A new Homological Invariant for Modules

TL;DR

This paper introduces a new homological invariant called $zeta$-dimension for modules over Noetherian local rings, which refines existing dimensions and characterizes Gorenstein rings.

Contribution

It defines the $zeta$-dimension using Vogel cohomology, providing a finer invariant that unifies and extends classical homological dimensions.

Findings

$zeta$-dimension characterizes Gorenstein rings.

$zeta$-dimension is finer than Gorenstein dimension.

Provides a new perspective on projective and complete intersection dimensions.

Abstract

Let $R$ be a commutative Noetherian local ring with residue field $k$. Using the structure of Vogel cohomology, for any finitely generated module $M$, we introduce a new dimension, called $\zeta$-dimension, denoted by $\zeta-dim_R M$. This dimension is finer than Gorenstein dimension and has nice properties enjoyed by homological dimensions. In particular, it characterizes Gorenstein rings in the sense that: a ring $R$ is Gorenstein if and only if every finitely generated $R$-module has finite $\zeta$-dimension. Our definition of $\zeta$-dimension offer a new homological perspective on the projective dimension, complete intersection dimension of Avramov et al. and $G$-dimension of Auslander and Bridger.

Equations

$$\mathrm{apd}_{R}M=\text{sup}\Bigg{\{}i\in\mathbb{N}_{0}\begin{array}[]{c|c}&\text{Ext}_{R}^{i}(M,T)\neq 0~{}\text{for some finitely generated}\\ &~{}R\text{-}\text{module}~{}T~{}\text{with}~{}\mathrm{pd}_{R}T<\infty\end{array}\Bigg{\}}.$$

$${\mathrm{apd}}_{R}M\le \zeta \text{-}{\dim }_{R}M\le \text{G}\text{-}{\dim }_R(M)\le \text{CI}\text{-}{\dim }_R(M)\le {\mathrm{pd}}_R(M),$$

$$(\widetilde{P_M,P_N})=(P_M,P_N)/(P_M,P_N{)}_b.$$

$$0\to (P_M,P_N{)}_b\to (\widetilde{P_M,P_N})\to 0,$$

$$\zeta\text{-}{\mathrm{dim}}_{R}M=\text{inf}\Bigg{\{}~{}n\begin{array}[]{c|c}&\zeta^{n}(M)~{}\text{is epimorphism and}~{}\zeta^{i}(M)~{}\text{is isomorphism}\\ &~{}\text{for all}~{}i>n\end{array}\Bigg{\}}.$$

$$0\to k\to M\to N\to 0,$$

$$\cdots$$

$$0\to {\Gamma }_m(M)\to M\to M/{\Gamma }_m(M)\to 0$$

$${\text{Ext}}_R^{i-1}({\Gamma }_m(M),k)$$

$$0\to M\stackrel{r}{\to }M\to M/rM\to 0$$

$${\text{Ext}}_R^i(M/rM,k)$$

$${\widehat{\text{Ext}}}_R^n(M,N)={\text{H}}^n{\text{Hom}}_R(T,N).$$

$$\zeta \text{-}{\dim }_{R}M\le \text{G}\text{-}{\dim }_{R}M$$

$$0\to P_g\to P_{g-1}\to \cdots \to P_1\to G_0\to M\to 0$$

$${\text{Ext}}_R^i(M,N/rN)$$

$${\text{Ext}}_R^i(M,N)$$

$$\zeta \text{-}{\dim }_{R_{\mathfrak{p}}}{M}_{\mathfrak{p}}\le \zeta \text{-}{\dim }_{R}M.$$

$$\mathrm{apd}_{R}M=\text{sup}\Bigg{\{}i\in\mathbb{N}_{0}\begin{array}[]{c|c}&\text{Ext}_{R}^{i}(M,T)\neq 0~{}\text{for some finitely generated}\\ &~{}R\text{-}\text{module}~{}T~{}\text{with}~{}\mathrm{pd}_{R}T<\infty\end{array}\Bigg{\}}.$$

$${\mathrm{apd}}_{R}M\le \zeta \text{-}{\dim }_{R}M$$

$${\text{Ext}}_R^{n-1}(M,k)$$

$${\mathrm{apd}}_R(M)\le \zeta \text{-}{\dim }_{R}M\le G\text{-}{\dim }_{R}M\le CI\text{-}{\dim }_{R}M\le {\mathrm{pd}}_{R}M,$$

