Cohen-Macaulay homological dimensions

TL;DR

This paper introduces new Cohen-Macaulay homological dimensions for complexes, characterizes Cohen-Macaulay rings, and relates these dimensions to existing invariants, advancing the understanding of homological properties in commutative algebra.

Contribution

It defines Cohen-Macaulay projective, injective, and flat dimensions for complexes and explores their relationships with existing homological invariants.

Findings

Cohen-Macaulay dimensions characterize Cohen-Macaulay rings.

Cohen-Macaulay flat dimension lies between Gorenstein flat and large restricted flat dimensions.

Cohen-Macaulay injective dimension lies between Gorenstein injective dimension and Chouinard invariant.

Abstract

We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.

Equations

$$\text{mbox}CM-dim{}_R(M):=$$

$$\displaystyle\inf\left\{\mbox{G-dim}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{G-dim}\,_{Q}(R^{\prime})\bigg{|}\begin{array}[]{l}R\rightarrow R^{\prime}\leftarrow Q\text{ is a }\\ \mbox{CM}\,\text{-quasi-deformation }\end{array}\right\}$$

$$\mbox{Rfd}\,_{R}(M)\leq\mbox{CM${}_{*}$\text{-}fd}\,_{R}(M)\leq\mbox{Gfd}\,_{R}(M),$$

$$\mbox{CM${}_{*}$\text{-}fd}\,_{R}(M)=\sup\{0ptR_{\mathfrak{p}}-0pt_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\mid\mathfrak{p}\in\mbox{Spec}\,(R)\},$$

$$\sup\{0ptR_{\mathfrak{p}}-0pt_{R_{\mathfrak{p}}}M_{\mathfrak{p}}|\mathfrak{p}\in\mbox{Spec}\,(R)\}\leq\mbox{CM${}_{*}$\text{-}id}\,_{R}(M)\leq\mbox{Gid}\,_{R}(M),$$

$$\displaystyle\mbox{CM${}_{*}$\text{-}id}\,_{R}(M)=$$

$$=$$

$$\text{mbox}G-dim{}_R(M):=-\inf (\mathbf{R}\text{mbox}Hom{}_R(M,R)),$$

$$X=\cdots \stackrel{{\partial }_2^X}{\to }{P}_1\stackrel{{\partial }_1^X}{\to }{P}_0\stackrel{{\partial }_0^X}{\to }{P}_{-1}\stackrel{{\partial }_{-1}^X}{\to }{P}_{-2}\stackrel{{\partial }_{-2}^X}{\to }\cdots$$

$$Y=\cdots \stackrel{{\partial }_2^Y}{\to }{F}_1\stackrel{{\partial }_1^Y}{\to }{F}_0\stackrel{{\partial }_0^Y}{\to }{F}_{-1}\stackrel{{\partial }_{-1}^Y}{\to }{F}_{-2}\stackrel{{\partial }_{-2}^Y}{\to }\cdots$$

$$Z=\cdots \stackrel{{\partial }_2^Z}{\to }{I}_1\stackrel{{\partial }_1^Z}{\to }{I}_0\stackrel{{\partial }_0^Z}{\to }{I}_{-1}\stackrel{{\partial }_{-1}^Z}{\to }{I}_{-2}\stackrel{{\partial }_{-2}^Z}{\to }\cdots$$

$$\text{mbox}Gpd{}_R(M):=\inf \{\sup \{n\mid H_n\ne 0\}\mid H\simeq M\text{ is a }G\text{-projective resolution}\}.$$

$$\text{mbox}CM-dim{}_R(M):=$$

$$\displaystyle\inf\left\{\mbox{G-dim}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{G-dim}\,_{Q}(R^{\prime})\bigg{|}\begin{array}[]{l}R\rightarrow R^{\prime}\leftarrow Q\text{ is a }\\ \mbox{CM}\,\text{-quasi-deformation }\end{array}\right\},$$

$$\text{mbox}CM-dim{}_R(M)=0ptR-0p{t}_R(M).$$

$$\mbox{CM${}_{*}$\text{-}pd}\,_{R}(M):=\inf\left\{\mbox{Gpd}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime})\bigg{|}\begin{array}[]{l}R\rightarrow R^{\prime}\leftarrow Q\text{ is a }\\ \mbox{CM}\,\text{-quasi-deformation}\end{array}\right\}$$

$$\mbox{CM${}_{*}$\text{-}fd}\,_{R}(M):=\inf\left\{\mbox{Gfd}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime})\bigg{|}\begin{array}[]{l}R\rightarrow R^{\prime}\leftarrow Q\text{ is a }\\ \mbox{CM}\,\text{-quasi-deformation}\end{array}\right\}$$

$$\mbox{CM${}_{*}$\text{-}id}\,_{R}(M):=\inf\left\{\mbox{Gid}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime})\bigg{|}\begin{array}[]{l}R\rightarrow R^{\prime}\leftarrow Q\text{ is a }\\ \mbox{CM}\,\text{-quasi-deformation}\end{array}\right\}$$

$$\mbox{CM${}_{*}$\text{-}pd}\,_{R}(M)\leq\mbox{Gpd}\,_{R}(M),$$

$$\mbox{CM${}_{*}$\text{-}fd}\,_{R}(M)\leq\mbox{Gfd}\,_{R}(M),$$

$$\mbox{CM${}_{*}$\text{-}id}\,_{R}(M)\leq\mbox{Gid}\,_{R}(M).$$

$$\mathrm{ht}I=$$

$$=$$

$$\text{mbox}Gid{}_{\hat{R}}(M{\otimes }_R\hat{R})=0pt\hat{R}-\inf (M{\otimes }_R\hat{R})=0ptR-\inf (M)=\text{mbox}Gid{}_R(M).$$

$$\mbox{CM${}_{*}$\text{-}id}\,_{R}(M)=\inf\left\{\mbox{Gid}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime})\bigg{|}\begin{array}[]{l}R\rightarrow R^{\prime}\leftarrow Q\text{ is a }\\ \mbox{CM}\,\text{-quasi-deformation}\\ \text{such that }Q\text{ is complete}\end{array}\right\}.$$

