-module. A finitely generated $R$-module $G$ is *totally $C$-reflexive* if the natural map $G\to{\operatorname{Hom}_{R}({\operatorname{Hom}_{R}(G,C)},C)}$ is an isomorphism, and ${\operatorname{Ext}_{R}^{{\geqslant}1}(G,C)}=0={\operatorname{Ext}_{R}^{{\geqslant}1}({\operatorname{Hom}_{R}(G,C)},C)}$. Let ${{\mathcal{G}}_C}(R)$ denote the class of totally $C$-reflexive $R$-modules, and set ${{\mathcal{G}}}(C)={{\mathcal{G}}_C}(R)\cap{{\mathcal{B}}_C}(R)$. In the case $C=R$ we use the more common terminology “totally reflexive” and the notation ${{\mathcal{G}}}(R)={{\mathcal{G}}}_R(R)={{\mathcal{G}}}(R)\cap{{\mathcal{B}}}_R(R)$. We abbreviate as follows: $$\begin{aligned}
{{\mathrm{G}_{C}\text{-}\!\dim}}_R(-)
&=\text{the homological dimension obtained from resolutions  in ${{\mathcal{G}}_C}(R)$}
\\
{{\mathcal{G}}}(C)\text{-}{\operatorname{pd}}_R(-)
&=\text{the homological dimension obtained from resolutions  in ${{\mathcal{G}}}(C)$}
\\
{\mathrm{G}\text{-}\!\dim}_R(-)
&=\text{the homological dimension obtained from resolutions  in ${{\mathcal{G}}}(R)$.}\end{aligned}$$

The following facts are included for perspective.

Let $C$ be a semidualizing $R$-module, and let $M$ be a finitely generated $R$-module. Because of the containments ${{\mathcal{P}}_C}(R)\subseteq{{\mathcal{G}}}(C)\subseteq{{\mathcal{G}}_C}(R)$, one has ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(M){\leqslant}{{\mathcal{G}}}(C)\text{-}{\operatorname{pd}}_R(M){\leqslant}{{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M)$ with equality to the left of any finite quantity. In particular, the case $C=R$ says that ${\mathrm{G}\text{-}\!\dim}_R(M)={{\mathcal{G}}}(R)\text{-}{\operatorname{pd}}_R(M){\leqslant}{\operatorname{pd}}_R(M)$ since ${{\mathcal{G}}}(R)={{\mathcal{G}}}_R(R)$, with equality holding when ${\operatorname{pd}}_R(M)<\infty$.

The next lemma explains how these homological dimensions are connected.

\[lem0701x\] Let $C$ be a semidualizing $R$-module. For a finitely generated $R$-module $M$, the following conditions are equivalent:

1.  \[lem0701xi\] ${{\mathcal{G}}}(C)\text{-}{\operatorname{pd}}_R(M)<\infty$;

2.  \[lem0701xii\] ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(M)<\infty$ and $M\in{{\mathcal{B}}_C}(R)$; and

3.  \[lem0701xiii\] ${\mathrm{G}\text{-}\!\dim}_R({{\operatorname{Hom}_{R}(},C)}M)<\infty$ and $M\in{{\mathcal{B}}_C}(R)$.

When these conditions are satisfied, we have $$\begin{aligned}
{{\mathcal{G}}}(C)\text{-}{\operatorname{pd}}_R(M)
&={{\mathrm{G}_{C}\text{-}\!\dim}}_R(M)
={\mathrm{G}\text{-}\!\dim}_R({{\operatorname{Hom}_{R}(},C)}M).\end{aligned}$$

Now we are in position to complete the proof of Theorem \[thm110527b\].

\[proof110530b\] Assume that $M$ is a non-zero finitely generated $R$-module such that ${{\mathrm{G}_{C}\text{-}\!\dim}}_R(M) = r < \infty$ and $M\in{{\mathcal{B}}_C}(R)$. Then Lemma \[lem0701x\] implies that ${\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(C,M)})=r<\infty$. Since $M\neq 0$, we have ${\operatorname{Hom}_{R}(C,M)}\neq 0$, by Fact \[fact110527a\](b), so Theorem \[thm110527a\] provides an $R$-regular sequence $\underline z = z_1, \ldots, z_r$, integers $n_0, n_1, \ldots , n_r  {\geqslant}0$ such that $n{\geqslant}1$, and an exact sequence $$\label{eq110530c}
0 \to {\operatorname{Hom}_{R}(C,M)} \to Z' \to N' \to 0$$ with $Z' = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ and ${\mathrm{G}\text{-}\!\dim}_R(N') {\leqslant}r$.