$${\text{Ext}}_R^i(M,k)$$

Full text

1. Introduction

Let $M$ be a finitely generated module over a commutative Noetherian ring $R$. There are several homological invariants assigned to $M$. The most important one is projective dimension $\text{pd}_{R}M$. Auslander and Bridger [AB] singled out the class of R-modules of finite Gorenstein dimension ($G$-dimension) as a generalization of modules of finite projective dimension. Avramov et al. [AGP] introduced the concept of complete intersection (CI)-dimension. The purpose of this paper is to offer a new dimension that we believe gives a new homological perspective on the aforementioned concepts. We show that the above dimensions can be considered as special cases of a much more general dimension that we shall call it $\zeta$-dimension. To introduce this dimension, we use the structure of Vogel cohomology developed by Pierre Vogel in 1980. The cohomology theory that he developed, associates to each pair $(M,N)$ of modules, a sequence of $R$-modules $\widetilde{\text{Ext}}_{R}^{n}(M,N)$ for $n\in{\mathbb{Z}}$, and comes equipped with a natural transformation $\zeta^{*}(M,N):\text{Ext}_{R}^{n}(M,N)\rightarrow\widetilde{\text{Ext}}_{R}^{n}(M,N)$ of cohomology functors. For any $R$-module $M$ and any integer $i$, we let $\zeta^{i}(M)$ denote the natural map $\text{Ext}_{R}^{i}(M,k)\rightarrow\widetilde{\text{Ext}}_{R}^{i}(M,k)$, where $k$ is the residue field of $R$, and define $\zeta\text{-}\text{dim}_{R}M$ to be the infimum $n\in{\mathcal{N}}$, such that $\zeta^{n}(M)$ is epimorphism and $\zeta^{n}(M)$ is isomorphism for all $i>n$. Note that $\text{pd}_{R}M$, when it is finite, is equal to the supremum of $i$’s such that $\text{Ext}_{R}^{i}(M,k)\neq 0$. On the other hand, $\widetilde{\text{Ext}}_{R}^{i}(M,~{})$ will vanish, for all $i\in{\mathbb{Z}}$. So in fact $\mathrm{pd}_{R}M$ is equal to the infimum of $i$’s such that $\text{Ext}_{R}^{i}(M,k)\cong\widetilde{\text{Ext}}_{R}^{i}(M,K)$, that is $\mathrm{pd}_{R}M=\zeta\text{-}\text{dim}_{R}M$. Moreover, if $G\text{-}\mathrm{dim}_{R}M$ is finite, we shall show that it can be interpreted as the vanishing of certain $\zeta^{i}$’s (see Theorem 1.4).

Moreover we examine the ability of $\zeta$-dimension to detect Gorensteinness of the underlying ring: it is finite for all modules over a Gorenstein ring, conversely if $\zeta\text{-}\mathrm{dim}_{R}k<\infty$ the ring is Gorenstein. $\zeta$-dimension shares many basic properties with other homological dimensions. In particular, it localizes. In an attempt to find a lower bound for $\zeta$-dimension, in Theorem 1.8, for any finitely generated Rmodule $M$ we obtain the following inequality $\text{apd}_{R}M\leq\zeta\text{-}\mathrm{dim}_{R}M$, with equality if $\zeta-\mathrm{dim}_{R}M$ is finite. We recall that $\mathrm{apd}_{R}M$ is defined by the formula

[TABLE]

So the place of $\zeta-di_{R}(M)$ in the hierarchy of homological dimensions is determined as

[TABLE]

with equality to the left of any finite ones. No example of a module $M$ with $\zeta\text{-}{\mathrm{dim}}_{R}M<\text{G-}{\mathrm{dim}}_{R}M$ is known at present to the authors.

Towards the end of the paper, we deal with the resolving property of the category of modules of finite $\zeta$-dimension. Throughout the paper $(R,\mathfrak{m},k)$ is a commutative Noetherian local ring with residue field $k$.

2. $\zeta$-DIMENSION

We begin by recalling the construction of Vogel cohomology. First let us mention that, abusing notation we shall use the symbol $(A,B)$ for the graded Hom-functor applied to graded $R$-modules $A$ and $B$. Thus $(A,B)_{n}=\Pi_{i\in{\mathbb{Z}}}\text{Hom}_{R}(A_{i},B_{i+n})$. The differential is defined on $(A,B)$ by the formula $\partial(f)(x)=\partial(f(x))-(-1)^{\text{deg}f}f(\partial(x))$, where $x\in A$, thus making $(A,B)$ into a complex.