$$\text{mbox}Gid{}_Q(M{\otimes }_R{R}^{\prime })=\text{mbox}Gid{}_{\hat{Q}}(M{\otimes }_R{R}^{\prime }{\otimes }_Q\hat{Q})=\text{mbox}Gid{}_{\hat{Q}}(M{\otimes }_R\widehat{R^{\prime }}),$$

$$\mbox{CM${}_{*}$\text{-}fd}\,_{R}(M)=\inf\left\{\mbox{Gfd}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime})\bigg{|}\begin{array}[]{l}R\rightarrow R^{\prime}\leftarrow Q\text{ is a }\\ \mbox{CM}\,\text{-quasi-deformation}\\ \text{such that }Q\text{ is complete}\end{array}\right\}.$$

$$\text{mbox}Gpd{}_{\hat{R}}(M{\otimes }_R\hat{R})\le \text{mbox}Gpd{}_R(M)$$

$$\mbox{CM${}_{*}$\text{-}pd}\,_{R}(M)=\inf\left\{\mbox{Gpd}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime})\bigg{|}\begin{array}[]{l}R\rightarrow R^{\prime}\leftarrow Q\text{ is a }\\ \mbox{CM}\,\text{-quasi-deformation}\\ \text{such that }Q\text{ is complete}\end{array}\right\}.$$

$$\text{mbox}Gfd{}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\le \text{mbox}Gfd{}_R(M),$$

Full text

1. Introduction

A commutative Noetherian local ring $R$ is regular if the residue field $k$ has finite projective dimension and only if all $R$-modules have finite projective dimension [2] [25]. This theorem of Auslander, Buchsbaum and Serre is a main motivation of studing homological dimensions. The injective and flat dimensions have similar behavior.

Auslander and Bridger [1], introduced a homological dimension for finitely generated modules designed to single out modules with properties similar to those of modules over Gorenstein rings. They called it G-dimension and it is a refinement of the projective dimension and showed that a local Noetherian ring $(R,\mathfrak{m},k)$ is Gorenstein if the residue field $k$ has finite G-dimension and only if all finitely generated $R$-modules have finite G-dimension.

To extend the G-dimension beyond the realm of finitely generated modules over Noetherian rings, Enochs and Jenda [12] introduced the notion of Gorenstein projective module. Then the notion of Gorenstein projective dimension was studied in [7].

The notion of Gorenstein injective module is dual to that of Gorenstein projective module and were introduced in the same paper by Enochs and Jenda [12]. Then the notion of Gorenstein injective dimension was studied in [7].

Another extension of the G-dimension is based on Gorenstein flat modules, a notion due to Enochs, Jenda, and Torrecillas [13]. Then the notion of Gorenstein flat dimension was studied in [7].

More recently, the complete intersection dimension has been introduced for finitely generated $R$-modules, using quasi-deformations and projective dimension, to characterize the complete intersection property of local rings [3]. Parallel to Gorenstein projective, injective and flat dimensions, the complete intersection projective, injective and flat dimensions have been introduced and studied in [22], [23], [24] and [21].

The Cohen-Macaulay dimension of a finitely generated $R$-module $M$, as defined by Gerko [16] is

[TABLE]

(see Section 2 for the definition of CM -quasi-deformation).

The purpose of this paper is to develop a similar theory of projective, injective and flat analogue for Cohen-Macaulay case. Thus we introduce Cohen-Macaulay projective dimension (\mbox{CM{}{*}\text{-}pd}\,), Cohen-Macaulay injective dimension (\mbox{CM{}{}\text{-}id}\,) and Cohen-Macaulay flat dimension (\mbox{CM{}_{}\text{-}fd}\,) for homologically bounded complexes over commutative Noetherian local rings $(R,\mathfrak{m},k)$ with identity (see Definition 3.1). In particular \mbox{CM-dim}\,_{R}(M)=\mbox{CM{}{*}\text{-}pd}\,_{R}(M)=\mbox{CM{}{*}\text{-}fd}\,_{R}(M), for a finitely generated $R$-module $M$. Among other things, we show that these invariants characterize the Cohen-Macaulay property for local rings. We also show that if $M$ is a homologically bounded $R$-complex, then we have the inequalities

[TABLE]

with equality to the left of any finite value. In particular if $\mbox{Gfd}\,_{R}(M)<\infty$, then \mbox{CM{}{*}\text{-}fd}\,_{R}(M)=\mbox{Gfd}\,_{R}(M), and if \mbox{CM{}{*}\text{-}fd}\,_{R}(M)<\infty, then

[TABLE]

where $\mbox{Rfd}\,_{R}(M)$ is the large restricted flat dimension. Also, we show that there are inequalities

[TABLE]

such that if $\mbox{Gid}\,_{R}(M)<\infty$, then \mbox{CM{}{*}\text{-}id}\,_{R}(M)=\mbox{Gid}\,_{R}(M), and if \mbox{CM{}{*}\text{-}id}\,_{R}(M)<\infty for a homologically finite $R$-complex $M$, then

[TABLE]

Finally we compare our Cohen-Macaulay homological dimensions with the homological dimensions of Holm and Jørgension [17].

2. Definitions and Notations

Let $(R,\mathfrak{m},k)$ and $(S,\mathfrak{n},l)$ be commutative local Noetherian rings.

We work in the derived category $\mathcal{D}(R)$ of complexes of $R$-modules, indexed homologically. A complex $M$ is homologically bounded if $\mbox{H}_{i}(M)=0$ for all $|i|\gg 0$; and it is homologically finite if $\oplus_{i}\mbox{H}_{i}(M)$ is finitely generated.

Fix $R$-complexes $M$ and $N$. Let $M\otimes_{R}^{\mathbf{L}}N$ and $\mathbf{R}\mbox{Hom}\,_{R}(M,N)$ denote the left-derived tensor product and right-derived homomorphism complexes, respectively. Let $\inf(M)$ and $\sup(M)$ denote the infimum and supremum, respectively, of the set $\{n\in\mathbb{Z}\mid\mbox{H}_{n}(M)\neq 0\}$.