The condition $M\in{{\mathcal{B}}_C}(R)$ implies that ${\operatorname{Hom}_{R}(C,M)}\in{{\mathcal{A}}_C}(R)$ by Fact \[disc01xy\](c), and Fact \[disc01xy\](a) implies that $Z'\in{{\mathcal{A}}_C}(R)$. Also from Fact \[disc01xy\](a)–(b), we conclude that $N'\in{{\mathcal{A}}_C}(R)$, which implies that ${\operatorname{Tor}^{R}_{1}(C,N')}=0$. Thus, an application of ${C\otimes_{R}-}$ to the sequence  yields the next exact sequence: $$\label{eq110530d}
0 \to {C\otimes_{R}{\operatorname{Hom}_{R}(C,M)}} \to {C\otimes_{R}Z'} \to {C\otimes_{R}N'} \to 0.$$ The assumption $M\in{{\mathcal{B}}_C}(R)$ gives an isomorphism ${C\otimes_{R}{\operatorname{Hom}_{R}(C,M)}}\cong M$. The equality $Z' = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ implies that $Z:={C\otimes_{R}Z'} = \oplus_{i=0}^r(C/(\underline z_{[i]})C)^{n_i}$.

The condition $N'\in{{\mathcal{A}}_C}(R)$ implies that $N:={C\otimes_{R}N'}\in{{\mathcal{B}}_C}(R)$ by Fact \[disc01xy\](c), and $N'\cong{\operatorname{Hom}_{R}(C,N)}$ by the definition of what it means for $N'$ to be in ${{\mathcal{A}}_C}(R)$. Thus, because of Lemma \[lem0701x\], we have $${{\mathrm{G}_{C}\text{-}\!\dim}}_R(N)={\mathrm{G}\text{-}\!\dim}({\operatorname{Hom}_{R}(C,N)})={\mathrm{G}\text{-}\!\dim}_R(N') {\leqslant}r.$$ Thus, the sequence  satisfies the conclusion of Theorem \[thm110527b\].

Proofs of Theorems \[thm110608a\] and \[thm110530b\] {#sec110608a}
====================================================

We continue with a definition due to Enochs and Jenda [@enochs:gipm].

\[notation110609a\] A *complete injective resolution* is an exact complex $Y$ of injective $R$-modules such that ${\operatorname{Hom}_{R}(J,Y)}$ is exact for each injective $R$-module $J$. An $R$-module $N$ is *G-injective* if there exists a complete injective resolution $Y$ such that $N\cong{\operatorname{Ker}}(\partial^Y_0)$, and $Y$ is a *complete injective resolution of $N$*. Let ${{\mathcal{GI}}}(R)$ denote the class of G-injective $R$-modules, and let ${\operatorname{Gid}}_R(-)$ denote the homological dimension obtained from coresolutions in ${{\mathcal{GI}}}(R)$. We say that an $R$-module $M$ has *locally finite G-injective dimension* provided that ${\operatorname{Gid}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

\[disc110726a\] Using existing technology, the modules of finite G-injective dimension behave best when $R$ is Cohen-Macaulay with a dualizing module.[^7] For instance, in general it is not known whether the G-injective dimension localizes; but it is known to localize when $R$ has a dualizing module. This is due to the following connection with Bass classes, the local case of which is from [@enochs:fdgipm].[^8]

\[lem110610a\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module. Let $M$ be an $R$-module. Then an $R$-module $M$ has locally finite G-injective dimension if and only if $M\in{{\mathcal{B}}}_{\omega}(R)$.

If $R$ is local, then the result follows from [@enochs:fdgipm Proposition 1.4 and Theorems 1.6 and 2.5]. By definition, the condition “$M$ has locally finite G-injective dimension” is a local condition. Fact \[disc01xy\](b) shows that the condition “$M\in{{\mathcal{B}}}_{\omega}(R)$” is also a local condition. Note that Fact \[fact110527a\](a) implies that for each maximal ideal ${{\mathfrak{m}}}\subset R$, the localization $\omega_{{{\mathfrak{m}}}}$ is a dualizing module for the Cohen-Macaulay local ring $R_{{{\mathfrak{m}}}}$. Thus, the general result follows from the local case.

