#  on flat and gorenstein flat dimensions of local cohomology modules

Majid Rahro Zargar  and  Hossein Zakeri  M.R. Zargar and H. Zakeri, Faculty of
mathematical sciences and computer, Kharazmi University, 599 Taleghani Avenue,
Tehran 15618, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and let $M$ be a
relative Cohen-Macaulay $R$–module with respect to a proper ideal
$\mathfrak{a}$ of $R$ and set $n:=\mbox{ht}\,_{M}\mathfrak{a}$. We prove that
$\mbox{fd}\,_{R}M<\infty$ if and only if
$\mbox{fd}\,_{R}\mbox{H}^{n}_{\mathfrak{a}}(M)<\infty$, and that
$\mbox{fd}\,_{R}\mbox{H}^{n}_{\mathfrak{a}}(M)=\mbox{fd}\,_{R}M+n$. This
result provides some characterizations of Gorenstein local rings which, under
some additional assumptions, have been exposed in the literature. We also
prove that $\mbox{Gpd}\,_{R}M<\infty$ if and only if
$\mbox{Gfd}\,_{R}\mbox{H}^{n}_{\mathfrak{m}}(M)<\infty$ and that
$\mbox{Gfd}\,_{R}\mbox{H}^{n}_{\mathfrak{m}}(M)=\mbox{Gpd}\,_{R}M+n$ whenever
$M$ is Cohen-Macaulay with $\mbox{dim}\,M=n$. As an application of this
result, we show that its $G_{C}$–Gorenstein projective version holds true,
where $C$ is a semidualizing $R$–module.

###### Key words and phrases:

Flat dimension, Gorenstein injective dimension, Gorenstein flat dimension,
Local cohomology, Relative Cohen-Macaulay module, Semidualizing module.

###### 2000 Mathematics Subject Classification:

13D05, 13D45, 18G20

##  1\. introduction

Throughout this paper, $R$ is a commutative Noetherian ring, $\mathfrak{a}$ is
a proper ideal of $R$ and $M$ is an $R$-module. From section 3, we assume that
$R$ is local with maximal ideal $\mathfrak{m}$. In this case, $\hat{R}$
denotes the $\mathfrak{m}$–adic completion of $R$ and
$\mbox{E}(R/\mathfrak{m})$ denotes the injective hull of the residue field
$R/\mathfrak{m}$. For each non-negative integer $i$, let
$\mbox{H}_{\mathfrak{a}}^{i}(M)$ denotes the $i$-th local cohomology module of
$M$ with respect to $\mathfrak{a}$ (see [3] for its definition and basic
results). Also, we use $\mbox{id}\,_{R}(M)$, $\mbox{pd}\,_{R}(M)$ and
$\mbox{fd}\,_{R}(M)$, respectively, to denote the usual injective, projective
and flat dimensions of $M$ respectively. The notions of Gorenstein injective,
Gorenstein projective and Gorenstein flat, were introduced by Enochs and Jenda
in [9]. Notice that, the classes of Gorenstein injective, Gorenstein
projective and Gorenstein flat modules include, respectively the classes of
injective, projective and flat modules. Recently, the authors proved, in [17,
Theorem 2.5], that if $M$ is a certain module over a local ring $R$, then
$\mbox{id}\,_{R}(M)$ and
$\mbox{id}\,_{R}(\mbox{H}_{\mathfrak{a}}^{\small{\mbox{ht}\,_{M}\mathfrak{a}}}(M))$
are simultaneously finite and the equality
$\mbox{id}\,_{R}(\mbox{H}_{\mathfrak{a}}^{\small{\mbox{ht}\,_{M}\mathfrak{a}}}(M))=\mbox{id}\,_{R}(M)-\mbox{ht}\,_{M}\mathfrak{a}$
holds. Also, a counterpart of this result was established in Gorenstein
homological algebra. Indeed, under the additional assumption that $R$ has a
dualizing complex, it was proved that $\mbox{Gid}\,_{R}M<\infty$ implies
$\mbox{Gid}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)<\infty$ and that the converse
holds whenever both $R$ and $M$ are Cohen-Macaulay. As an application of this
result, it was shown that the equality
$\mbox{Gid}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)=\mbox{Gid}\,_{R}M-n$ holds,
whenever $M$ is a Cohen-Macaulay module over the Cohen-Macaulay local ring
$(R,\mathfrak{m})$ and $\mbox{dim}\,M=n$.

The principal aim of this paper is to study, in like manner, the projective
(resp. Gorenstein projective) dimension of certain $R$-modules in terms of
flat (resp. Gorenstein flat) dimension of its local cohomology modules.

In this paper we will use the concept of relative Cohen-Macaulay modules which
has been studied in [12] under the title of cohomologically complete
intersections and continued in [17] and [16]. The organization of this paper
is as follows. In section 3, we prove, in 3.1, that if $M$ is relative Cohen-
Macaulay with respect to $\mathfrak{a}$ and $\mbox{ht}\,_{M}{\mathfrak{a}}=n$,
then $\mbox{fd}\,_{R}M$ and $\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)$
are simultaneously finite and
$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)=\mbox{fd}\,_{R}M+n$. Next, in
3.3, we prove that a $d$-dimensional finitely generated $R$–module $M$ with
finite projective dimension is Cohen-Macaulay if and only if
$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{pd}\,_{R}M+d$. Notice
that, this result is a generalization of [9, Proposition 9.5.22]. In 3.5, 3.6
and 3.7, we also generalize some results of the author and H. Zakeri, which
have been proved in [16] and [17] under some additional assumptions. In 3.8,
we generalize the well-known result that if the local ring $R$ admits a non-
zero Cohen-Macaulay module of finite projective dimension, then $R$ is Cohen-
Macaulay. Indeed, it is shown that $R$ is Cohen-Macaulay if it admits a non-
zero Cohen-Macaulay $R$–module with finite $\mbox{C-pd}\,_{R}(M)$, where $C$
is a semidualizing $R$–module. In section 4, as a main result, a Gorenstein
projective version of 3.1 is demonstrated in a certain case. Indeed, it is
shown, in 4.2, that if $M$ is Cohen-Macaulay with $\mbox{dim}\,M=n$, then
$\mbox{Gpd}\,_{R}M<\infty$ if and only if
$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)<\infty$ and, moreover, the
equality $\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)=\mbox{Gpd}\,_{R}M+n$
holds. Also, in 4.3, we provide a Gorenstein projective version of 3.3.
Finally, with the aid of the above result, it is proved that if $C$ is a
semidualizing $R$-module and $M$ is Cohen-Macaulay with $\mbox{dim}\,M=n$,
then the quantities $\mbox{$G_{C}$-pd}\,_{R}M$ and
$\mbox{$G_{C}$-fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)$ are simultaneously
finite and the equality
$\mbox{$G_{C}$-fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)=\mbox{$G_{C}$-pd}\,_{R}M+n$
holds.

##  2\. preliminaries

In this section we recall some definitions and facts which are needed
throughout this paper.