Definition/Notation 2.1**.**

A homologically finite $R$-complex $M$ is reflexive if the complex $\mathbf{R}\mbox{Hom}\,_{R}(M,R)$ is homologically bounded and the biduality morphism $\delta_{M}:M\to\mathbf{R}\mbox{Hom}\,_{R}(\mathbf{R}\mbox{Hom}\,_{R}(M,R),R)$ is an isomorphism in $\mathcal{D}(R)$. Set

[TABLE]

if $M$ is reflexive, and $\mbox{G-dim}\,_{R}(M):=\infty$ otherwise. Set also $\mbox{G-dim}\,_{R}(0)=-\infty$. This is the G-dimension of Auslander and Bridger [1] and Yassemi [27]. **

Definition/Notation 2.2**.**

An $R$-module $G$ is $G$-projective if there exists an exact sequence of $R$-modules

[TABLE]

such that $G\cong\mbox{Coker}(\partial^{X}_{1})$, each $P_{i}$ is projective, and $\mbox{Hom}\,_{R}(X,Q)$ is exact for each projective $R$-module $Q$.

An $R$-module $G$ is $G$-flat if there exists an exact sequence of $R$-modules

[TABLE]

such that $G\cong\mbox{Coker}(\partial^{Y}_{1})$, each $F_{i}$ is flat, and $I\otimes_{R}Y$ is exact for each injective $R$-module $I$.

An $R$-module $G$ is $G$-injective if there exists an exact sequence of $R$-modules

[TABLE]

such that $G\cong\mbox{Coker}(\partial^{Z}_{1})$, each $I_{i}$ is injective, and $\mbox{Hom}\,_{R}(I,Z)$ is exact for each injective $R$-module $I$.

Let $M$ be a homologically bounded $R$-complex. A $G$-projective resolution of $M$ is an isomorphism $H\simeq M$ in $\mathcal{D}(R)$ where $H$ is a complex of $G$-projective $R$-modules such that $H_{i}=0$ for all $i\ll 0$. The $G$-projective dimension of $M$ is

[TABLE]

The $G$-flat dimension of $M$ is defined similarly and denoted $\mbox{Gfd}\,_{R}(M)$, while the $G$-injective dimension $\mbox{Gid}\,_{R}(M)$ is dual [7]. These are the $G$-projective, $G$-flat, and $G$-injective dimensions of Enochs, Jenda and Torrecillas (which they consider only in the case of modules) [12] and [13]. **

Remark 2.3**.**

(1) It is known that, for a homologically bounded $R$-complex $M$, $\mbox{Gpd}\,_{R}(M)$ and $\mbox{Gfd}\,_{R}(M)$ are simultaneously finite [24, Proposition 4.3].

(2) Let $R\to S$ be a flat local homomorphism and $M$ a finitely generated $R$-module. Then it is well-known that, $\mbox{G-dim}\,_{R}(M)=\mbox{G-dim}\,_{S}(M\otimes_{R}S)$ and $\mbox{G-dim}\,_{R}(M)=\mbox{Gfd}\,_{R}(M)=\mbox{Gpd}\,_{R}(M)$ [9].

(3) The finiteness of $G$-projective, $G$-flat, and $G$-injective dimensions characterize the Gorenstein property of local rings [7]. **

Definition/Notation 2.4**.**

A finitely generated $R$-module $M$ is called G-perfect if $\mbox{G-dim}\,_{R}M=\mbox{grade}\,_{R}M:=\inf\{i\mid\mbox{Ext}\,^{i}_{R}(M,R)\neq 0\}$. Let $Q$ be a local ring and $J$ an ideal of $Q$. By abuse of language we say that $J$ is G-perfect if the $Q$-module $Q/J$ has the corresponding property.

A CM -deformation of $R$ is a surjective local homomorphism $Q\to R$ such that $J=\ker(Q\to R)$ is a G-perfect ideal in $Q$. A CM -quasi-deformation of $R$ is a diagram of local homomorphisms $R\rightarrow R^{\prime}\leftarrow Q$, with $R\rightarrow R^{\prime}$ a flat extension and $R^{\prime}\leftarrow Q$ a CM -deformation.

The Cohen-Macaulay dimension of a nonzero finitely generated $R$-module $M$, as defined by Gerko [16] is

[TABLE]

and set $\mbox{CM-dim}\,_{R}(0)=-\infty$. **

Remark 2.5**.**

By [16, Theorems 3.8 and 3.9, and Proposition 3.10] we have

(1) $R$ is Cohen-Macaulay if and only if $\mbox{CM-dim}\,_{R}(k)<\infty$.

(2) If $M$ is a finitely generated $R$-module such that $\mbox{CM-dim}\,_{R}(M)<\infty$, then

[TABLE]

(3) For each prime ideal $\mathfrak{p}$ of $R$, $\mbox{CM-dim}\,_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\leq\mbox{CM-dim}\,_{R}(M)$. **

Definition/Notation 2.6**.**

A finitely generated $R$-module $C$ is semidualizing if the homothety morphism $\chi_{C}^{R}:R\to\mathbf{R}\mbox{Hom}\,_{R}(C,C)$ is an isomorphism in $\mathcal{D}(R)$. A finitely generated $R$-module $D$ is canonical if it is semidualizing and $\mbox{id}\,_{R}(D)$ is finite.**

Let $\varphi:R\to S$ be a local ring homomorphism. We denote $\widehat{R}$ the completion of $R$ at its maximal ideal and let $\varepsilon_{R}:R\to\widehat{R}$ denote the natural map. The completion of $\varphi$ is the unique local ring homomorphism $\widehat{\varphi}:\widehat{R}\to\widehat{S}$ such that $\widehat{\varphi}\circ\varepsilon_{R}=\varepsilon_{S}\circ\varphi$. The semi-completion of $\varphi$ is the composition $\varepsilon_{S}\circ\varphi:R\to\widehat{S}$.

3. Cohen-Macaulay projective, flat and injective dimensions

In this section we introduce a Cohen-Macaulay projective dimension, Cohen-Macaulay flat dimension, and Cohen-Macaulay injective dimension for homologically bounded $R$-complexes and derive their basic properties. When $M$ is a module, Definition 3.1 is from [22], which is in turn modeled on [3] and [16].