The next result shows that the G-dimension of a finitely generated module behaves like its projective dimension.

\[lem110610b\] Let $M$ be a finitely generated $R$-module. The following conditions are equivalent:

1.  \[lem110610b1\] ${\mathrm{G}\text{-}\!\dim}_{R}(M)<\infty$;

2.  \[lem110610b2\] ${\mathrm{G}\text{-}\!\dim}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

When $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$, these conditions are equivalent to the following:

1.  \[lem110610b3\] $M\in{{\mathcal{A}}}_{\omega}(R)$.

The implication $\eqref{lem110610b1}\implies\eqref{lem110610b2}$ follows from the inequality ${\mathrm{G}\text{-}\!\dim}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}}){\leqslant}{\mathrm{G}\text{-}\!\dim}_R(M)$, which is straightforward to verify. The implication $\eqref{lem110610b2}\implies\eqref{lem110610b1}$ is from [@avramov:rrc1 Theorem 3.3]. When $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$, the equivalence $\eqref{lem110610b2}\iff$(iii) follows from the local results [@enochs:fdgipm Proposition 1.3 and Theorems 1.6 and 2.1] as in the proof of Lemma \[lem110610a\].[^9]

\[proof110609a\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$ and that $M$ is a non-zero finitely generated $R$-module with locally finite G-injective dimension. Lemma \[lem110610a\] implies that $M \in \mathcal B_{\omega}(R)$. Using Foxby equivalence (from Fact \[disc01xy\](c)) we conclude that ${\operatorname{Hom}_{R}(\omega,M)}\in{{\mathcal{A}}}_{\omega}(R)$, so Lemma \[lem110610b\] implies that $r:={\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,M)})<\infty$. The fact that $R$ is noetherian implies that ${\operatorname{Hom}_{R}(\omega,M)}$ is finitely generated. (The proof concludes as in Proof \[proof110530b\]. We include the details for the sake of thoroughness.)

Since $M\neq 0$, we have ${\operatorname{Hom}_{R}(\omega,M)}\neq 0$, by Fact \[fact110527a\](b), so Theorem \[thm110527a\] implies that there are an $R$-regular sequence $\underline z = z_1, \ldots, z_r$, integers $n_0, n_1, \ldots , n_r  {\geqslant}0$ with $n_r{\geqslant}1$, and an exact sequence $$\label{proof110609a1}
0 \to {\operatorname{Hom}_{R}(\omega,M)} \to Z' \to N' \to 0$$ with $Z' = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ and ${\mathrm{G}\text{-}\!\dim}_R(N') {\leqslant}r$. Fact \[disc01xy\](a) implies that $Z'\in{{\mathcal{A}}}_\omega(R)$. From Fact \[disc01xy\](b), we conclude that $N'\in{{\mathcal{A}}}_\omega(R)$, which implies that ${\operatorname{Tor}^{R}_{1}(\omega,N')}=0$. Thus, an application of ${\omega\otimes_{R}-}$ to the sequence  yields the next exact sequence: $$\label{proof110609a2}
0 \to {\omega\otimes_{R}{\operatorname{Hom}_{R}(\omega,M)}} \to {\omega\otimes_{R}Z'} \to {\omega\otimes_{R}N'} \to 0.$$ The assumption $M\in{{\mathcal{B}}}_\omega(R)$ gives an isomorphism ${\omega\otimes_{R}{\operatorname{Hom}_{R}(\omega,M)}}\cong M$. The equality $Z' = \oplus_{i=0}^r(R/(\underline z_{[i]}))^{n_i}$ implies that $Z:={\omega\otimes_{R}Z'} = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$.

The condition $N'\in{{\mathcal{A}}}_\omega(R)$ implies that $N:={\omega\otimes_{R}N'}\in{{\mathcal{B}}}_\omega(R)$ by Fact \[disc01xy\](c), and $N'\cong{\operatorname{Hom}_{R}(\omega,N)}$ by the definition of what it means for $N'$ to be in ${{\mathcal{A}}}_\omega(R)$. Thus, we have ${\mathrm{G}\text{-}\!\dim}({\operatorname{Hom}_{R}(\omega,N)})={\mathrm{G}\text{-}\!\dim}_R(N') {\leqslant}r$. So, the sequence  satisfies the conclusion of Theorem \[thm110608a\].

For our next results, we need a better understanding of the relationship between semidualizing modules and a pointwise dualizing module.

\[lem110611a\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$. If $C$ is a semidualizing $R$-module, then so is ${\operatorname{Hom}_{R}(C,\omega)}$.

In the local case, this result is standard; see, e.g., [@sather:bnsc Facts 1.18–1.19]. Since the semidualizing and (pointwise) dualizing conditions are local by Fact \[fact110527a\](a), the general case of the result follows.

