Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 26, 2007, 69–77

© Wei Jiaqun

Abstract. We prove that two categories ${\mathcal{G}}_{\omega }$ and ${\mathcal{X}}_{\omega }$, introduced for the faithfully balanced selforthogonal module $\omega$ by Auslander and Reiten in [2] and [3] respectively, coincide with each other. As an application we give a generalization of a main theorem in [6].

1. Introduction

Throughout this note, we assume that $R$ ($S$ respectively) is a left (right respectively) noetherian ring and modules are finitely generated left modules. By a subcategory we mean a full subcategory closed under isomorphisms. We fix $\omega$ a faithfully balanced selforthogonal $R$-module with $S={End}_R\omega$ (i.e. a generalized tilting module in sense of [8]), which is defined to satisfy the conditions (a) $R\simeq End{\omega }_S$ and, (b) ${Ext}_R^i\left(\omega ,\omega \right)=0={Ext}_S^i\left(\omega ,\omega \right)$ for all $i\ge 1$.

The notion of the Gorenstein dimension was introduced in [1] for a two-sided noetherian ring $R$. An $R$-module $M$ has Gorenstein dimension zero, written $G$-$dimM=0$, if $M$ is a reflexive $R$-module and ${Ext}_R^i\left(M,R\right)=0={Ext}_R^i\left(M^{\ast },R\right)$ for all $i\ge 1$, where $M^{\ast }={Hom}_R\left(M,R\right)$. Further, $G$-$dimM=n$ if $n$ is the minimal integer such that there is an exact sequence $0\to L_n\to \cdots \to L_0\to M\to 0$ with $G$-$dim{L}_i=0$ for all $0\le i\le n$. Otherwise, $G$-$dimM=\infty$.

In [2], the authors studied a generalization of this dimension. An $R$-module $M$ is said to be of generalized Gorenstein dimension zero, written $G$-${dim}_{\omega }M=0$, if ${Ext}_R^i\left(M,\omega \right)=0={Ext}_S^i\left({Hom}_R\left(M,\omega \right),\omega \right)$ for all $i\ge 1$ and the canonical map ${\sigma }_M:M\to {Hom}_S\left({Hom}_R\left(M,\omega \right),\omega \right)$ is an isomorphism. $G$-${dim}_{\omega }M=n$ if $n$ is the minimal integer such that there is an exact sequence $0\to L_n\to \cdots \to L_0\to M\to 0$ with $G$-${dim}_{\omega }{L}_i=0$ for all $0\le i\le n$, otherwise, $G$-${dim}_{\omega }M=\infty$. We denoted by ${\mathcal{G}}_{\omega }$ the subcategory of $R$-modules of generalized Gorenstein dimension zero.

It was also introduced in [3] a subcategory, denoted by ${\mathcal{X}}_{\omega }$, whose objects are $R$-modules $M$ such that there is an exact sequence $0\to M\to {\omega }_0{\to }^{f_0}{\omega }_1{\to }^{f_1}\dots$ with ${\omega }_i\in {add}_R\omega$, i.e. the subcategory of the direct summands of finite direct sum of $\omega$, with $Im{f}_j{\in }^{\perp }\omega =\left\{X\in R-mod\mid {Ext}_R^i\left(X,\omega \right)=0\text{ for all }i\ge 1\right\}$ for all $j\ge 0$.

One purpose of this note is to study the relation between the above-referenced two subcategories. It is well known that ${\mathcal{G}}_{\omega }\subseteq {\mathcal{X}}_{\omega }$ and ${\mathcal{G}}_{\omega }={\mathcal{X}}_{\omega }$ when $\omega$ is a cotilting bimodule (see [2] for the case of artin algebras and [6] for the case of two-sided noetherian ring). In this note, we will show that ${\mathcal{G}}_{\omega }={\mathcal{X}}_{\omega }$ always holds for the faithfully balanced selforthogonal module $\omega$. Note that the cotilting bimodule (in artin algebras [3] and in [6]) is faithfully balanced selforthogonal, but conversely the faithfully balanced selforthogonal module need not be cotilting (see for instance [9, Example 3.1]). As the Gorenstein dimension and its generalizations were recently studied by many mathematicians (e.g. [4, 5, 7] etc.), this result would be helpful for us to understand the (generalized) Gorenstein dimension more precisely.