Definition 3.1**.**

Let $(R,\mathfrak{m})$ be a local ring. For each homologically bounded $R$-complex $M$, define the Cohen-Macaulay projective dimension, Cohen-Macaulay flat dimension and Cohen-Macaulay injective dimension of $M$ as,

[TABLE]

[TABLE]

[TABLE]

respectively. **

Remark 3.2**.**

(1) It is known that $\mbox{Gpd}\,_{R}(M)$ and $\mbox{Gfd}\,_{R}(M)$ are simultaneously finite by Remark 2.3(1). Hence \mbox{CM{}{*}\text{-}pd}\,_{R}(M) and \mbox{CM{}{*}\text{-}fd}\,_{R}(M) are simultaneously finite.

(2) By taking the trivial CM -quasi-deformation $R\rightarrow R\leftarrow R$, one has

[TABLE]

[TABLE]

[TABLE]

(3) By Remark 2.3(2) it can be seen that if $M$ is a finitely generated $R$-module then, \mbox{CM{}{*}\text{-}pd}\,_{R}(M)=\mbox{CM{}{*}\text{-}fd}\,_{R}(M)=\mbox{CM}-$\mbox{dim}\,_{R}(M)$. **

The following two theorems show that the finiteness of these dimensions characterize the Cohen-Macaulay rings.

Theorem 3.3**.**

The following conditions are equivalent:

• (1)

The ring $R$ is Cohen-Macaulay.

• (2)

\mbox{CM{}_{}\text{-}pd}\,_{R}(M)<\infty for every homologically bounded $R$-complex $M$.*

• (3)

\mbox{CM{}_{}\text{-}pd}\,_{R}(k)<\infty.*

• (4)

\mbox{CM{}_{}\text{-}fd}\,_{R}(M)<\infty for every homologically bounded $R$-complex $M$.*

• (5)

\mbox{CM{}_{}\text{-}fd}\,_{R}k<\infty.*

Proof.

(1)$\Rightarrow$(2) Let $\widehat{R}$ be the $\mathfrak{m}$-adic completion of $R$. Since $R$ is Cohen-Macaulay, so is $\widehat{R}$. Therefore by Cohen’s structure theorem, $\widehat{R}$ is isomorphic to $Q/J$, where $Q$ is a regular local ring. By Cohen-Macaulay-ness of $\widehat{R}$ and regularity of $Q$, the ideal $J$ is G-perfect. Thus $R\rightarrow\widehat{R}\leftarrow Q$ is a CM -quasi-deformation. Since $Q$ is regular $\mbox{Gpd}\,_{Q}(M\otimes_{R}\widehat{R})<\infty$ for every homologically bounded $R$-complex $M$. Thus \mbox{CM{}_{*}\text{-}pd}\,_{R}(M) is finite.

(2)$\Rightarrow$(3) and (4)$\Rightarrow$(5) are trivial.

(2)$\Rightarrow$(4) and (3)$\Rightarrow$(5) are trivial since \mbox{CM{}{*}\text{-}fd}\,_{R}(M)\leq\mbox{CM{}{*}\text{-}pd}\,_{R}(M).

(5)$\Rightarrow$(1) It follows from Remark 3.2(3) that \mbox{CM-dim}\,_{R}(k)=\mbox{CM{}_{*}\text{-}fd}\,_{R}(k)<\infty. Now Remark 2.5(1), completes the proof. ∎

Theorem 3.4**.**

The following conditions are equivalent.

• (1)

The ring $R$ is Cohen-Macaulay.

• (2)

\mbox{CM{}_{}\text{-}id}\,_{R}(M)<\infty for every homologically bounded $R$-complex $M$.*

• (3)

\mbox{CM{}_{}\text{-}id}\,_{R}(k)<\infty.*

Proof.

(1)$\Rightarrow$(2) is the same as proof of part (1)$\Rightarrow$(2) of Theorem 3.3.

(2)$\Rightarrow$(3) is trivial.

(3)$\Rightarrow$(1) Suppose \mbox{CM{}_{*}\text{-}id}\,_{R}(k)<\infty. So that there is a CM -quasi-deformation $R\rightarrow R^{\prime}\leftarrow Q$, such that $\mbox{Gid}\,_{Q}(k\otimes_{R}R^{\prime})$ is finite. It is clear that $k\otimes_{R}R^{\prime}$ is a cyclic $Q$-module. Consequently $Q$ is a Gorenstein ring by [15, Theorem 4.5]. We plan to show that $R^{\prime}$ is a Cohen-Macaulay ring. Let $I=\ker(Q\rightarrow R^{\prime})$ which is G-perfect by definition. We have

[TABLE]

in which the equalities follow from Cohen-Macaulay-ness of $Q$; G-perfectness of $I$; Auslander-Buchsbaum formula; [4, Exercise 1.2.26]; Cohen-Macaulay-ness of $Q$; and [4, Corollary 2.1.4] respectively. Therefore we obtain that $\mbox{dim}\,R^{\prime}-0ptR^{\prime}=0$, that is $R^{\prime}$ is Cohen-Macaulay. Now [4, Theorem 2.1.7] gives us the desired result. ∎

The proof of the above theorem says some thing more, viz., a local ring $R$ is Cohen-Macaulay if and only if there exists a cyclic $R$-module of finite Cohen-Macaulay injective dimension.

Corollary 3.5**.**

Assume that $C\neq 0$ is a cyclic $R$-module. Then $R$ is a Cohen-Macaulay ring if and only if \mbox{CM{}_{}\text{-}id}\,_{R}C<\infty.*

Remark 3.6**.**

Let $M$ be a homologically finite $R$-complex such that $\mbox{Gid}\,_{R}(M)<\infty$. Then by [15, Theorem 3.6], we obtain that $\mbox{Gid}\,_{\widehat{R}}(M\otimes_{R}\widehat{R})<\infty$. Hence using [11, Corollary 2.3], we have

[TABLE]

Proposition 3.7**.**

Let $M$ be a homologically finite $R$-complex. Then

[TABLE]

Proof.

It is clear that the left hand side is less than or equal to the right hand side. Now let $R\rightarrow R^{\prime}\leftarrow Q$ be a CM -quasi-deformation. Then note that $R\rightarrow\widehat{R^{\prime}}\leftarrow\widehat{Q}$ is also a CM -quasi-deformation such that

[TABLE]

and $\mbox{Gfd}\,_{Q}(R^{\prime})=\mbox{Gfd}\,_{\widehat{Q}}(\widehat{R^{\prime}})$, where the first equality holds by Remark 3.6. So we can assume in the CM -quasi-deformation $R\rightarrow R^{\prime}\leftarrow Q$ that, $Q$ is a complete local ring. This shows the equality. ∎

Proposition 3.8**.**

Let $M$ be a homologically bounded $R$-complex. Then

[TABLE]

Proof.