The next two lemmas elaborate on the local-global behavior for our invariants.

\[lem110611d\] Let $M$ be a finitely generated $R$-module, and let $C$ be a semidualizing $R$-module. Then ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_{R}(M)<\infty$ if and only if $\text{P}_{C_{{{\mathfrak{m}}}}}\text{-}{\operatorname{pd}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

The forward implication follows from the inequality $\text{P}_{C_{{{\mathfrak{m}}}}}\text{-}\dim_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}}){\leqslant}{{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_R(M)$, which is straightforward to verify. For the converse, assume that $\text{P}_{C_{{{\mathfrak{m}}}}}\text{-}{\operatorname{pd}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$. It follows that the module ${\operatorname{Hom}_{R}(C,M)}_{{{\mathfrak{m}}}}\cong{\operatorname{Hom}_{R_{{{\mathfrak{m}}}}}(C_{{{\mathfrak{m}}}},M_{{{\mathfrak{m}}}})}$ has finite projective dimension over $R_{{{\mathfrak{m}}}}$ for all ${{\mathfrak{m}}}$. Hence, the finitely generated $R$-module ${\operatorname{Hom}_{R}(C,M)}$ has finite projective dimension, so Fact \[disc01xyz\] implies that ${{{\mathcal{P}}_C}\text{-}{\operatorname{pd}}}_{R}(M)$ is finite.

The next lemma is a souped-up version of a special case of a result of Takahashi and White that is documented in [@sather:tate1].

\[lem110612a\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$. For each finitely generated $R$-module $M$, one has ${{\mathcal{P}}}_{\omega}\text{-}{\operatorname{pd}}_R(M)<\infty$ if and only if $M$ has locally finite injective dimension.

When $\omega$ is a dualizing module for $R$, i.e., when $R$ has finite Krull dimension, this result is from [@sather:tate1 Lemma 2.7]. In particular, this takes care of the local case. The general case follows from the local case, by Lemma \[lem110611d\].

Here are some more notions from [@holm:smarghd].

\[notation08b\] Let $C$ be a semidualizing $R$-module, and let ${{\mathcal{I}}_C}(R)$ denote the class of modules $N\cong {\operatorname{Hom}_{R}(C,I)}$ with $I$ injective. Modules in ${{\mathcal{I}}_C}(R)$ are called *$C$-injective*. We let ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R(-)$ denote the homological dimension obtained from coresolutions in ${{\mathcal{I}}_C}(R)$. We say that an $R$-module $M$ has *locally finite ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$* provided that $\text{I}_{C_{{{\mathfrak{m}}}}}\text{-}{\operatorname{id}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

\[fact110612a\] Let $C$ be a semidualizing $R$-module. The Auslander class ${{\mathcal{A}}_C}(R)$ contains every $R$-module of finite ${{\mathcal{I}}}_C$-injective dimension; see [@holm:fear Lemmas 4.1 and 5.1]. Given an $R$-module $M$, one has ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R(M)={\operatorname{id}}_R({C\otimes_{R}M})$ and ${\operatorname{id}}_R(M)={{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R({\operatorname{Hom}_{R}(C,M)})$ by [@takahashi:hasm Theorem 2.11]. In particular, one has ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R(M)<\infty$ if and only if ${\operatorname{id}}_R({C\otimes_{R}M})<\infty$, and one has ${\operatorname{id}}_R(M)<\infty$ if and only if ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}_R({\operatorname{Hom}_{R}(C,M)})<\infty$. It follows that $M$ has locally finite ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$ if and only if ${C\otimes_{R}M}$ has locally finite injective dimension, and $M$ has locally finite injective dimension if and only if ${\operatorname{Hom}_{R}(C,M)}$ has locally finite ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$.

\[lem110612b\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$. Let $C$ be a semidualizing $R$-module, and let $x=x_1,\ldots,x_i\in R$ be an $R$-regular sequence. Then ${\operatorname{Hom}_{R}(C,\omega/(\underline x)\omega)}\cong{C^{\dagger}}/(\underline x){C^{\dagger}}$ where ${C^{\dagger}}={\operatorname{Hom}_{R}(C,\omega)}$.