We in turn compare the generalized Gorenstein dimension with the left orthogonal dimension related to $\omega$ defined in [6]. Accordantly, an $R$-module $M$ has left orthogonal dimension related to $\omega$ equal to $n$, written ${}^{\perp }\omega -dimM=n$, if $n$ is the minimal integer such that there is an exact sequence $0\to K_n\to \cdots \to K_0\to M\to 0$ with $K_i{\in }^{\perp }\omega$ for all $0\le i\le n$, otherwise, ${}^{\perp }\omega -dimM=\infty$. We show that ${}^{\perp }\omega -dimM\le G$-${dim}_{\omega }M$ in general and they agree when the latter is finite. Consequently, we get a method to compute the generalized Gorenstein dimension when it is known to be finite. We also study the subcategory of modules of finite generalized Gorenstein dimension, denoted by $\widehat{{\mathcal{G}}_{\omega }}$, and some known results were extended.

We denote by $R-mod$ ($mod-S$ respectively) the category of all finitely generated left $R$-modules (right $S$-modules respectively). Let $\mathcal{C}$ be a subcategory of $A-mod$, we denote by $\hat{\mathcal{C}}$ the category of all modules $M$ such that there is an exact sequence $0\to C_m\to \cdots \to C_0\to M\to 0$ for some integer $m$ with each $C_i\in \mathcal{C}$. Assume $\mathcal{C}\supset \mathcal{D}$ are two subcategories of $R$-mod. Let $C\in \mathcal{C}$ and $D\in {add}_R\mathcal{D}$. A homomorphism $D\to C$ is said to be a right $\mathcal{D}$-approximation of $C$ if ${Hom}_R\left(X,D\right)\to {Hom}_R\left(X,C\right)$ is an epimorphism for any $X\in {add}_R\mathcal{D}$. $\mathcal{D}$ is said to be a contravariantly finite subcategory of $\mathcal{C}$ (or $\mathcal{D}$ is contravariantly finite in $\mathcal{C}$) if every $C\in \mathcal{C}$ has a right $\mathcal{D}$-approximation. Dually, A homomorphism $C\to D$ is said to be a left $\mathcal{D}$-approximation of $C$ if ${Hom}_R\left(D,X\right)\to {Hom}_R\left(C,X\right)$ is an epimorphism for any $X\in {add}_R\mathcal{D}$. $\mathcal{D}$ is said to be a covariantly finite subcategory of $\mathcal{C}$ (or $\mathcal{D}$ is covariantly finite in $\mathcal{C}$) if every $C\in \mathcal{C}$ has a left $\mathcal{D}$-approximation. $\mathcal{D}$ is said to be a functionally finite subcategory of $\mathcal{C}$ (or $\mathcal{D}$ is functionally finite in $\mathcal{C}$) if $\mathcal{D}$ is both a contravariantly finite subcategory and a covariantly finite subcategory of $\mathcal{C}$.

Assume that $\mathcal{D}$ is contravariantly finite subcategory in $\mathcal{C}$ and ${\mathcal{D}}^{\prime }$ is covariantly finite subcategory in $\mathcal{C}$. If every kernel of right $\mathcal{D}$-approximations is in ${\mathcal{D}}^{\prime }$, then ${\mathcal{D}}^{\prime }$ is called the associated (with $\mathcal{D}$) covariantly finite subcategory. Similarly, if every cokernel of left ${\mathcal{D}}^{\prime }$-approximations is in $\mathcal{D}$, then $\mathcal{D}$ is called the associated (with ${\mathcal{D}}^{\prime }$) contravariantly finite subcategory.

For $A\in R-mod$ ($mod-S$ respectively), we put $A^{\omega }={Hom}_R\left(A,\omega \right)$ (${Hom}_S\left(A,\omega \right)$ respectively). An $R$-module $M$ is said to be $\omega$-reflexive if the natural homomorphism ${\sigma }_M:M\to M^{\omega \omega }$ is an isomorphism.

2. Main Results

Proof. It’s sufficient to show ${\mathcal{X}}_{\omega }\subseteq {\mathcal{G}}_{\omega }$ by [2, Proposition 4.3]. Obviously we have ${\mathcal{X}}_{\omega }{\subseteq }^{\perp }\omega$. Now let $M\in {\mathcal{X}}_{\omega }$. By the definition we see that there exists an exact sequence $0\to M\to {\omega }_0\to N\to 0$ with ${\omega }_0\in {add}_R\omega$ and $N\in {\mathcal{X}}_{\omega }$. Note that $N{\in }^{\perp }\omega$, so that the induced sequence $0\to N^{\omega }\to {\omega }_0^{\omega }\to M^{\omega }\to 0$ is exact. Hence we have the following exact commutative diagram, where ${\sigma }_N$, ${\sigma }_{{\omega }_0}$, ${\sigma }_M$ are the natural homomorphisms.