######  Definition 2.1.

_Following[20, Definition 2.1], let $\mathcal{X}$ be a class of $R$-modules
and let $M$ be an $R$-module. An $\mathcal{X}$-coresolution of $M$ is a
complex of $R$-modules in $\mathcal{X}$ of the form_

$X=0\longrightarrow
X_{0}\stackrel{{\scriptstyle\partial_{0}^{X}}}{{\longrightarrow}}X_{-1}\stackrel{{\scriptstyle\partial_{-1}^{X}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\partial_{n+1}^{X}}}{{\longrightarrow}}X_{n}\stackrel{{\scriptstyle\partial_{n}^{X}}}{{\longrightarrow}}X_{n-1}\stackrel{{\scriptstyle\partial_{n-1}^{X}}}{{\longrightarrow}}\cdots$

_such that $\mbox{H}_{0}(X)\cong M$ and $\mbox{H}_{n}(X)=0$ for all $n\leq-1$.
The $\mathcal{X}$-injective dimension of $M$ is the quantity_

${\mbox{$\mathcal{X}$-id}\,_{R}(M)}=\inf\\{\sup\\{-n\geq 0|X_{n}\neq
0\\}~{}|~{}X~{}\text{is an $\mathcal{X}$-coresolution of $M$~{}}\\}.$

_The modules of $\mathcal{X}$-injective dimension zero are precisely the non-
zero modules of $\mathcal{X}$ and also
$\mbox{$\mathcal{X}$-id}\,_{R}(0)=-\infty$._

_Dually, an $\mathcal{X}$-resolution of $M$ is a complex of $R$-modules in
$\mathcal{X}$ of the form_

$X=\cdots\longrightarrow
X_{n}\stackrel{{\scriptstyle\partial_{n}^{X}}}{{\longrightarrow}}X_{n-1}\stackrel{{\scriptstyle\partial_{n-1}^{X}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle\partial_{2}^{X}}}{{\longrightarrow}}X_{1}\stackrel{{\scriptstyle\partial_{1}^{X}}}{{\longrightarrow}}X_{0}\longrightarrow
0$

_such that $\mbox{H}_{0}(X)\cong M$ and $\mbox{H}_{n}(X)=0$ for all $n\geq 1$.
The $\mathcal{X}$-projective dimension of $M$ is the quantity_

${\mbox{$\mathcal{X}$-pd}\,_{R}(M)}=\inf\\{\sup\\{n\geq 0|X_{n}\neq
0\\}~{}|~{}X~{}\text{is an $\mathcal{X}$-resolution of $M$~{}}\\}.$

_The modules of $\mathcal{X}$-projective dimension zero are precisely the non-
zero modules of $\mathcal{X}$ and also
$\mbox{$\mathcal{X}$-pd}\,_{R}(0)=-\infty$._

_The following notion of semidualizing modules goes back at least to
Vasconcelos[23], but was rediscovered by others. The reader is referred to
[19] for more details about semidualizing modules._

######  Definition 2.2.

_A finitely generated $R$-module $C$ is called semidualizing if the natural
homomorphism $R\rightarrow\mbox{Hom}\,_{R}(C,C)$ is an isomorphism and
$\mbox{Ext}\,_{R}^{i}(C,C)=0$ for all $i\geq 1$. An $R$-module $D$ is said to
be a dualizing $R$-module if it is semidualizing and has finite injective
dimension. For a semidualizing $R$-module $C$, we set_

$\begin{array}[]{rl}&\mathcal{I}_{C}(R)=\\{~{}\mbox{Hom}\,_{R}(C,I)|~{}~{}I~{}~{}\text{is
an injective $R$-module}\\},\\\
&\mathcal{P}_{C}(R)=\\{~{}C\otimes_{R}P|~{}~{}P~{}~{}\text{is a projective
$R$-module}\\},\\\
&\mathcal{F}_{C}(R)=\\{~{}C\otimes_{R}F|~{}~{}F~{}~{}\text{is a flat
$R$-module}\\}.\end{array}$

_The $R$–modules in $\mathcal{I}_{C}(R)$, $\mathcal{P}_{C}(R)$ and
$\mathcal{F}_{C}(R)$ are called $C$–injective, $C$–projective and $C$–flat,
respectively. For convenience the quantities
$\mathcal{I_{C}}(R)$-$\mbox{id}\,_{R}M$ and
$\mathcal{P_{C}}(R)$-$\mbox{pd}\,_{R}M$, which are defined as in 2.1, are
denoted by $\mbox{C-id}\,_{R}(M)$ and $\mbox{C-pd}\,_{R}(M)$ respectively.
Notice that when $C=R$ these notions recover the concepts of injective and
projective dimensions, respectively. _

Based on the work of E.E. Enochs and O.M.G. Jenda [9], the following notions
were introduced and studied by H. Holm and P. J${\o}$rgensen [14].

######  Definitions 2.3.

_Let $C$ be a semidualizing $R$-module. A complete
$\mathcal{I}_{C}\mathcal{I}$-coresolution is a complex $Y$ of $R$-modules such
that_

  * (i)

$Y$ is exact and $\mbox{Hom}\,_{R}(I,Y)$ is exact for each
$I\in\mathcal{I}_{C}(R)$, and that

  * (ii)

$Y_{i}\in\mathcal{I}_{C}(R)$ for all $i>0$ and $Y_{i}$ is injective for all
$i\leq 0$.

_An $R$-module $M$ is called $G_{C}$-injective if there exists a complete
$\mathcal{I}_{C}\mathcal{I}$-coresolution $Y$ such that
$M\cong\ker(\partial_{0}^{Y})$. In this case $Y$ is a complete
$\mathcal{I}_{C}\mathcal{I}$-coresolution of $M$. The class of
$G_{C}$-injective $R$-modules is denoted by $\mathcal{GI_{C}}(R)$._

_A complete $\mathcal{P}_{C}\mathcal{P}$-resolution is a complex $X$ of
$R$–modules such that_

  * (i)

$X$ is exact and $\mbox{Hom}\,_{R}(X,P)$ is exact for each
$P\in\mathcal{P}_{C}(R)$, and that

  * (ii)

$X_{i}\in\mathcal{P}_{C}(R)$ for all $i<0$ and $X_{i}$ is projective for all
$i\geq 0$.

_An $R$-module $M$ is called $G_{C}$-projective if there exists a complete
$\mathcal{P}_{C}\mathcal{P}$-resolution $X$ such that
$M\cong\mbox{Coker}\,(\partial_{1}^{X})$. In this case $X$ is a complete
$\mathcal{P}_{C}\mathcal{P}$-resolution of $M$. The class of
$G_{C}$-projective $R$-modules is denoted by $\mathcal{GP_{C}}(R)$._

_A complete $\mathcal{F}_{C}\mathcal{F}$-resolution is a complex $Z$ of
$R$–modules such that_

  * (i)

$Z$ is exact and $Z\otimes_{R}I$ is exact for each $I\in\mathcal{I}_{C}(R)$,
and that

  * (ii)

$Z_{i}\in\mathcal{F}_{C}(R)$ for all $i<0$ and $Z_{i}$ is flat for all $i\geq
0$.

_An $R$-module $M$ is called $G_{C}$-flat if there exists a complete
$\mathcal{F}_{C}\mathcal{F}$-resolution $Z$ such that
$M\cong\mbox{Coker}\,(\partial_{1}^{Z})$. In this case $Z$ is a complete
$\mathcal{F}_{C}\mathcal{F}$-resolution of $M$. The class of $G_{C}$-flat
$R$-modules is denoted by $\mathcal{GF_{C}}(R)$._

_For convenience the $\mathcal{GI_{C}}(R)$-$\mbox{id}\,_{R}M$,
$\mathcal{GP_{C}}(R)$-$\mbox{pd}\,_{R}M$ and
$\mathcal{GF_{C}}(R)$-$\mbox{pd}\,_{R}M$ of $M$ which are defined as in 2.1
are denoted by $\mbox{$G_{C}$-id}\,_{R}(M)$, $\mbox{$G_{C}$-pd}\,_{R}M$ and
$\mbox{$G_{C}$-fd}\,_{R}M$, respectively. Note that when $C=R$ these notions
are exactly the concepts of Gorenstein injective, Gorenstein projective and
Gorenstein flat dimensions which were introduced in [9]._

######  Definition 2.4.

_We say that a finitely generated $R$-module $M$ is relative Cohen Macaulay
with respect to $\mathfrak{a}$ if there is precisely one non-vanishing local
cohomology module of $M$ with respect to $\mathfrak{a}$. Clearly this is the
case if and only if
$\mbox{grade}\,(\mathfrak{a},M)=\mbox{cd}\,(\mathfrak{a},M)$, where
$\mbox{cd}\,(\mathfrak{a},M)$ is the largest integer $i$ for which
$\mbox{H}_{\mathfrak{a}}^{i}(M)\neq 0$. Observe that the notion of relative
Cohen-Macaulay module is connected with the notion of cohomologically complete
intersection ideal which has been studied in [12]. _

######  Remark 2.5.

_Let $M$ be a relative Cohen-Macaulay module with respect to $\mathfrak{a}$
and let $\mbox{cd}\,(\mathfrak{a},M)=n$. Then, in view of [3, theorems 6.1.4,
4.2.1, 4.3.2], it is easy to see that
$\mbox{Supp}\,\mbox{H}^{n}_{\mathfrak{a}}(M)=\mbox{Supp}\,({M}/{\mathfrak{a}M})$
and $\mbox{ht}\,_{M}\mathfrak{a}=\mbox{grade}\,(\mathfrak{a},M)$, where
$\mbox{ht}\,_{M}\mathfrak{a}=\inf\\{\
\mbox{dim}\,_{R_{\mathfrak{p}}}M_{\mathfrak{p}}|~{}\mathfrak{p}\in\mbox{Supp}\,(M/\mathfrak{a}M)~{}\\}$._

Next, we recall some elementary results about the trivial extension of a ring
by a module.