Let $K$ be the Koszul complex $K^R(\underline x)$, and let $K^+$ denote the augmented Koszul complex $$K^+=(0\to R\to\cdots\to R\to \underbrace{R/(\underline x)}_{\text{deg. $-1$}}\to 0).$$ Since $x$ is $R$-regular, $K^+$ is an exact sequence of $R$-modules of finite projective dimension. Thus, each module $K^+_j$ is in ${{\mathcal{A}}}_{\omega}(R)$, so the induced complex $${\omega\otimes_{R}K^+}=(0\to \omega\to\cdots\to \omega\to \underbrace{\omega/(\underline x)\omega}_{\text{deg. $-1$}}\to 0)$$ is an exact sequence of $R$-modules locally of finite injective dimension. In particular, the modules in this exact sequence are in ${{\mathcal{B}}_C}(R)$, so the next sequence is also exact: $$\begin{aligned}
{\operatorname{Hom}_{R}(C,{\omega\otimes_{R}K^+})}\hspace{-2cm}\\
&=(0\to {\operatorname{Hom}_{R}(C,\omega)}\to\cdots\to {\operatorname{Hom}_{R}(C,\omega)}\to \underbrace{{\operatorname{Hom}_{R}(C,\omega/(\underline x)\omega)}}_{\text{deg. $-1$}}\to 0).\end{aligned}$$ It is straightforward to show that the differential on this sequence in positive degrees is the same as the differential on ${{C^{\dagger}}\otimes_{R}K}$. It follows that $${\operatorname{Hom}_{R}(C,\omega/(\underline x)\omega)}
\cong{\operatorname{H}}_0({{C^{\dagger}}\otimes_{R}K})
\cong{C^{\dagger}}/(\underline x){C^{\dagger}}$$ as desired.

\[proof110609b\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$ and that $M$ is a non-zero finitely generated $R$-module with locally finite ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$. Fact \[fact110612a\] implies that ${C\otimes_{R}M}$ has locally finite injective dimension. Since $M\neq 0$, we have ${C\otimes_{R}M}\neq 0$, by Fact \[fact110527a\](b), so Theorem \[thm110530a\] provides an integer $r = {\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}M})})$, a proper $R$-regular sequence $\underline z = z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$\label{proof110609b1}
0 \to {C\otimes_{R}M} \to Z' \to N' \to 0$$ with $Z' = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$, where $N'$ has locally finite injective dimension and ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,N')}){\leqslant}r$.

As ${C\otimes_{R}M}$ has locally finite injective dimension, we have ${C\otimes_{R}M}\in{{\mathcal{B}}_C}(R)$ by Fact \[disc01xy\](b), so ${\operatorname{Ext}_{R}^{1}(C,{C\otimes_{R}M})}=0$ and ${\operatorname{Hom}_{R}(C,{C\otimes_{R}M})}\cong M$. Thus, an application of ${\operatorname{Hom}_{R}(C,-)}$ to the sequence  yields the next exact sequence: $$\label{proof110609b2}
0 \to M \to {\operatorname{Hom}_{R}(C,Z')} \to {\operatorname{Hom}_{R}(C,N')} \to 0.$$ Since $N'$ has locally finite injective dimension, Fact \[fact110612a\] implies that the $R$-module $N:={\operatorname{Hom}_{R}(C,N')}$ has locally finite ${{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$. With $Z:={\operatorname{Hom}_{R}(C,Z')}$, it remains to show that $Z \cong \oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}$ and ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})}){\leqslant}r$. The first of these follows from the next sequence of isomorphisms $$\begin{aligned}
Z
&\cong{\operatorname{Hom}_{R}(C,\oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i})}\\
&\cong\oplus_{i=0}^r{\operatorname{Hom}_{R}(C,{\omega\otimes_{R}(R/(\underline z_{[i]})})}^{n_i}\\
&\cong\oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}\end{aligned}$$ where the last isomorphism is from Lemma \[lem110612b\]. To complete the proof, observe that since $N'$ has locally finite injective dimension, it is in ${{\mathcal{B}}_C}(R)$, so we have $N'\cong {C\otimes_{R}N}$. This implies that ${\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})})={\operatorname{pd}}_R({\operatorname{Hom}_{R}(\omega,N')})
{\leqslant}r$, as desired.