The proof is the same as proof of Proposition 3.7, but here use [19, Corollary 8.9] instead of Remark 3.6. ∎

Let $M$ be homologically bounded $R$-complex. Then Foxby showed that

[TABLE]

(see [10, Ascent table II(b)]).

Proposition 3.9**.**

Let $M$ be a homologically bounded $R$-complex. Then

[TABLE]

Proof.

The proof is the same as proof of Proposition 3.7, but here use the comment just before the proposition instead of Remark 3.6. ∎

A homological dimension should not grow under localization. Let $\mathfrak{p}$ be a prime ideal of $R$ and $M$ a homologically bounded $R$-complex. It is well known that

[TABLE]

and Foxby showed that (when $R$ has finite Krull dimension)

[TABLE]

(see [9, Page 262]). On the other hand if $R$ has a dualizing complex then,

[TABLE]

by [9, Proposition 5.5].

Theorem 3.10**.**

Let $M$ be a homologically finite $R$-complex. For each prime ideal $\mathfrak{p}\in\mbox{Spec}\,(R)$ there is an inequality

[TABLE]

Proof.

Assume that \mbox{CM{}{*}\text{-}id}\,_{R}(M)<\infty. Let $R\rightarrow R^{\prime}\leftarrow Q$ be a CM -quasi-deformation with $Q$ a complete local ring, such that $\mbox{Gid}\,_{Q}(M\otimes R^{\prime})<\infty$ and \mbox{CM{}{*}\text{-}id}\,_{R}(M)=\mbox{Gid}\,_{Q}(M\otimes R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime}) by Proposition 3.7. Hence $Q$ admits a dualizing complex.

Let $\mathfrak{p}$ be a prime ideal of $R$. Since $R\rightarrow R^{\prime}$ is a faithfully flat extension of rings, there is a prime ideal $\mathfrak{p}^{\prime}$ in $R^{\prime}$ lying over $\mathfrak{p}$. Let $\mathfrak{q}$ be the inverse image of $\mathfrak{p}^{\prime}$ in $Q$. The map $R_{\mathfrak{p}}\rightarrow R^{\prime}_{\mathfrak{p}^{\prime}}$ is flat, and $R^{\prime}_{\mathfrak{p}^{\prime}}\leftarrow Q_{\mathfrak{q}}$ is a CM -deformation and note that $\mbox{Gfd}\,_{Q_{\mathfrak{q}}}(R^{\prime}_{\mathfrak{p}^{\prime}})=\mbox{Gfd}\,_{Q}R^{\prime}$. Therefore the diagram $R_{\mathfrak{p}}\rightarrow R^{\prime}_{\mathfrak{p}^{\prime}}\leftarrow Q_{\mathfrak{q}}$ is a CM -quasi-deformation with

[TABLE]

where the inequality holds by [9, Proposition 5.5]. Hence \mbox{CM{}_{*}\text{-}id}\,_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})<\infty. So we obtain

[TABLE]

Thus the desired inequality follows. ∎

We do not know when the inequality \mbox{CM{}{*}\text{-}id}\,_{R_{\mathfrak{p}}}{(M_{\mathfrak{p}})}\leq\mbox{CM{}{}\text{-}id}\,_{R}(M) holds in general. However for \mbox{CM{}_{}\text{-}pd}\,_{R}(M) and \mbox{CM{}_{*}\text{-}fd}\,_{R}(M) we have

Theorem 3.11**.**

Let $M$ be a homologically bounded $R$-complex. For each prime ideal $\mathfrak{p}\in\mbox{Spec}\,(R)$ there is an inequality

• (1)

\mbox{CM{}{*}\text{-}pd}\,_{R_{\mathfrak{p}}}{(M_{\mathfrak{p}})}\leq\mbox{CM{}{}\text{-}pd}\,_{R}(M).*

• (2)

\mbox{CM{}{*}\text{-}fd}\,_{R_{\mathfrak{p}}}{(M_{\mathfrak{p}})}\leq\mbox{CM{}{}\text{-}fd}\,_{R}(M).*

Proof.

The proof is the same as proof of Theorem 3.10, but here we do not need $Q$ is a complete local ring. ∎

Proposition 3.12**.**

Let $M$ be a homologically finite $R$-complex. Then there is an equality

[TABLE]

Proof.

It is clear that the left hand side is less than or equal to the right hand side. Let $R\rightarrow R^{\prime}\leftarrow Q$ be a CM -quasi-deformation with $Q$ a complete local ring, such that \mbox{CM{}_{*}\text{-}id}\,_{R}(M)=\mbox{Gid}\,_{Q}(M\otimes R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime}) by Proposition 3.7. Hence $Q$ admits a dualizing complex. Now choose $\mathfrak{p}^{\prime}\in\mbox{Spec}\,(R^{\prime})$ such that it is a minimal prime ideal containing $\mathfrak{m}R^{\prime}$; thus $\mathfrak{m}=\mathfrak{p}^{\prime}\cap R$ and $\mathfrak{p}^{\prime}=\mathfrak{q}/J$ for some $\mathfrak{q}\in\mbox{Spec}\,(Q)$, where $J=\ker(Q\rightarrow R^{\prime})$. Now the diagram $R\rightarrow R^{\prime}_{\mathfrak{p}^{\prime}}\leftarrow Q_{\mathfrak{q}}$ is a CM -quasi-deformation such that the closed fiber of $R\rightarrow R^{\prime}_{\mathfrak{p}^{\prime}}$ is Artinian. It is clear that $\mbox{Gfd}\,_{Q}R^{\prime}=\mbox{Gfd}\,_{Q_{\mathfrak{q}}}R^{\prime}_{\mathfrak{p}^{\prime}}$. Also we have

[TABLE]

where the inequality holds by [9, Proposition 5.5]. Hence \mbox{Gid}\,_{Q_{\mathfrak{q}}}(M\otimes_{R}R^{\prime}_{\mathfrak{p}})-\mbox{Gfd}\,_{Q_{\mathfrak{q}}}(R^{\prime}_{\mathfrak{p}})\leq\mbox{CM{}_{*}\text{-}id}\,_{R}(M). So the proof is complete. ∎

Proposition 3.13**.**

Let $M$ be a homologically bounded $R$-complex. Then there are equalities

[TABLE]

[TABLE]

Proof.