######  Definition and Facts 2.6.

_Let $C$ be an $R$-module. Then the direct sum $R\oplus C$ has the structure
of a commutative ring with respect to the multiplication defined by_

$(a,c)(a^{\prime},c^{\prime})=(aa^{\prime},ac^{\prime}+a^{\prime}c),$

_for all $(a,c),(a^{\prime},c^{\prime})\in R\oplus C$. This ring is called
trivial extension of $R$ by $C$ and is denoted by $R\ltimes C$. The following
properties of $R\ltimes C$ are needed in this paper._

  * (i)

_There are natural ring homomorphisms $R\rightleftarrows R\ltimes C$ which
enable us to consider $R$-modules as $R\ltimes C$-modules, and vice versa._

  * (ii)

_For any ideal $\mathfrak{a}$ of $R$, $\mathfrak{a}\oplus C$ is an ideal of
$R\ltimes C$._

  * (iii)

$(R\ltimes C,\mathfrak{m}\oplus C)$ is a Noetherian local ring whenever
$(R,\mathfrak{m})$ is a Noetherian local ring and $C$ is a finitely generated
$R$-module. Also, in this case, $\mbox{dim}\,R=\mbox{dim}\,R\ltimes C$.

The classes defined next is collectively known as Foxby classes. The reader is
referred to [19] for some basic results about those classes.

######  Definition 2.7.

_Let $C$ be a semidualizing $R$-module. The Bass class with respect to $C$ is
the class $\mathcal{B_{C}}(R)$ of $R$–modules $M$ such that _

  * (i)

_$\mbox{Ext}\,_{R}^{i}(C,M)=0=\mbox{Tor}\,^{R}_{i}(C,\mbox{Hom}\,_{R}(C,M))$
for all $i\geq 1$_, and that

  * (ii)

_the natural evaluation map $C\otimes_{R}\mbox{Hom}\,_{R}(C,M)\rightarrow M$
is an isomorphism_.

_Dually, the Auslander class with respect to $C$, denoted by
$\mathcal{A}_{C}(R)$, consists of all $R$-modules $M$ such that_

  * (i)

_$\mbox{Tor}\,^{R}_{i}(C,M)=0=\mbox{Ext}\,_{R}^{i}(C,C\otimes_{R}M)$ for all
$i\geq 1$_, and that

  * (ii)

_the natural map $M\rightarrow\mbox{Hom}\,_{R}(C,C\otimes_{R}M)$ is an
isomorphism_.

##  3\. local cohomology and flat dimension

The following theorem, which is one of the main results of this section,
provides a comparison between the flat dimensions of a relative Cohen-Macaulay
module and its non-zero local cohomology module. Here we adopt the convention
that the flat dimension of the zero module is to be taken as $-\infty$.

######  Theorem 3.1.

Let $n$ be a non-negative integer such that $\emph{H}^{i}_{\mathfrak{a}}(M)=0$
for all $i\neq n$.

  * (i)

If _$\mbox{fd}\,_{R}M <\infty$_, then
_$\mbox{fd}\,_{R}\mbox{H}^{n}_{\mathfrak{a}}(M) <\infty$._

  * (ii)

The converse holds whenever $M$ is finitely generated.

Furthermore, if $M$ is non-zero finitely generated, then
_$\mbox{fd}\,_{R}\mbox{H}^{n}_{\mathfrak{a}}(M)=\mbox{fd}\,_{R}M+n=\mbox{pd}\,_{R}M+n$_.

###### Proof.

(i): First notice that we may assume $\mbox{H}^{n}_{\mathfrak{a}}(M)\neq 0$.
Let $\mbox{fd}\,_{R}M=s$ and let $c$ be the arithmetic rank of $\mathfrak{a}$.
Then, there exists a sequence $x_{1},\ldots,x_{c}$ of elements of $R$ such
that $\sqrt{\mathfrak{a}}=\sqrt{(x_{1},\ldots,x_{c})}$. We notice that $n\leq
c$. Let $C(R)^{\bullet}$ denotes the $\check{C}$ech complex of $R$ with
respect to $x_{1},\ldots,x_{c}$. Let $N$ be an arbitrary $R-$module and let
$F_{\bullet}$ be a free resolution for $N$. For the first quadrant bicomplex
$\mathcal{M}=\\{M_{p,q}=F_{p}\otimes_{R}M\otimes_{R}C_{c-q}\\}$ we denote the
total complex of $\mathcal{M}$ by $\mbox{Tot}\,(\mathcal{M})$. Now, with the
notation of [18], $\mbox{E}^{1}$ is the bigraded module whose $(p,q)$ term is
$\mbox{H}^{{}^{\prime\prime}}_{q}(M_{p,*})$, the q-th homology of the p-th
column. Since $F_{p}$ is flat, by assumption we have

${}^{I}E_{p,q}^{1}=\mbox{H}^{{}^{\prime\prime}}_{q}(M_{p,*})=\begin{cases}0&\text{if
$q\neq c-n$}\\\ F_{p}\otimes_{R}\mbox{H}^{n}_{\mathfrak{a}}(M)&\text{if
$q=c-n$},\end{cases}$

therefore

${}^{I}E_{p,q}^{2}=\mbox{H}^{{}^{\prime}}_{p}\mbox{H}^{{}^{\prime\prime}}_{q}(\mathcal{M})=\begin{cases}0&\text{if
$q\neq c-n$}\\\
\mbox{Tor}\,^{R}_{p}(N,\mbox{H}^{n}_{\mathfrak{a}}(M))&\text{if
$q=c-n$};\end{cases}$

and hence the spectral sequence collapses. Note that, in view of [18, Theorem
10.16] we have
${}^{I}E_{p,q}^{2}\underset{p}{\Longrightarrow}\mbox{H}_{p+q}(\mbox{Tot}\,(\mathcal{M}))$
for all $p,q$. Thus, for all $t=p+q$, there is the following filtration

${0}=\Phi^{-1}H_{t}\subseteq\Phi^{0}H_{t}\subseteq\ldots\subseteq\Phi^{t-1}H_{t}\subseteq\Phi^{t}H_{t}=H_{t}$

such that ${}^{I}E_{p,q}^{\infty}\cong\Phi^{p}H_{t}/\Phi^{p-1}H_{t}$.
Therefore, one can use the above filtration to see that

(3.1)
$\mbox{Tor}\,^{R}_{p}(N,\mbox{H}^{n}_{\mathfrak{a}}(M))\cong\mbox{H}_{p+c-n}(\mbox{Tot}\,(\mathcal{M}))$

for all $p$.