In the next definition, the special case $C=R$ recovers the complete injective resolutions and Gorenstein injective modules.

\[notation08b’\] Let $C$ be a semidualizing $R$-module. A *complete ${{\mathcal{I}}_C}{{\mathcal{I}}}$-coresolu-tion* is a complex $Y$ of $R$-modules such that $Y$ is exact and ${\operatorname{Hom}_{R}(U,Y)}$ is exact for each $U\in{{\mathcal{I}}_C}(R)$, $Y_i$ is injective for $i{\leqslant}0$, and $Y_i$ is $C$-injective for $i>0$. An $R$-module $N$ is *$\text{G}_C$-injective* if there exists a complete ${{\mathcal{I}}_C}{{\mathcal{I}}}$-coresolution $Y$ such that $N\cong{\operatorname{Ker}}(\partial^Y_0)$, and $Y$ is a *complete ${{\mathcal{I}}_C}{{\mathcal{I}}}$-coresolution of $N$*. Let ${{\mathcal{GI}}_C}(R)$ denote the class of $\text{G}_C$-injective $R$-modules, and set ${{\mathcal{G}}}({{\mathcal{I}}_C})={{\mathcal{GI}}_C}(R)\cap{{\mathcal{A}}_C}(R)$. We let ${{\mathcal{GI}}_C}\text{-}{\operatorname{id}}_R(-)$ and ${{\mathcal{G}}}({{\mathcal{I}}_C})\text{-}{\operatorname{id}}_R(-)$ denote the homological dimensions obtained from coresolutions in ${{\mathcal{GI}}_C}(R)$ and ${{\mathcal{G}}}({{\mathcal{I}}_C})$, respectively.

An $R$-module $M$ has *locally finite ${{\mathcal{G}}}({{\mathcal{I}}_C})\text{-}{\operatorname{id}}$* provided that ${{\mathcal{G}}}({{\mathcal{I}}}_{C_{{{\mathfrak{m}}}}})\text{-}{\operatorname{id}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$. An $R$-module $M$ has *locally finite ${{\mathcal{G}}}{{\mathcal{I}}_C}\text{-}{\operatorname{id}}$* provided that ${{\mathcal{G}}}{{\mathcal{I}}}_{C_{{{\mathfrak{m}}}}}\text{-}{\operatorname{id}}_{R_{{{\mathfrak{m}}}}}(M_{{{\mathfrak{m}}}})<\infty$ for each maximal ideal ${{\mathfrak{m}}}\subset R$.

The next lemma explains the relation between the different Gorenstein injective dimensions.

\[lem0701’\] Let $C$ be a semidualizing $R$-module. For an $R$-module $M$, the following conditions are equivalent:

1.  \[lem0701’i\] ${{\mathcal{G}}}({{\mathcal{I}}_C})\text{-}{\operatorname{id}}_R(M)<\infty$;

2.  \[lem0701’ii\] ${{\mathcal{G}}}{{\mathcal{I}}_C}\text{-}{\operatorname{id}}_R(M)<\infty$ and $M\in{{\mathcal{A}}_C}(R)$; and

3.  \[lem0701’iii\] ${\operatorname{Gid}}_R(C\otimes_RM)<\infty$ and $M\in{{\mathcal{A}}_C}(R)$.

When these conditions are satisfied, we have $$\begin{aligned}
 {{\mathcal{G}}}({{\mathcal{I}}_C})\text{-}{\operatorname{id}}_R(M)
  &
  ={{\mathcal{G}}}{{\mathcal{I}}_C}\text{-}{\operatorname{id}}_R(M)
  ={\operatorname{Gid}}_R(C\otimes_RM).  \end{aligned}$$

We actually need the following corollary of the previous result.

\[lem110613a\] Let $C$ be a semidualizing $R$-module. For an $R$-module $M$, the following conditions are equivalent:

1.  \[lem110613ai\] $M$ has locally finite ${{\mathcal{G}}}({{\mathcal{I}}_C})\text{-}{\operatorname{id}}$;

2.  \[lem110613aii\] $M$ has locally finite ${{\mathcal{G}}}{{\mathcal{I}}_C}\text{-}{\operatorname{id}}$ and $M\in{{\mathcal{A}}_C}(R)$; and

3.  \[lem110613aiii\] ${C\otimes_{R}M}$ has locally finite ${\operatorname{Gid}}$ and $M\in{{\mathcal{A}}_C}(R)$.

This follows from the definitions of the locally finite Gorenstein injective dimensions and the local-global principal for Auslander classes from Fact \[disc01xy\](b).

\[proof110609c\] Assume that $R$ is Cohen-Macaulay with a pointwise dualizing module $\omega$ and that $M$ is a non-zero finitely generated $R$-module in ${{\mathcal{A}}_C}(R)$ with locally finite ${{\mathcal{G}}}{{{\mathcal{I}}_C}\text{-}{\operatorname{id}}}$. Lemma \[lem110613a\] implies that ${C\otimes_{R}M}$ has locally finite G-injective dimension. Since $M\neq 0$, we have ${C\otimes_{R}M}\neq 0$, by Fact \[fact110527a\](b), so Theorem \[thm110608a\] provides an integer $r = {\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}M})})$, a proper $R$-regular sequence $\underline z = z_1, \ldots , z_r$, non-negative integers $n_0, n_1, \ldots , n_r$ with $n_r{\geqslant}1$, and an exact sequence $$\label{proof110609c1}
0 \to {C\otimes_{R}M} \to Z' \to N' \to 0$$ such that $Z' = \oplus_{i=0}^r(\omega/(\underline z_{[i]})\omega)^{n_i}$, $N'\in{{\mathcal{B}}}_{\omega}(R)$, and ${\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,N')}){\leqslant}r$.