The proof is the same as proof of Proposition 3.12, but here we do not need $Q$ is a complete local ring. ∎

Remark 3.14**.**

(1) Let $M$ be a homologically finite $R$-complex. Then, one can combine the proofs of Propositions 3.7 and 3.12, to obtain an equality

[TABLE]

(2) Likewise for a homologically bounded $R$-complex $M$, one can combine the proofs of Propositions 3.8, 3.9 and 3.12, to obtain the equalities

[TABLE]

[TABLE]

4. Large restricted flat dimension and Chouinard’s invariant

Recall from [8], that the large restricted flat dimension is defined by

[TABLE]

This number is finite, as long as $\mbox{H}(M)$ is nonzero and the Krull dimension of $R$ is finite; see [8, Proposition 2.2]. It is useful to keep in mind an alternative formula [8, Theorem 2.4] for computing this invariant:

[TABLE]

Recall here that the depth of a homologically bounded $R$-complex $M$ is defined by

[TABLE]

and it is shown that $0pt_{R}(M)\geq-\sup(M)$.

It is proved in [19, Theorem 8.8] that for an $R$-complex $M$, $\mbox{Rfd}\,_{R}(M)$ is a refinement of $\mbox{Gfd}\,_{R}(M)$, that is

[TABLE]

with equality if $\mbox{Gfd}\,_{R}(M)$ is finite.

First, we plan to show that, when the Cohen-Macaulay flat dimension of a homologically bounded $R$-complex $M$ is finite, then it is equal to the large restricted flat dimension of $M$. The following proposition is the main tool.

Proposition 4.1**.**

Let $R\rightarrow S\leftarrow Q$ be a CM -quasi-deformation, and let $M$ be a homologically bounded $R$-complex. Then

[TABLE]

Proof.

First we prove the equality

[TABLE]

for a homologically bounded $S$-complex $Y$. To this end, choose by [8, Theorem 2.4(b)] a prime ideal $\mathfrak{p}$ of $S$ such that the first equality below holds. Let $\mathfrak{q}$ be the inverse image of $\mathfrak{p}$ in $Q$. Therefore there is an isomorphism $Y_{\mathfrak{p}}\cong Y_{\mathfrak{q}}$ of $Q_{\mathfrak{q}}$-modules and a CM -deformation $Q_{\mathfrak{q}}\rightarrow S_{\mathfrak{p}}$. Hence

[TABLE]

The second equality holds since $Q_{\mathfrak{q}}\rightarrow S_{\mathfrak{p}}$ is surjective and [18, Proposition 5.2(1)]; the third equality holds by Auslander-Bridger formula [1]; the fourth equality is due to the G-perfectness assumption of $S$ over $Q$; while the inequality follows from [8, Theorem 2.4(b)]. Now by [26, Proposition 3.5] we have

[TABLE]

which is the desired equality.

Now we have

[TABLE]

where the inequality is in [26, Proposition 3.5], the first equality follows from the hypotheses, and the second equality follows from the above observation. Hence

[TABLE]

where the second equality holds by [19, Lemma 8.5(1)]. ∎

Corollary 4.2**.**

Let $M$ be a homologically bounded $R$-complex. Then we have the inequalities

[TABLE]

with equality to the left of any finite value. In particular if \mbox{CM{}_{}\text{-}fd}\,_{R}(M)<\infty, then*

[TABLE]

Now using Corollary 4.2, we investigate the effect of change of ring on Cohen-Macaulay flat dimension.

Proposition 4.3**.**

Let $M$ be a homologically bounded $R$-complex. Let $R\rightarrow R^{\prime}$ be a local flat extension, and $M^{\prime}=M\otimes_{R}R^{\prime}$. Then

[TABLE]

with equality when \mbox{CM{}_{}\text{-}fd}\,_{R^{\prime}}(M^{\prime}) is finite.*

Proof.

Suppose that \mbox{CM{}{*}\text{-}fd}\,_{R^{\prime}}(M^{\prime})<\infty, and let $R^{\prime}\rightarrow R^{\prime\prime}\leftarrow Q$ be a CM -quasi-deformation with $\mbox{Gfd}\,_{Q}(M^{\prime}\otimes_{R^{\prime}}R^{\prime\prime})<\infty$. Since $R\rightarrow R^{\prime}$ and $R^{\prime}\rightarrow R^{\prime\prime}$ are flat extensions, the local homomorphism $R\rightarrow R^{\prime\prime}$ is also flat. Hence $R\rightarrow R^{\prime\prime}\leftarrow Q$ is a CM -quasi-deformation with $\mbox{Gfd}\,_{Q}(M\otimes_{R}R^{\prime\prime})<\infty$. It follows that \mbox{CM{}{*}\text{-}fd}\,_{R}(M) is finite. Now by Corollary 4.2 and [19, Lemma 8.5(1)], we have

[TABLE]

to complete the proof. ∎

Proposition 4.4**.**

For every homologically bounded $R$-complex $M$

[TABLE]

Proof.