A similar argument applies to the second iterated homology, using the fact
that each $C_{c-q}$ is flat, yields

${}^{II}E_{p^{{}^{\prime}},q^{{}^{\prime}}}^{2}=\mbox{H}^{{}^{\prime\prime}}_{p^{{}^{\prime}}}\mbox{H}^{{}^{\prime}}_{q^{{}^{\prime}}}(\mathcal{M})=\begin{cases}0&\text{if
$q^{{}^{\prime}}>s$}\\\
\mbox{H}_{\mathfrak{a}}^{c-p^{{}^{\prime}}}(\mbox{Tor}\,^{R}_{q^{{}^{\prime}}}(N,M))&\text{if
$q^{{}^{\prime}}\leq s.$}\end{cases}$

Now, we claim that ${}^{II}E_{p^{{}^{\prime}},q^{{}^{\prime}}}^{\infty}=0$ for
all $p^{{}^{\prime}},q^{{}^{\prime}}$ such that
$p^{{}^{\prime}}+q^{{}^{\prime}}=p+c-n$ and that $p>s+n$. To this end, first
notice that, by [18, Theorem 10.16], we have
${}^{II}E_{p^{{}^{\prime}},q^{{}^{\prime}}}^{2}\underset{p_{{}^{\prime}}}{\Longrightarrow}\mbox{H}_{p^{{}^{\prime}}+q^{{}^{\prime}}}(\mbox{Tot}\,(\mathcal{M}))$.
If $q^{{}^{\prime}}>s$, there is nothing to prove. Let $q^{{}^{\prime}}\leq
s$. Then $0>c-p^{{}^{\prime}}$ and hence
${}^{II}E_{p^{{}^{\prime}},q^{{}^{\prime}}}^{2}=0$; which in turn yields
${}^{II}E_{p^{{}^{\prime}},q^{{}^{\prime}}}^{\infty}=0$. Now, by using a
similar filtration as above, one can see that
$\mbox{H}_{p+c-n}(\mbox{Tot}\,(\mathcal{M}))=0$ for all $p>s+n$. Therefore
$\mbox{Tor}\,^{R}_{p}(N,\mbox{H}^{n}_{\mathfrak{a}}(M))=0$ for all $p>s+n$;
and hence $\mbox{fd}\,_{R}(\mbox{H}_{\mathfrak{a}}^{n}(M))\leq s+n$.

(ii): First, notice that $\mbox{Tor}\,^{R}_{i}(R/\mathfrak{m},M)$ is
$\mathfrak{a}$-torsion for all $i$. Therefore, by using the same arguments as
above, one can deduce that

${}^{II}E_{p^{{}^{\prime}},q^{{}^{\prime}}}^{2}=\mbox{H}^{{}^{\prime\prime}}_{p^{{}^{\prime}}}\mbox{H}^{{}^{\prime}}_{q^{{}^{\prime}}}(\mathcal{M})=\begin{cases}0&\text{if
$p^{{}^{\prime}}\neq c$}\\\
\mbox{Tor}\,^{R}_{q^{{}^{\prime}}}(R/\mathfrak{m},M)&\text{if
$p^{{}^{\prime}}=c$.}\end{cases}$

Thus, the spectral sequence collapses at the c-th column; and hence we get the
isomorphism
$\mbox{Tor}\,^{R}_{q^{{}^{\prime}}}(R/\mathfrak{m},M)\cong\mbox{H}_{q^{{}^{\prime}}+c}(\mbox{Tot}\,(\mathcal{M}))$
for all $q^{{}^{\prime}}$. It therefore follows, by the isomorphism (3.1),
that

(3.2)
$\mbox{Tor}\,_{p}^{R}(R/\mathfrak{m},\mbox{H}_{\mathfrak{a}}^{n}(M))\cong\mbox{Tor}\,_{p-n}^{R}(R/\mathfrak{m},M)$

for all $p$. Now, suppose for a moment that $M$ is non-zero and finitely
generated. Then, one can use [18, Corollary 8.54], to see that
$\mbox{fd}\,_{R}M<\infty$. The final assertion is a consequence of (i) and
(ii). ∎

######  Corollary 3.2.

Suppose that $R$ is relative Cohen-Macaulay with respect to $\mathfrak{a}$ and
that _$\mbox{ht}\,_{R}\mathfrak{a}=n$_. Then, for every non-zero faithfully
flat $R$–module $M$ we have
_$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)=n$._

###### Proof.

Let $M$ be a non-zero faithfully flat $R$–module. Since the functor
$\mbox{H}_{\mathfrak{a}}^{n}(-)$ is right exact, we have
$\mbox{H}_{\mathfrak{a}}^{n}(M)\cong\mbox{H}_{\mathfrak{a}}^{n}(R)\otimes_{R}M$;
and hence by assumption $\mbox{H}_{\mathfrak{a}}^{n}(M)\neq 0$ and
$\mathfrak{m}M\neq M$. By, [18, Theorem 5.40], there is a directed index set
$I$ and a family of finitely generated free $R$–modules $\\{M_{i}\\}_{i\in I}$
such that $M=\underset{{i\in I}}{\varinjlim}M_{i}$. Notice that each $M_{i}$
is relative Cohen-Macaulay with respect to $\mathfrak{a}$ and that
$\mbox{ht}\,_{M_{i}}\mathfrak{a}=n$. Therefore,
$\mbox{H}_{\mathfrak{a}}^{j}(M)=\underset{i\in
I}{\varinjlim}\mbox{H}_{\mathfrak{a}}^{j}(M_{i})=0$ for all $j\neq n$; and
hence, in view of 3.1(i), we get
$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)\leq n$. Now, if
$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)<n$, then
$\mbox{Tor}\,^{R}_{n}(R/\mathfrak{m},\mbox{H}_{\mathfrak{a}}^{n}(M))=0$. But,
by the isomorphism (3.2), which is proved without finitely generated
assumption on $M$, we have
$\mbox{Tor}\,_{n}^{R}(R/\mathfrak{m},\mbox{H}_{\mathfrak{a}}^{n}(M))\cong
M/\mathfrak{m}M\neq 0$ which is a contradiction. ∎

The next proposition is a generalization of [9, Proposition 9.5.22].

######  Proposition 3.3.

Let $M$ be a $d$-dimensional finitely generated $R$–module of finite
projective dimension. Then the following statements are equivalent.

  * (i)

_$M$ is Cohen-Macaulay._

  * (ii)

_$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{pd}\,_{R}M+d$_.

  * (iii)

_$\mbox{pd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{pd}\,_{R}M+d$_.

Moreover, if one of the above statements holds, then $R$ is Cohen-Macaulay.

###### Proof.

We first notice that the Artinian $R$–module $\mbox{H}_{\mathfrak{m}}^{d}(M)$
has a natural $\hat{R}$–module structure and that
$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{fd}\,_{\hat{R}}(\mbox{H}_{\mathfrak{m}}^{d}(M))$.
Now, assume that $\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)<\infty$. Then,
in view of [15, Proposition 6] and [11, Theorem 3.2.6], we see that
$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)\leq\mbox{pd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)\leq\mbox{dim}\,R$.
Next, by [4, Theorem 3.1.17], [6, Theorem 4.16] and the New Intersection
Theorem, one can deduce that

$\mbox{fd}\,_{\hat{R}}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{id}\,_{\hat{R}}\mbox{Hom}\,_{\hat{R}}(\mbox{H}_{\mathfrak{m}}^{d}(M),\mbox{E}_{\hat{R}}(\hat{R}/\mathfrak{m}\hat{R}))=0pt\hat{R}=\mbox{dim}\,R.$

It therefore follows that
$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{pd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{dim}\,R$
and that $R$ is Cohen-Macaulay.

Now, the implications (ii)$\Leftrightarrow$(iii) follows immediately from the
above argument.

(ii)$\Rightarrow$(i): Since
$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)<\infty$, one can use the
conclusion of the above argument in conjunction with the Auslander-Buchsbaum
Theorem [4, Theorem 1.3.3] to see that $M$ is Cohen-Macaulay. Finally the
implication (i)$\Rightarrow$(ii) follows from 3.1.

∎

The following corollary, which is an immediate consequence of the pervious
proposition, has been proved in [9, Proposition 9.5.22] under the extra
condition that the underlying ring admits a canonical module.