Since $M$ is in ${{\mathcal{A}}_C}(R)$, we have ${\operatorname{Ext}_{R}^{1}(C,{C\otimes_{R}M})}=0$ and ${\operatorname{Hom}_{R}(C,{C\otimes_{R}M})}\cong M$. Thus, an application of ${\operatorname{Hom}_{R}(C,-)}$ to the sequence  yields the next exact sequence: $$\label{proof110609c2}
0 \to M \to {\operatorname{Hom}_{R}(C,Z')} \to {\operatorname{Hom}_{R}(C,N')} \to 0.$$ Note that $Z'$ has locally finite injective dimension, so we have $Z'\in{{\mathcal{B}}_C}(R)$, which implies that ${\operatorname{Hom}_{R}(C,Z')}$ and ${\operatorname{Hom}_{R}(C,N')}$ are in ${{\mathcal{A}}_C}(R)$ by Fact \[disc01xy\](b)–(c). With $Z:={\operatorname{Hom}_{R}(C,Z')}$, we have ${\mathrm{G}\text{-}\!\dim}_R({\operatorname{Hom}_{R}(\omega,{C\otimes_{R}N})}){\leqslant}r$ and $Z \cong \oplus_{i=0}^r({C^{\dagger}}/(\underline z_{[i]}){C^{\dagger}})^{n_i}$, as in Proof \[proof110609b\].

Acknowledgments {#acknowledgments .unnumbered}
===============

We are grateful to the referee for her/his thoughtful comments.

[10]{}

M. Auslander, *Anneaux de [G]{}orenstein, et torsion en algèbre commutative*, Séminaire d’Algèbre Commutative dirigé par Pierre Samuel, vol. 1966/67, Secrétariat mathématique, Paris, 1967. [MR ]{}[37 \#1435]{}

M. Auslander and M. Bridger, *Stable module theory*, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. [MR ]{}[42 \#4580]{}

L. L. Avramov and R.-O. Buchweitz, *Support varieties and cohomology over complete intersections*, Invent. Math. **142** (2000), no. 2, 285–318. [MR ]{}[1794064 (2001j:13017)]{}

L. L. Avramov, S. B. Iyengar, and J. Lipman, *Reflexivity and rigidity for complexes. [I]{}. [C]{}ommutative rings*, Algebra Number Theory **4** (2010), no. 1, 47–86. [MR ]{}[2592013]{}

L. L. Avramov and A. Martsinkovsky, *Absolute, relative, and [T]{}ate cohomology of modules of finite [G]{}orenstein dimension*, Proc. London Math. Soc. (3) **85** (2002), 393–440. [MR ]{}[2003g:16009]{}

L. W. Christensen, A. Frankild, and H. Holm, *On [G]{}orenstein projective, injective and flat dimensions—a functorial description with applications*, J. Algebra **302** (2006), no. 1, 231–279. [MR ]{}[2236602]{}

E. E. Enochs and O. M. G. Jenda, *Gorenstein injective and projective modules*, Math. Z. **220** (1995), no. 4, 611–633. [MR ]{}[1363858 (97c:16011)]{}

E. E. Enochs, O. M. G. Jenda, and J. Z. Xu, *Foxby duality and [G]{}orenstein injective and projective modules*, Trans. Amer. Math. Soc. **348** (1996), no. 8, 3223–3234. [MR ]{}[1355071 (96k:13010)]{}

H.-B. Foxby, *Gorenstein modules and related modules*, Math. Scand. **31** (1972), 267–284 (1973). [MR ]{}[48 \#6094]{}

[to3em]{}, *Gorenstein dimensions over [C]{}ohen-[M]{}acaulay rings*, Proceedings of the international conference on commutative algebra (W. Bruns, ed.), Universität Osnabrück, 1994, pp. 59–63.