If \mbox{CM{}{*}\text{-}fd}\,_{R}(M)=\infty, then we obtain that \mbox{CM{}{}\text{-}fd}\,_{\widehat{R}}(M\otimes_{R}\widehat{R})=\infty by Proposition 4.3. Now assume that \mbox{CM{}_{}\text{-}fd}\,_{R}(M)<\infty. Using Proposition 4.3, it is sufficient to prove that \mbox{CM{}_{*}\text{-}fd}\,_{\widehat{R}}(M\otimes_{R}\widehat{R}) is finite. To this end, choose a CM -quasi-deformation $R\rightarrow R^{\prime}\leftarrow Q$ of $R$ such that $\mbox{Gfd}\,_{Q}(M\otimes_{R}R^{\prime})<\infty$. So we have $\widehat{R}\rightarrow\widehat{R^{\prime}}\leftarrow\widehat{Q}$ is a CM -quasi-deformation of $\widehat{R}$ with respect to their maximal ideal-adic completions. Now using [19, Corollary 8.9] we obtain

[TABLE]

Hence $\mbox{Gfd}\,_{\widehat{Q}}((M\otimes_{R}\widehat{R})\otimes_{\widehat{R}}\widehat{R^{\prime}})$ is finite which in turn implies that \mbox{CM{}_{*}\text{-}fd}\,_{\widehat{R}}(M\otimes_{R}\widehat{R}) is finite. ∎

Next, recall that the width of an $R$-complex $M$ is defined by

[TABLE]

and that $0pt_{R}(M)\geq\inf(M)$. Also, if $M$ is homologically finite, then

[TABLE]

It is the dual notion for $0pt_{R}(M)$. In particular by [8, Proposition 4.8], we have

[TABLE]

where $\mbox{E}_{R}(k)$ denotes the injective envelope of $k$ over $R$.

The Chouinard invariant [6, Corollary 3.1] is denoted by $\mbox{Ch}_{R}(M)$ and

[TABLE]

It is proved in [11, Theorem 2.2] that for an $R$-complex $M$, $\mbox{Ch}_{R}(M)$ is a refinement of $\mbox{Gid}\,_{R}(M)$, that is

[TABLE]

with equality if $\mbox{Gid}\,_{R}(M)$ is finite. Now we want to show that the Cohen-Macaulay injective dimension is bounded below by the Chouinard’s invariant.

Lemma 4.5**.**

Suppose that $Q\rightarrow S$ is a surjective local homomorphism and $Y$ is an $S$-complex. Then we have

[TABLE]

Proof.

We have the following equalities:

[TABLE]

where the first one is by [8, Proposition 4.8]; the second one is by [5, Lemma 10.1.15]; the third one is by adjointness of Hom  and tensor; the fourth one is true since $Q\rightarrow S$ is surjective and [18, Proposition 5.2(1)]; while the last one is again by [8, Proposition 4.8]. Here we used $k$ for the residue fields of $Q$ and $S$, and $\mbox{E}_{Q}(k)$ and $\mbox{E}_{S}(k)$ for the injective envelopes of $k$ over respectively $Q$ and $S$. ∎

Lemma 4.6**.**

Suppose that $R\rightarrow S$ is a flat local ring homomorphism, and $M$ is a homologically bounded $R$-complex. Then we have

[TABLE]

Proof.

A standard application of the Künneth formula yields the equality. ∎

Proposition 4.7**.**

Let $R\rightarrow S$ be a flat local homomorphism and let $M$ be a homologically bounded $R$-complex. Then

[TABLE]

Proof.

Let $\mathfrak{p}\in\mbox{Spec}\,(R)$ such that $\mbox{Ch}_{R}(M)=0ptR_{\mathfrak{p}}-0pt_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})$. Let $\mathfrak{q}\in\mbox{Spec}\,(S)$ contain $\mathfrak{p}S$ minimally. Since $R\rightarrow S$ is a flat local homomorphism we have $\mathfrak{p}=\mathfrak{q}\cap R$. Hence:

[TABLE]

in which the second equality holds by Lemma 4.6 and the fact that $R_{\mathfrak{p}}\rightarrow S_{\mathfrak{q}}$ has Artinian closed fibre. ∎

Proposition 4.8**.**

Let $Q\rightarrow S$ be a CM -deformation, and $Y$ be a homologically bounded $S$-complex. Then

[TABLE]

Proof.

Choose a prime ideal $\mathfrak{p}$ of $S$ such that the first equality below holds. Let $\mathfrak{q}$ be the inverse image of $\mathfrak{p}$ in $Q$. Therefore there is an isomorphism $Y_{\mathfrak{p}}\cong Y_{\mathfrak{q}}$ of $Q_{\mathfrak{q}}$-complexes and a CM -deformation $Q_{\mathfrak{q}}\rightarrow S_{\mathfrak{p}}$. Hence

[TABLE]

The second equality holds since $Q_{\mathfrak{q}}\rightarrow S_{\mathfrak{p}}$ is surjective; the third equality holds by Auslander-Bridger formula [1]; the fourth equality is due to the G-perfectness assumption of $S$ over $Q$. ∎

Theorem 4.9**.**

Let $M$ be a homologically bounded $R$-complex. Then there is the inequality

[TABLE]

Proof.

We can assume that \mbox{CM{}{*}\text{-}id}\,_{R}(M)<\infty. Choose a CM -quasi-deformation $R\rightarrow R^{\prime}\leftarrow Q$, such that \mbox{CM{}{*}\text{-}id}\,_{R}(M)=\mbox{Gid}\,_{Q}(M\otimes_{R}R^{\prime})-\mbox{Gfd}\,_{Q}(R^{\prime}). Hence we have

[TABLE]

in which the second equality comes by [11, Theorem 2.2], and inequalities follow Propositions 4.8 and 4.7 respectively. ∎

Corollary 4.10**.**

Let $M$ be a homologically bounded $R$-complex. Then there are inequalities

[TABLE]

such that if $\mbox{Gid}\,_{R}(M)<\infty$, then \mbox{Gid}\,_{R}(M)=\mbox{CM{}_{}\text{-}id}\,_{R}(M).*

Proof.

The inequalities hold by Theorem 4.9 and Remark 3.2(2). And if $\mbox{Gid}\,_{R}(M)<\infty$, then the equality holds by [11, Theorem 2.2]. ∎

Corollary 4.11**.**

Let $M$ be a homologically finite $R$-complex such that \mbox{CM{}_{}\text{-}id}\,_{R}(M) is finite. Then*

[TABLE]

Proof.

By Proposition 3.12 there is a CM -quasi-deformation $R\rightarrow R^{\prime}\leftarrow Q$ such that the closed fibre of $R\rightarrow R^{\prime}$ is Artinian and the first equality below holds. So that

[TABLE]

The second equality holds by [11, Corollary 2.3] and the Auslander-Bridger formula [1], while the lase equality holds, because the closed fiber of $R\to R^{\prime}$ is Artinian and [4, Proposition 1.2.16].