######  Corollary 3.4.

Suppose that _$\mbox{dim}\,R=d$_. Then the following statements are
equivalent.

  * (i)

_$R$ is a Cohen-Macaulay ring._

  * (ii)

_$\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(R)=d$_.

  * (iii)

_$\mbox{pd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(R)=d$_.

The next proposition has been proved in [16, Proposition 3.3] under the extra
conditions that the underlying ring is Cohen-Macaulay and admits a dualizing
complex.

######  Proposition 3.5.

Let $C$ be a semidualizing $R$-module. Then the following statements are
equivalent.

  * (i)

$C$ is a dualizing $R$-module.

  * (ii)

_$\mbox{$G_{C}$-id}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(R) <\infty$ for all
ideals $\mathfrak{a}$ of $R$ such that $R$ is relative Cohen-Macaulay with
respect to $\mathfrak{a}$ and that $\mbox{ht}\,_{R}\mathfrak{a}=n.$ _

  * (iii)

$\emph{\mbox{$G_{C}$-id}\,}_{R}\emph{\mbox{H}}^{n}_{\mathfrak{a}}(R)<\infty$
for some ideal $\mathfrak{a}$ of $R$ such that $R$ is relative Cohen-Macaulay
with respect to $\mathfrak{a}$ and that
$\emph{\mbox{ht}\,}_{R}\mathfrak{a}=n.$

###### Proof.

The implication (i)$\Rightarrow$(ii) follows from [16, Theorem 3.2(ii)] and
the implication (ii)$\Rightarrow$(iii) is clear.

(iii)$\Rightarrow$(i): Suppose that
$\mbox{$G_{C}$-id}\,_{R}\mbox{H}^{n}_{\mathfrak{a}}(R)<\infty$, where
$\mathfrak{a}$ is an ideal of $R$ such that $R$ is relative Cohen-Macaulay
with respect to $\mathfrak{a}$ and that $\mbox{ht}\,_{R}\mathfrak{a}=n$. Then,
in view of 3.1, $\mbox{fd}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(R)<\infty$. Hence,
one can use [15, Proposition 6] to see that
$\mbox{pd}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(R)<\infty$. Therefore, by [20,
Theorem 2.3], we have
$\mbox{$G_{C}$-id}\,_{R}\mbox{H}^{n}_{\mathfrak{a}}(R)=\mbox{C-id}\,_{R}\mbox{H}^{n}_{\mathfrak{a}}(R)$.
Hence, one can use [16, Theorem 3.2(ii)] to complete the proof. ∎

An immediate consequence of the previous proposition is the next Corollary,
which has been proved in [17, Corollary 3.10] with additional assumptions that
$R$ is Cohen-Macaulay and admits a dualizing complex.

######  Corollary 3.6.

The following statements are equivalent.

  * (i)

_$R$ is a Gorenstein ring._

  * (ii)

_$\mbox{Gid}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(R) <\infty$ for all ideals
$\mathfrak{a}$ of $R$ such that $R$ is relative Cohen-Macaulay with respect to
$\mathfrak{a}$ and that $\mbox{ht}\,_{R}\mathfrak{a}=n.$ _

  * (iii)

_$\mbox{Gid}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(R) <\infty$ for some ideal
$\mathfrak{a}$ of $R$ such that $R$ is relative Cohen-Macaulay with respect to
$\mathfrak{a}$ and that $\mbox{ht}\,_{R}\mathfrak{a}=n.$ _

It follows from the proof of [16, Theorem 3.2(i)] that if $n$ is a non-
negative integer and $M$ is an $R$–module( not necessarily finitely generated)
such that ${\mbox{H}}^{i}_{\mathfrak{a}}(M)=0$ for all $i\neq n$ and that
$\mbox{C-id}\,_{R}M$ is finite, then
$\mbox{C-id}\,_{R}{\mbox{H}}^{n}_{\mathfrak{a}}(M)$ is finite. This fact leads
us to the following proposition which recovers [16, Theorem 3.8].

######  Proposition 3.7.

Let $C$ be a semidualizing $R$-module. Consider the following statements.

  * (i)

$R$ is Gorenstein.

  * (ii)

_$\mbox{C-id}\,_{R}{\mbox{H}}^{n}_{\mathfrak{a}}(C) <\infty$_ for all ideals
$\mathfrak{a}$ of $R$ such that $R$ is relative Cohen-Macaulay with respect to
$\mathfrak{a}$ and that $\emph{\mbox{ht}\,}_{R}\mathfrak{a}=n$.

  * (iii)

_$\mbox{C-id}\,_{R}{\mbox{H}}^{n}_{\mathfrak{a}}(C) <\infty$_ for some ideal
$\mathfrak{a}$ of $R$ such that $R$ is relative Cohen-Macaulay with respect to
$\mathfrak{a}$ and that $\emph{\mbox{ht}\,}_{R}\mathfrak{a}=n$.

_Then, the implications (i) $\Rightarrow$(ii)$\Rightarrow$(iii) hold true, and
the implication (iii)$\Rightarrow$(i) holds true whenever $R$ is Cohen-
Macaulay._

###### Proof.

First, notice that $R\cong C$ whenever $R$ is Gorenstein. Hence, the
implication (i)$\Rightarrow$(ii) follows from [17, Theorem 2.5(i)]. The
implication (ii)$\Rightarrow$(iii) is clear.

(iii)$\Rightarrow$(i): Let $\mathfrak{a}$ be an ideal of $R$ such that $R$ is
relative Cohen-Macaulay with respect to $\mathfrak{a}$ and that
$\mbox{ht}\,_{R}\mathfrak{a}=n$. Since
$\mbox{Supp}\,_{R}(C)=\mbox{Spec}\,(R)$, in view of [7, Theorem 2.2], we get
$\mbox{cd}\,(\mathfrak{a},R)=\mbox{cd}\,(\mathfrak{a},C)$. On the other hand,
by [19, Theorem 2.2.6(c)],
$\mbox{grade}\,(\mathfrak{a},R)=\mbox{grade}\,(\mathfrak{a},C)$. Hence, 2.4
implies that $C$ is relative Cohen-Macaulay with respect to $\mathfrak{a}$.
Also, by [3, Theorem 3.4.10], the local cohomology functor
$\mbox{H}_{\mathfrak{a}}^{i}(-)$ commutes with direct limits and any
$R$-module can be viewed as a direct limit of its finitely generated
submodules. It therefore follows that the functor
$\mbox{H}_{\mathfrak{a}}^{n}(-)$ is right exact. Thus, in view of [3, Exercise
6.1.9], we have
$\mbox{H}_{\mathfrak{a}}^{n}(R)\otimes_{R}C\cong\mbox{H}_{\mathfrak{a}}^{n}(C)$.
Since
$\mbox{H}_{\mathfrak{m}}^{0}(\mbox{E}(R/\mathfrak{m}))=\mbox{E}(R/\mathfrak{m})$
and for any non-maximal prime ideal $\mathfrak{p}$ of $R$,
$\mbox{H}_{\mathfrak{m}}^{0}(\mbox{E}(R/\mathfrak{p}))=0$, we may apply [17,
Proposition 2.8] to see that
$\mbox{H}_{\mathfrak{m}}^{i}(\mbox{H}_{\mathfrak{a}}^{n}(C))=\mbox{H}_{\mathfrak{m}}^{n+i}(C)$
for all $i\geq 0$. Therefore, by considering the additional assumption that
$R$ is Cohen-Macaulay, one can deduce that

$\mbox{H}_{\mathfrak{m}}^{i}(\mbox{H}_{\mathfrak{a}}^{n}(C))=\begin{cases}0&\text{if
$i\neq\mbox{dim}\,R/\mathfrak{a}$}\\\ \mbox{H}^{d}_{\mathfrak{m}}(C)&\text{if
$i=\mbox{dim}\,R/\mathfrak{a},$}\end{cases}$

where $d=\mbox{dim}\,R$. Thus, by the assumption and [16, Theorem 3.2(i)], we
see that $\mbox{C-id}\,_{R}{\mbox{H}}^{d}_{\mathfrak{m}}(C)$ is finite. Now,
one can use [16, Theorem 3.8] to complete the proof. ∎

It is known that if a local ring admits a non-zero Cohen-Macaulay $R$–module
of finite projective dimension, then it is a Cohen-Macaulay ring. The
following theorem is a generalization of this result.