It is easy to see that ${\sigma }_{{\omega }_0}$ is a natural isomorphism, so we have that ${\sigma }_M$ is a monomorphism. Note that $N$ has properties same as $M$, so ${\sigma }_N$ is a monomorphism, too. Therefore ${\sigma }_M$ must be an isomorphism. In the same way we obtain that ${\sigma }_N$ is an isomorphism. It follows that ${Ext}_S^1\left(M^{\omega },\omega \right)=0$. Since ${Ext}_S^{i+1}\left(M^{\omega },\omega \right)\simeq {Ext}_S^i\left(N^{\omega },\omega \right)$ for $i\ge 1$, we see that ${Ext}_S^i\left(M^{\omega },\omega \right)=0$ for all $i\ge 1$, by repeating the process. Hence $M$ is an $R$-module of generalized Gorenstein dimension zero and it follows ${\mathcal{X}}_{\omega }\subseteq {\mathcal{G}}_{\omega }$. □

The following proposition compares the generalized Gorenstein dimension with the left orthogonal dimension relative to $\omega$.

Consequently, we get a method to compute the generalized Gorenstein dimension when it is known to be finite.

The following lemma generalizes [1, Theorem 3.13].

Proof. It’s easy to see that, for any integer $t\ge 1$, there is the following exact commutative diagram, where $P_i,Q_i$, $0\le i\le t-1$, are finitely generated projective $R$-modules and ${\Omega }^t\left(A\right)$, ${\Omega }^t\left(B\right)$, ${\Omega }^t\left(C\right)$ are some $t$-syzygies of $A$, $B$, $C$ respectively.

If two of ${\Omega }^t\left(A\right)$, ${\Omega }^t\left(B\right)$, ${\Omega }^t\left(C\right)$ are modules of generalized Gorenstein dimension zero, then the generalized Gorenstein dimension of the third is clearly not more than one. It follows from Lemma 2.5 that, if two of $A,B,C$ in the exact sequence $0\to A\to B\to C\to 0$ have finite generalized Gorenstein dimension, then the third also has finite generalized Gorenstein dimension.

For any $n>G$-${dim}_{\omega }C$ we easily see that ${\Omega }^n\left(C\right)\in {\mathcal{G}}_{\omega }$. By Corollary 2.2, ${\mathcal{G}}_{\omega }$ is closed under extensions and kernels of epimorphisms. Hence ${\Omega }^n\left(A\right)\in {\mathcal{G}}_{\omega }$ if and only if ${\Omega }^n\left(B\right)\in {\mathcal{G}}_{\omega }$. It follows again by Lemma 2.5 that $G$-${dim}_{\omega }A\le n$ if and only if $G$-${dim}_{\omega }B\le n$.

Assume that $G$-${dim}_{\omega }B=0$. If $G$-${dim}_{\omega }C=1$, then by Lemma 2.5 we have $G$-${dim}_{\omega }A=0$. If $G$-${dim}_{\omega }C>1$, then $sup\left\{k\mid {Ext}_R^k\left(C,\omega \right)\ne 0\right\}=sup\left\{k\mid {Ext}_R^k\left(A,\omega \right)\ne 0\right\}+1$ since ${Ext}_R^{i+1}\left(C,\omega \right)\simeq {Ext}_R^i\left(A,\omega \right)$ for all $i\ge 1$. By Proposition 2.3 we see that $G$-${dim}_{\omega }A=G$-${dim}_{\omega }C-1$. □

The following lemma is obvious (cf. [6, Lemma 1]).

Proof. For any $N\in \widehat{{\mathcal{G}}_{\omega }}$, let $t\left(N\right)=G$-${dim}_{\omega }N<\infty$. We will proceed by induction on $t\left(N\right)$.

If $t\left(N\right)=0$ i.e., $N\in {\mathcal{G}}_{\omega }$, then let $L=0$ and $G=N$, and the first desired exact sequence follows. By Theorem 1, we see that there is an exact sequence $0\to N\to {\omega }_0\to G^{\prime }\to 0$ such that ${\omega }_0\in {add}_R\omega$ and $G^{\prime }\in {\mathcal{G}}_{\omega }$. It’s easy to see that this is just the second desired exact sequence by Lemma 2.7.