A. Frankild and S. Sather-Wagstaff, *Reflexivity and ring homomorphisms of finite flat dimension*, Comm. Algebra **35** (2007), no. 2, 461–500. [MR ]{}[2294611]{}

E. S. Golod, *[$G$]{}-dimension and generalized perfect ideals*, Trudy Mat. Inst. Steklov. **165** (1984), 62–66, Algebraic geometry and its applications. [MR ]{}[85m:13011]{}

R. Hartshorne, *Residues and duality*, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin, 1966. [MR ]{}[36 \#5145]{}

M. Hochster and Y. Yao, *An embedding theorem for modules of finite ([G]{}-)projective dimension*, preprint (2009).

H. Holm and P. J[ø]{}rgensen, *Semi-dualizing modules and related [G]{}orenstein homological dimensions*, J. Pure Appl. Algebra **205** (2006), no. 2, 423–445. [MR ]{}[2203625]{}

H. Holm and D. White, *Foxby equivalence over associative rings*, J. Math. Kyoto Univ. **47** (2007), no. 4, 781–808. [MR ]{}[2413065]{}

S. Sather-Wagstaff, *Bass numbers and semidualizing complexes*, Commutative algebra and its applications, Walter de Gruyter, Berlin, 2009, pp. 349–381. [MR ]{}[2640315]{}

S. Sather-Wagstaff, T. Sharif, and D. White, *Tate cohomology with respect to semidualizing modules*, J. Algebra **324** (2010), no. 9, 2336–2368. [MR ]{}[2684143]{}

R. Takahashi and D. White, *Homological aspects of semidualizing modules*, Math. Scand. **106** (2010), no. 1, 5–22. [MR ]{}[2603458]{}

W. V. Vasconcelos, *Divisor theory in module categories*, North-Holland Publishing Co., Amsterdam, 1974, North-Holland Mathematics Studies, No. 14, Notas de Matemática No. 53. \[Notes on Mathematics, No. 53\]. [MR ]{}[0498530 (58 \#16637)]{}

D. White, *Gorenstein projective dimension with respect to a semidualizing module*, J. Commut. Algebra **2** (2010), no. 1, 111–137. [MR ]{}[2607104]{}

[^1]: This material is based on work supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442. The author was supported in part by a grant from the NSA

[^2]: This is also now called “Gorenstein projective dimension” after the work of Enochs, Jenda, and others. However, this is a generalization of Auslander and Bridger’s original notion for non-finitely-generated modules. Since we are focused on finitely generated modules here, we continue with Auslander and Bridger’s terminology.

[^3]: It is worth noting that these notions appeared (more or less) implicitly in several places. However, to the best of our knowledge these are the first places where ${\text{G}_C}$-dimension and ${{\mathcal{P}}_C}$-projective dimension were studied explicitly.

[^4]: Ths can be partially explained by the fact that, at the time of the writing of this article, the localization question for G-injective dimension is still not completely answered. Remark \[disc110726a\].

[^5]: In [@hochster:etmf] the term “global canonical module” is used in place of “pointwise dualizing module”. Our choice follows the terminology of Grothendieck and Hartshorne [@hartshorne:rad].

[^6]: We observe the same convention for any homological dimension of the zero module.

[^7]: Most good behavior is also known when $R$ has a dualizing complex, but we restrict our attention to the Cohen-Macaulay case. For instance, the conclusion of Lemma \[lem110610a\] holds when $R$ is only assumed to have a pointwise dualizing complex; the proof is the same, using results from [@christensen:ogpifd] in place of the results from [@enochs:fdgipm].

[^8]: It is worth noting that the results in [@enochs:fdgipm] assume that $R$ is local, hence with finite Krull dimension; also, the results of [@christensen:ogpifd] assume implicitly that $R$ has finite Krull dimension since the definition of a dualizing complex used there includes an assumption of finite injective dimension. Contrast this with our definition of pointwise dualizing module, and with Grothendieck’s definition of a pointwise dualizing complex from [@hartshorne:rad].

[^9]: Note that this uses the characterization of totally reflexive modules in terms of “complete resolutions” found in [@avramov:svcci (4.4.4)]; see also [@avramov:aratc Theorem 3.1]. Specifically, an $R$-module $M$ is totally reflexive if and only if there is an exact complex $F=\cdots{\xrightarrow}{\partial^F_1}F_0{\xrightarrow}{\partial^F_0}F_{-1}{\xrightarrow}{\partial^F_{-1}}\cdots$ such that ${\operatorname{Hom}_{R}(F,R)}$ is exact and $M\cong{\operatorname{Im}}(\partial^F_0)$. Note that the local result was also announced in [@foxby:gdcmr Corollary 3.3].