######  Theorem 3.8.

Let $C$ be a semidualizing $R$–module. If there exists a non-zero Cohen-
Macaulay $R$-module $M$ with finite _$\mbox{C-pd}\,_{R}M$_ , then $R$ is
Cohen-Macaulay.

###### Proof.

Let $M$ be a non-zero Cohen-Macaulay $R$–module of dimension $n$ such that
$\mbox{C-pd}\,_{R}M$ is finite. Notice that, in view of [22, Theorem 2.11(c)],
we have $\mbox{C-pd}\,_{R}M=\mbox{pd}\,_{R}\mbox{Hom}\,_{R}(C,M)$. Also, since
$C\otimes_{R}\hat{R}$ is a semidualizing $\hat{R}$–module and
$\mbox{Hom}\,_{\hat{R}}(\hat{C},\hat{M})\cong\mbox{Hom}\,_{R}(C,M)\otimes_{R}\hat{R}$,
we may assume that $R$ is complete. Now, by using [22, Corollary 2.9(a)], we
have $M\in\mathcal{B}_{C}(R)$. Therefore,
$\mbox{Tor}\,_{i}^{R}(C,\mbox{Hom}\,_{R}(C,M))=0$ for all $i>0$ and
$C\otimes_{R}\mbox{Hom}\,_{R}(C,M)\cong M$. Hence, one can use [1, Theorem
1.2] to obtain the following equalities

$\begin{array}[]{rl}0pt_{R}M&=0pt_{R}(C\otimes_{R}\mbox{Hom}\,_{R}(C,M))\\\
&=0pt_{R}C-0ptR+0pt_{R}\mbox{Hom}\,_{R}(C,M)\\\
&=0pt_{R}\mbox{Hom}\,_{R}(C,M).\end{array}$

On the other hand, since
$\mbox{Ass}\,_{R}(\mbox{Hom}\,_{R}(C,M))=\mbox{Ass}\,_{R}(M)$ and $M$ is
Cohen-Macaulay, we see that
$\mbox{dim}\,_{R}(M)=\mbox{dim}\,_{R}(\mbox{Hom}\,_{R}(C,M))$. Therefore,
$\mbox{Hom}\,_{R}(C,M)$ is Cohen-Macaulay. Hence, one can use 3.1 to see that
the injective dimension of the finitely generated $R$–module
$\mbox{Hom}\,_{R}(\mbox{H}_{\mathfrak{m}}^{n}(\mbox{Hom}\,_{R}(C,M)),\mbox{E}_{R}(R/\mathfrak{m}))$
is finite. Therefore, by the New Intersection Theorem, $R$ is Cohen-Macaulay.
∎

Applying Theorem 3.8 to the semidualizing $R$–module $C=R$, we immediately
obtain the following well-known result.

######  Corollary 3.9.

If $R$ admits a non-zero Cohen-Macaulay module of finite projective dimension,
then $R$ is Cohen-Macaulay.

######  Proposition 3.10.

Let $M$ be relative Cohen-Macaulay with respect to $\mathfrak{a}$ and set
_$\mbox{ht}\,_{M}\mathfrak{a}=n$_. Suppose that
_$\mbox{Gid}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)$_ and
_$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)$_ are finite. Then
_$\mbox{pd}\,_{R}M$_ is finite if and only if _$\mbox{id}\,_{R}M$_ is finite.

###### Proof.

Let $\mbox{pd}\,_{R}M<\infty$. Then, in view of 3.1, we have
$\mbox{fd}\,_{R}(\mbox{H}_{\mathfrak{a}}^{n}(M))<\infty$. Therefore, by the
assumption and [13, Theorem 2.6(i)],
$\mbox{id}\,_{R}\mbox{H}_{\mathfrak{a}}^{n}(M)<\infty$. Hence, in view of [17,
Theorem 2.5], $\mbox{id}\,_{R}M<\infty$. Now, to prove the reverse, suppose
that $\mbox{id}\,_{R}M<\infty$. Then, by [17, Theorem 2.5],
$\mbox{id}\,_{R}(\mbox{H}_{\mathfrak{a}}^{n}(M))<\infty$. Therefore, by the
assumption and [6, Proposition 4.21],
$\mbox{fd}\,_{R}(\mbox{H}_{\mathfrak{a}}^{n}(M))<\infty$. Hence, in view of
3.1, we have $\mbox{pd}\,_{R}M<\infty$. ∎

######  Remark 3.11.

_If $R$ is Gorenstein, then, by [6, theorems 3.14 and 4.11],
$\mbox{Gfd}\,_{R}M$ and $\mbox{Gid}\,_{R}M$ are finite for any $R$–module $M$.
Therefore 3.10 implies the well-known fact that, for a finitely generated
$R$–module $M$, $\mbox{pd}\,_{R}M$ is finite if and only if $\mbox{id}\,_{R}M$
is finite._

##  4\. local cohomology and gorenstein flat dimension

The starting point of this section is the next lemma which has been proved, in
[17, Lemma 3.7] and [17, Corollary 3.9], under the extra assumption that $R$
is Cohen-Macaulay.

######  Lemma 4.1.

Suppose that $M$ is a non-zero finitely generated $R$–module. Then the
following statements hold true.

  * (i)

_Suppose that $x\in\mathfrak{m}$ is both $R$–regular and $M$–regular. Then
$\mbox{Gid}\,_{R}M<\infty$ if and only if
$\mbox{Gid}\,_{{R/xR}}(M/xM)<\infty$._

  * (ii)

_Assume that $M$ is Cohen-Macaulay of dimension $n$. Then
$\mbox{Gid}\,_{R}M<\infty$ if and only if
$\mbox{Gid}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)<\infty$._

###### Proof.

First notice that, by [6, Theorem 3.24],
$\mbox{Gid}\,_{R}M=\mbox{Gid}\,_{\hat{R}}\hat{M}$. On the other hand, since
$\mbox{H}_{\mathfrak{m}}^{i}(M)$ is Artinian, in view of [21, Lemma 3.6], we
have
$\mbox{Gid}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)=\mbox{Gid}\,_{\hat{R}}\mbox{H}_{\mathfrak{m}}^{n}(M)=\mbox{Gid}\,_{\hat{R}}\mbox{H}_{\mathfrak{m}\hat{R}}^{n}(\hat{M})$.
Thus, we can assume that $R$ is complete; and hence it has a dualizing complex
$D$.

(i): Set $\overline{R}=R/xR$. We notice that
$\mbox{fd}\,_{R}\overline{R}<\infty$ and
${\mu}^{i+\small{0ptR}}(\mathfrak{m},R)=\mu^{i+\small{0pt\overline{R}}}(\overline{\mathfrak{m}},\overline{R})$
for all $i\in\mathbb{Z}$, where $\mu^{i}(\mathfrak{m},R)$ denotes the i-th
Bass number of $R$ with respect to $\mathfrak{m}$. Hence, by using [2, 2.11],
we see that $D\otimes_{R}^{\mathbf{L}}\overline{R}$ is a dualizing complex for
$\overline{R}$. On the other hand, by the assumption, one can deduce that
$\mbox{Tor}\,^{R}_{i}(\overline{R},M)=0$ for all $i>0$. Therefore
$\overline{M}\simeq M\otimes_{R}^{\mathbf{L}}\overline{R}$, in derived
category $\mathcal{D}(R)$. Now, we can use [5, Theorem 5.5] to complete the
proof.