Let $t\left(N\right)\ge 1$. Consider the exact sequence $0\to M\to P_0\to N\to 0$ with $P_0$ finitely generated projective. By Corollary 2.6, $t\left(M\right)=t\left(N\right)-1$ since $P_0$ is clearly in ${\mathcal{G}}_{\omega }$. Therefore, there is an exact sequence $0\to M\to B\to C\to 0$ with $B\in \widehat{{add}_R\omega }$ and $C\in {\mathcal{G}}_{\omega }$, by the induction assumptions. Now consider the following pushout diagram.

It follows from the middle column that $D\in {\mathcal{G}}_{\omega }$, since ${\mathcal{G}}_{\omega }$ is closed under extensions. By Lemma 2.7 we see that the middle row is just the desired first exact sequence. Moreover, note that there is an exact sequence $0\to D\to {\omega }_0^{\prime }\to A\to 0$ such that ${\omega }_0^{\prime }\in {add}_R\omega$ and $A\in {\mathcal{G}}_{\omega }$, so we have the following exact commutative diagram.

It follows from the middle row that $E\in \widehat{{add}_R\omega }$, since $B\in \widehat{{add}_R\omega }$. By Lemma 2.7 we see that the middle column is just the desired second exact sequence. □

By the above lemma and the definition of homologically finite subcategories, we have the following proposition.

Proof. As in Lemma 2.8, we obtain that, for any $M^{\omega }\in mod-S$, there exists an exact sequence $0\to A^{\prime }\to X{\to }^f{M}^{\omega }\to 0$ such that $f:X\to M^{\omega }$ is a right ${\mathcal{G}}_{\omega }$-approximation of $M^{\omega }$ and $A^{\prime }\in {\widehat{add\omega }}_S$ . Let $h=f^{\omega }\circ {\sigma }_M$ where ${\sigma }_M:M\to M^{\omega \omega }$ is the canonical homomorphism, we will show that $h:M\to X^{\omega }$ is a left ${\mathcal{G}}_{\omega }$-approximation of $M$ (note that $X^{\omega }$ has generalized Gorenstein dimension zero by the definition).

Let $g:M\to G$ be any homomorphism of $R$-modules with $G\in {\mathcal{G}}_{\omega }$. Consider the following diagram.

By Lemma 2.7, we have ${Ext}_S^1\left(G^{\omega },A^{\prime }\right)=0$. It follows that ${Hom}_S\left(G^{\omega },X\right)\to {Hom}_S\left(G^{\omega },M^{\omega }\right)$ is an epimorphism. Hence there is a homomorphism of right $S$-modules $\phi :G^{\omega }\to X$ such that $g^{\omega }=f\circ \phi$. Consequently we have $g^{\omega \omega }={\phi }^{\omega }\circ f^{\omega }$. Note that ${\sigma }_G$ is an isomorphism by the definition and that ${\sigma }_G\circ g=g^{\omega \omega }\circ {\sigma }_M$, so we have $g={\sigma }_G^{-1}\circ g^{\omega \omega }\circ {\sigma }_M={\sigma }_G^{-1}\circ {\phi }^{\omega }\circ f^{\omega }\circ {\sigma }_M={\sigma }_G^{-1}\circ {\phi }^{\omega }\circ h$. It follows that ${Hom}_R\left(X^{\omega },G\right)\to {Hom}_R\left(M,G\right)$ is an epimorphism. Then, by the definition $h:M\to G^{\omega }$ is a left ${\mathcal{G}}_{\omega }$-approximation of $M$. □

Combining Proposition 2.9 and Lemma 2.10, we obtain the following theorem.

An $R$-module $\omega$ is said to be a cotilting bimodule if it is faithfully balanced selforthogonal and ${id}_R\omega =id{\omega }_S<\infty$. It is well known that $\widehat{{\mathcal{G}}_{\omega }}=R-mod$ if $\omega$ is a cotilting bimodule (see [2] and [6]). Hence we have the following corollary as a special case of theorem 2.11.

References

Department of Mathematics, Nanjing Normal University, Nanjing 210097, China

E-mail address: weijiaqun@njnu.edu.cn

Received March 28, 2007; Revised version May 12, 2007