Now by Theorem 4.9, 0ptR-\inf(M)\leq\mbox{Ch}_{R}(M)\leq\mbox{CM{}{*}\text{-}id}\,_{R}(M)=0ptR-\inf(M). Therefore \mbox{CM{}{*}\text{-}id}\,_{R}(M)=\mbox{Ch}_{R}(M)=0ptR-\inf(M). ∎

In concluding, recall that there are notions of Cohen-Macaulay projective dimension, Cohen-Macaulay flat dimension and Cohen-Macaulay injective dimension of Holm and Jørgensen, which are different with our Definition 3.1.

Definition 4.12**.**

(cf., [17, Definition 2.3]) Let $(R,\mathfrak{m})$ be a local ring. For each homologically bounded $R$-complex $M$, the Cohen-Macaulay projective, flat and injective dimension, of $M$ is defined as, respectively,

[TABLE]

[TABLE]

[TABLE]

Here $R\ltimes C$ denotes the trivial extension ring of $R$ by $C$; it is the $R$-module $R\oplus C$ equipped with the multiplication $(r,c)(r^{\prime},c^{\prime})=(rr^{\prime},rc^{\prime}+r^{\prime}c)$. **

Remark 4.13**.**

(1) For each homologically bounded $R$-complex $M$, we have

[TABLE]

[TABLE]

[TABLE]

More precisely, assume that $\mbox{CM}\,\mbox{pd}\,_{R}(M)<\infty$ and choose a semidualizing $R$-module $C$ such that $\mbox{CM}\,\mbox{pd}\,_{R}(M)=\mbox{Gpd}\,_{R\ltimes C}(M)$. Then by [16, Lemma 3.6], we have the CM -quasi-deformation $R\to R\stackrel{{\scriptstyle\tau}}{{\leftarrow}}Q$ where $Q:=R\ltimes C$ and $\tau(r,c)=r$, such that $\mbox{G-dim}\,_{Q}(R)=0$. Thus we obtain

[TABLE]

This shows the first inequality. The proof of the other two inequalities are the same as the first one.

(2) The finiteness of the Cohen-Macaulay homological dimensions in Definition 4.12, characterize Cohen-Macaulay rings admitting a canonical module [17, Theorem 5.1].

(3) Assume that $(R,\mathfrak{m},k)$ is a Cohen-Macaulay ring, not admitting a canonical module (e.g., see [14] for such an example). Then \mbox{CM{}{*}\text{-}pd}\,_{R}(k)<\infty (and, \mbox{CM{}{}\text{-}fd}\,_{R}(k)<\infty, \mbox{CM{}_{}\text{-}id}\,_{R}(k)<\infty) but $\mbox{CM}\,\mbox{pd}\,_{R}(k)=\infty$ (and, $\mbox{CM}\,\mbox{fd}\,_{R}(k)=\infty$, $\mbox{CM}\,\mbox{id}\,_{R}(k)=\infty$). **

Lemma 4.14**.**

Assume that $C$ is a semidualizing $R$-module and let $M$ be a homologically bounded $R$-complex. Consider $M$ as a $R\ltimes C$-complex via the natural surjection $\tau:R\ltimes C\to R$.

• (1)

If $\mbox{Gfd}\,_{R\ltimes C}(M)<\infty$, then $\mbox{Gfd}\,_{R\ltimes C}(M)=\mbox{Rfd}\,_{R}(M)$.

• (2)

If $\mbox{Gid}\,_{R\ltimes C}(M)<\infty$, then $\mbox{Gid}\,_{R\ltimes C}(M)=\mbox{Ch}_{R}(M)$.

Proof.

Note that $\mbox{Spec}\,(R\ltimes C)=\{\mathfrak{p}\ltimes C\mid\mathfrak{p}\in\mbox{Spec}\,(R)\}$ and $(R\ltimes C)_{\mathfrak{p}\ltimes C}\cong R_{\mathfrak{p}}\ltimes C_{\mathfrak{p}}$ by [5, Exercise 6.2.12]. Let $L$ be an $R$-module which is an $R\ltimes C$-module via the surjection $\tau:R\ltimes C\to R$, and let $\mathfrak{p}$ be a prime ideal of $R$. Then $\varphi:L_{\mathfrak{p}\ltimes C}\to L_{\mathfrak{p}}$ sending $l/(r,c)$ to $l/r$ is an $R_{\mathfrak{p}}$-isomorphism. By [19, Theorem 8.8] we have the first equality below.

[TABLE]

The third equality holds since there is a surjection $R_{\mathfrak{p}}\ltimes C_{\mathfrak{p}}\to R_{\mathfrak{p}}$ and [18, Proposition 5.2(1)]. The fourth equality uses

[TABLE]

The proof of (2) is the same as (1) using [11, Theorem 2.2] instead of [19, Theorem 8.8], and Lemma 4.5, instead of [18, Proposition 5.2(1)]. ∎

Corollary 4.15**.**

Let $M$ be a homologically bounded $R$-complex.

• (1)

If $\mbox{CM}\,\mbox{fd}\,_{R}(M)<\infty$, then \mbox{CM}\,\mbox{fd}\,_{R}(M)=\mbox{CM{}_{}\text{-}fd}\,_{R}(M).*

• (2)

If $\mbox{CM}\,\mbox{id}\,_{R}(M)<\infty$, then \mbox{CM}\,\mbox{id}\,_{R}(M)=\mbox{CM{}_{}\text{-}id}\,_{R}(M).*

Proof.

Note that there are the inequalities

[TABLE]

(resp., \mbox{Ch}_{R}(M)\leq\mbox{CM{}_{*}\text{-}id}\,_{R}(M)\leq\mbox{CM}\,\mbox{id}\,_{R}(M)=\mbox{Ch}_{R}(M)) by Corollary 4.2 (resp., Theorem 4.9), and Lemma 4.14. ∎

Acknowledgement. The authors would like to thank the referee for his/her careful reading of the manuscript and several comments which greatly improved the paper. Parviz Sahandi would like to thank Sean Sather-Wagstaff for comments on an earlier version of this paper. Part of this work was completed while Siamak Yassemi was visiting the Institut des Hautes Etudes Scientifiques (IHES) in Bures-sur-Yvette, France. He wishes to express his gratitude to the Institute for its warm hospitality and for providing a stimulating research environment.