(ii): Let $M$ be Cohen-Macaulay with $\mbox{dim}\,M=n$. Then, the implication
($\Rightarrow$) follows from [17, Theorem 3.8(i)]. To prove the converse, we
proceed by induction on $n$. The case $n=0$ is obvious. Assume that $n>0$ and
that the result has been proved for $n-1$. Now, by using [17, Theorem
3.12(ii)] in conjunction with the assumption, one can choose an element $x$ in
$\mathfrak{m}$ which is both $R$-regular and $M$-regular. Next, we can use the
induced exact sequence

$0\longrightarrow{\mbox{H}}^{n-1}_{\mathfrak{m}}(M/xM)\longrightarrow{\mbox{H}}^{n}_{\mathfrak{m}}(M)\longrightarrow{\mbox{H}}^{n}_{\mathfrak{m}}(M)\longrightarrow
0$

and [6, Proposition 3.9] to see that
$\mbox{Gid}\,_{R}({\mbox{H}}^{n-1}_{\mathfrak{m}}(M/xM))$ is finite. Hence, by
the inductive hypothesis, $\mbox{Gid}\,_{R}M/xM$ is finite. Therefore, in view
of [6, Theorem 7.6(b)], $\mbox{Gid}\,_{R/xR}M/xM<\infty$. It therefore follows
from part(i) that $\mbox{Gid}\,_{R}M$ is finite. Now the result follows by
induction.

∎

The following theorem, which is the main result of this section, provides a
comparison between the Gorenstein projective dimensions of a Cohen-Macaulay
module and its non-zero local cohomology module with support in
$\mathfrak{m}$.

######  Theorem 4.2.

Suppose that the $R$–module $M$ is Cohen-Macaulay of dimension $n$. Then
_$\mbox{Gpd}\,_{R}(M)$_ and
_$\mbox{Gfd}\,_{R}(\mbox{H}_{\mathfrak{m}}^{n}(M))$_ are simultaneously finite
and when they are finite, there is an equality
_$\mbox{Gfd}\,_{R}(\mbox{H}_{\mathfrak{m}}^{n}(M))=\mbox{Gpd}\,_{R}(M)+n.$_

###### Proof.

First notice that, in view of [6, Theorem 4.27] and [6, propositions 2.16 and
1.26], we have
$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{Gfd}\,_{\hat{R}}\mbox{H}_{\mathfrak{m}\hat{R}}^{d}(\hat{M})$
and $\mbox{Gpd}\,_{R}(M)=\mbox{Gpd}\,_{\hat{R}}(\hat{M})$. Therefore, without
lose of generality, we can assume that $R$ is complete; and hence it is a
homomorphic image of a Gorenstein local ring $(S,\mathfrak{n})$ of dimension
$d$. Now, in view of the local duality theorem [3, Theorem 11.2.6], we have

(4.1)
$\mbox{H}_{\mathfrak{m}}^{n}(M)\cong\mbox{Hom}\,_{R}(\mbox{Ext}\,_{S}^{d-n}(M,S),\mbox{E}(R/\mathfrak{m})).$

Next, we notice that $M$ is a Cohen-Macaulay $S$–module of dimension $n$; and
hence, by [4, Theorem 3.3.10(c)(i)], the $S$–module
$\mbox{Ext}\,_{S}^{d-n}(M,S)$ is Cohen-Macaulay of dimension $n$. So that it
is a Cohen-Macaulay $R$–module. Therefore, again, we can use the local duality
theorem and [4, Theorem 3.3.10(c)(iii)] to obtain the following isomorphisms

(4.2)
$\begin{array}[]{rl}\mbox{H}_{\mathfrak{m}}^{n}(\mbox{Ext}\,_{S}^{d-n}(M,S))&\cong\mbox{Hom}\,_{R}(\mbox{Ext}\,_{S}^{d-n}(\mbox{Ext}\,_{S}^{d-n}(M,S),S),\mbox{E}(R/\mathfrak{m}))\\\
&\cong\mbox{Hom}\,_{R}(M,\mbox{E}(R/\mathfrak{m})).\end{array}$

Now, with the aid of the above arguments, there are equivalences

$\begin{array}[]{rl}\mbox{Gpd}\,_{R}(M)<\infty&\Leftrightarrow\mbox{Gid}\,_{R}(\mbox{Hom}\,_{R}(M,\mbox{E}(R/\mathfrak{m})))<\infty\Leftrightarrow\mbox{Gid}\,_{R}(\mbox{H}_{\mathfrak{m}}^{n}(\mbox{Ext}\,_{S}^{d-n}(M,S)))<\infty\\\
&\Leftrightarrow\mbox{Gid}\,_{R}(\mbox{Ext}\,_{S}^{d-n}(M,S))<\infty\Leftrightarrow\mbox{Gfd}\,_{R}(\mbox{H}_{\mathfrak{m}}^{n}(M))<\infty.\end{array}$

Notice that, in the above equivalences, the first step follows from [6,
Theorem 4.16 and Proposition 4.24], the second step is from (4.2), the third
step is clear by 4.1(ii), and the fourth step is an immediate consequence of
(4.1) and [6, Theorem 4.25].

For the final assertion, we notice that, since
$\mbox{H}_{\mathfrak{m}}^{n}(M)$ is Artinian and $R$ is complete, the
$R$–module
$\mbox{Hom}\,_{R}(\mbox{H}_{\mathfrak{m}}^{n}(M),\mbox{E}(R/\mathfrak{m}))$ is
finitely generated. Therefore, in view of [6, theorems 3.24 and 4.16], we get
the following equalities

$\begin{array}[]{rl}\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)&=\mbox{Gid}\,_{R}\mbox{Hom}\,_{R}(\mbox{H}_{\mathfrak{m}}^{n}(M),\mbox{E}(R/\mathfrak{m}))\\\
&=0ptR\\\ &=\mbox{Gpd}\,_{R}M+n,\end{array}$

where the last equality follows from [6, Proposition 4.24 and Theorem 1.25]. ∎

The following proposition is a Gorenstein projective version of 3.3.

######  Proposition 4.3.

Assume that $R$ is Cohen-Macaulay and that $M$ is a $d$–dimensional finitely
generated $R$–module of finite Gorenstein projective dimension. Then the
following statements are equivalent.

  * (i)

_$M$ is Cohen-Macaulay. _

  * (ii)

_$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{Gpd}\,_{R}M+d$_.

  * (iii)

_$\mbox{Gpd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{Gpd}\,_{R}M+d$_.

###### Proof.

The implication (i)$\Rightarrow$(ii) follows from 4.2.
(ii)$\Rightarrow$(i),(iii): Since $R$ has finite Kurll dimension and
$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M$) is finite, we have the
finitness of $\mbox{Gpd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)$ by [10, Theorem
3.4]. Hence, by [8, Corollary 2.4],
$\mbox{Gpd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)\leq\mbox{dim}\,R$. Therefore,
in view of [6, Theorem 4.23], we get the following inequalities

$\mbox{Gpd}\,_{R}M+d=\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)\leq\mbox{Gpd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)\leq\mbox{dim}\,R.$

Now, one can use [6, Proposition 2.16 and Theorem 1.25] to see that
$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{Gpd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)$
and that $0ptM=\mbox{dim}\,M$. Thus, $M$ is Cohen-Macaulay and (iii) holds
true.

(iii)$\Rightarrow$(ii): First, we notice that, by [6, Theorem 4.27],
$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{Gfd}\,_{\hat{R}}\mbox{H}_{\mathfrak{m}\hat{R}}^{d}(\hat{M})$
and that, in view of [6, propositions 4.23 and 2.20], the following
inequalities hold:

$\begin{array}[]{rl}\mbox{Gfd}\,_{\hat{R}}\mbox{H}_{\mathfrak{m}\hat{R}}^{d}(\hat{M})&\leq\mbox{Gpd}\,_{\hat{R}}\mbox{H}_{\mathfrak{m}\hat{R}}^{d}(\hat{M})\\\
&\leq\mbox{Gpd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}({M}).\end{array}$

Now, since $\mbox{H}_{\mathfrak{m}}^{d}(M)$ is an Artinian $\hat{R}$–module,
one can use [6, Theorem 4.16] to see that the finitely generated
$\hat{R}$–module
$\mbox{Hom}\,_{{R}}(\mbox{H}_{\mathfrak{m}}^{d}(M),\mbox{E}(R/\mathfrak{m}))$
is of finite Gorenstein injective dimension. Therefore, by [6, Theorem 3.24]
and [6, Theorem 4.16],
$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{Gid}\,_{\hat{R}}\mbox{Hom}\,_{{R}}(\mbox{H}_{\mathfrak{m}}^{d}(M),\mbox{E}(R/\mathfrak{m}))=\mbox{dim}\,R$.
Hence, one can use [8, Corollary 2.4] and above inequalities to complete the
proof.

∎

Next, we single out a certain case of 4.3 which is a Gorenstein projective
version of Corollary 3.4. Notice that the proof of the following corollary is
similar to the proof of 4.3(ii)$\Rightarrow$(i).

######  Corollary 4.4.

Suppose that _$\mbox{dim}\,R=d$_. Then the following statements are
equivalent.

  * (i)

_$R$ is Cohen-Macaulay._

  * (ii)

_$\mbox{Gfd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(R)=d$_.

The following proposition is a generalization of Theorem 4.2 in terms of
$G_{C}$–dimensions.

######  Proposition 4.5.

Let $C$ be a semidualizing $R$–module and let $M$ be a Cohen-Macaulay
$R$–module of dimension $n$. Then the following statements are equivalent.

  * (i)

_$\mbox{$G_{C}$-pd}\,_{R}M <\infty$_.

  * (ii)

_$\mbox{$G_{C}$-fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M) <\infty$_.

Furthermore,
_$\mbox{$G_{C}$-fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)=\mbox{$G_{C}$-pd}\,_{R}M+n$_.

###### Proof.

First, we notice that, in view of [3, Theorem 4.2.1],
$\mbox{H}_{\mathfrak{m}}^{i}(M)\cong\mbox{H}_{\mathfrak{m}\oplus C}^{i}(M)$
for all $i$. Therefore, $M$ is a Cohen-Macaulay $R\ltimes C$–module with
dimension $n$. On the other hand, by using [14, Theorem 2.16], we have
$\mbox{$G_{C}$-pd}\,_{R}M=\mbox{Gpd}\,_{R\ltimes C}M$ and
$\mbox{$G_{C}$-fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{n}(M)=\mbox{Gfd}\,_{R\ltimes
C}\mbox{H}_{\mathfrak{m}}^{n}(M)=\mbox{Gfd}\,_{R\ltimes
C}\mbox{H}_{\mathfrak{m}\oplus C}^{n}(M)$. Hence, by replacing $R$ with
$R\ltimes C$, one can use 4.2 to complete the proof.

∎

The following corollary is a consequence of pervious proposition and 4.3.

######  Corollary 4.6.

Let $R$ be Cohen-Macaulay, $C$ be a semidualizing $R$–module and let $M$ be a
$d$–dimensional finitely generated $R$–module of finite $G_{C}$–projective
dimension. Then the following statements are equivalent.

  * (i)

_$M$ is Cohen-Macaulay._

  * (ii)

_$\mbox{$G_{C}$-fd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{$G_{C}$-pd}\,_{R}M+d$_.

  * (iii)

_$\mbox{$G_{C}$-pd}\,_{R}\mbox{H}_{\mathfrak{m}}^{d}(M)=\mbox{$G_{C}$-pd}\,_{R}M+d$_.

###### Proof.

We notice that, by using [4, Exercise 1.2.26] and [19, Theorem 2.2.6], one can
deduce that $(R\ltimes C,\mathfrak{m}\oplus C)$ is a Cohen-Macaulay local
ring. Also, $M$ is a Cohen-Macaulay $R$–module if and only if $M$ is a Cohen-
Macaulay $R\ltimes C$–module. Therefore, the assertion follows from 4.3 and
4.5. ∎

## References

  * [1] M. Auslander, _Modules over unramified regular local rings,_ Illinois J. Math. 5 (1961) 631–647.
  * [2] L.L. Avramov and H-B. Foxby, _Ring homomorphisms and finite Gorenstein dimension,_ Proc. London Math. Soc. (3) 75 (1997) 241–270.
  * [3] M.P. Brodmann and R.Y. Sharp, _Local cohomology: An algebraic introduction with geometric applications_ , Cambridge University Press, Cambridge, 1998.
  * [4] W. Bruns and J. Herzog,_Cohen-Macaulay rings_ , Cambridge University Press, Cambridge, 1993\.
  * [5] L.W. Christensen, A. Frankild and H. Holm, _On Gorenstein projective, injective and flat dimensions A functorial description with applications_ , J. Algebra. 302 (2006) 231–279.
  * [6] L.W. Christensen, H-B. Foxby and H. Holm, _Beyond totally reflexive modules and back_ , In: Noetherian and Non-Noetherian Perspectives, edited by M. Fontana, S-E. Kabbaj, B. Olberding and I. Swanson, Springer Science+Business Media, LLC, New York, 2011 101–143.
  * [7] K. Divaani-Aazar, R. Naghipour and M. Tousi, _Cohomological dimension of certain algebraic varieties_ , Proc. Amer. Math. Soc. 130 (2002) 3537–3544.
  * [8] E.E. Enochs, O.M.G. Jenda, Jinzhong Xu, _Foxby duality and Gorenstein injective and projective modules_ , Trans. Amer. Math. Soc. 348(8) (1996) 3223–3234.
  * [9] E.E. Enochs and O.M.G. Jenda,_Relative homological algebra_ , de Gruyter, Berlin, 2000.
  * [10] M.A. Esmkhani, M. Tousi, _Gorenstein homological dimensions and Auslander categories_ , J. Algebra. 308(1) (2007) 321–329.
  * [11] L. Gruson and M. Raynaud, _Critères de platitude et de projectivité_. Techniques de ”platifi cation” d’un module, Invent. Math. 13 (1971) 1–89.
  * [12] M. Hellus and P. Schenzel, On cohomologically complete intersections, Journal of Algebra. 320 (2008) 3733–3748.
  * [13] H. Holm, _Rings with finite Gorenstein injective dimension_ , Proc. Amer. Math. Soc. 132 (2004) 1279–1283.
  * [14] H. Holm and P. Jørgensen, _Semidualizing modules and related Gorenstein homological dimension_ , J. Pure Appl. Algebra. 205 (2006) 423–445.
  * [15] C. U. Jensen, _On the Vanishing of $\underset{}{\varprojlim}^{(i)}$,_ J. Algebra 15 (1970) 151–166.
  * [16] M. Rahro Zargar,_Local cohomology modules and Gorenstein injectivity with respect to a semidualizing module_ , Arch. Math. (Basel) 100 (2013) 25–34.
  * [17] M. Rahro Zargar and H. Zakeri, _On injective and Gorenstein injective dimensions of local cohomology modules_ , to appear in Algebar Colloquium.
  * [18] J.J. Rotman, _An introduction to homological algebra_ , Second ed., Springer, New York, 2009.
  * [19] S. Sather-Wagstaff, _Semidualizing modules,_ http://www.ndsu.edu/pubweb/ ssatherw.
  * [20] S. Sather-Wagstaff and S. Yassemi,_Modules of finite homological dimension with respect to a semidualizing module_ , Arch. Math. (Basel) 93 (2009) 111–121.
  * [21] R. Sazeedeh, _Gorenstein injective of the section functor,_ Forum Mathematicum, 22 (2010) 1117–1127.
  * [22] R. Takahashi and D. White, _Homological aspects of semidualizing modules,_ Math. Scand. 106 (2010) 5–22.
  * [23] W. V. Vasconcelos, _Divisor theory in module categories_ , North-Holland Math. Stud. 14, North-Holland Publishing Co., Amsterdam (1974).