Let $M$ and $N$ be finitely generated $R$-modules and $P_{M}$ and $P_{N}$ denote their projective resolutions, respectively. We shall use $P_{M}$ (resp. $P_{N}$) to denote the corresponding underlying graded module. The subset $(P_{M},P_{N})_{b}$ of bounded homogeneous maps (a homogeneous map is called bounded if only finitely many components of that map are non-zero) is a graded submodule of $(P_{M},P_{N})$. The restriction of $\partial$ to $(P_{M},P_{N})_{b}$ make it into a subcomplex of $(P_{M},P_{N})$. The corresponding quotient complex will be of fundamental important to us. We denote by $(\widetilde{P_{M},P_{N}})$ the quotient complex

[TABLE]

Passing on to cohomology we obtain Vogel cohomology. It will be denoted by $\widetilde{\text{Ext}}_{R}^{*}(M,N)$. Moreover the short exact sequence

[TABLE]

where the cohomology of the middle term is just $\text{Ext}_{R}^{*}(M,N)$, yields upon passing to corresponding long cohomology exact sequence, a natural transformation $\zeta^{*}(M,N):\text{Ext}_{R}^{*}(M,N)\rightarrow\widetilde{\text{Ext}}_{R}^{*}(M,N).$

By essentially following the same argument analogous to ordinary cohomology, one can see that $\widetilde{\text{Ext}}_{R}^{*}$ is a cohomological functor, independent of the choice of projective resolutions of $M$ and $N[G,I]$. The following result that will be used latter, is easy to see.

Proposition 2.1**.**

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring. Then for any $R$-module $M$ the following are equivalent:

• i)

$\mathrm{pd}_{R}M$* is finite.*

• ii)

$\widetilde{\text{Ext}}_{R}^{i}(M,~{})=0$* for all integer $i$.*

• iii)

$\widetilde{\text{Ext}}_{R}^{i}(~{},M)=0$* for all integer $i$.*

For simplicity, for any $R$-module $M$ and any integer $i$ we let $\zeta^{i}(M)$ denote the natural transformation $\zeta^{i}(M,k):\text{Ext}_{R}^{n}(M,k)\rightarrow\widetilde{\text{Ext}}_{R}^{n}(M,k).$

Definition 2.2**.**

Let $M\neq 0$ be a finitely generated $R$-module. We assign an invariant to $M$, called $\zeta$-dimension of $M$, denoted $\zeta\text{-}{\mathrm{dim}}_{R}M$, by the formula

[TABLE]

We complement this by setting $\zeta\text{-}{\mathrm{dim}}_{R}0=-\infty$.

Next theorem is our first main result.

Theorem 2.3**.**

Let $(R,\mathfrak{m},k)$ be a commutative Noetherian local ring. Let $n\in\mathbb{N}$. Then the following conditions are equivalent:

• i)

$\zeta\text{-}{\mathrm{dim}}_{R}M\leq n$, for all finitely generated $R$-module $M$.

• ii)

$\zeta\text{-}{\mathrm{dim}}_{R}k\leq n$.

• iii)

$\zeta^{i}(k)$* is epimorphism, for all $i\geq n$.*

Proof.

The implications $(i)\Rightarrow(ii)$ and $(i)\Rightarrow(iii)$ are trivially hold.

$(ii)\Rightarrow(i)$. Let $M$ be a finitely generated $R$-module. We induce on ${\mathrm{dim}}M$. Suppose first ${\mathrm{dim}}M=0$. So $l(M)$, the lengths of $M$ is finite, say $s$. We use induction on s to prove the result in this case. If $s=1$, there is nothing to prove. So let $s>1$, and consider the short exact sequence

[TABLE]

where $l(N)=s-1$. For any integer $i$, there exists a commutative diagram of Rmodules and $R$-homomorphisms

[TABLE]

By induction assumption, $\zeta^{i}(k)$ and $\zeta^{i}(N)$ are both epimorphism for $i=n$ and isomorphism for $i>n$. This by a simple diagram chasing, in view of the Five Lemma, will implies that $\zeta^{i}(M)$ is epimorphism for $i=n$ and isomorphism for $i>n$. This completes the proof in this case. Now suppose, inductively, that ${\mathrm{dim}}M=n>0$ and the result has been proved for all $R$-modules of dimension less than $n$. Consider the short exact sequence

[TABLE]

of $R$-modules, where $\Gamma_{m}(M)$denotes the $\mathfrak{m}$-torsion functor $\cup_{n\in\mathbb{N}}0:_{(M)}\mathfrak{m}^{n}$. This in turn induces, for any integer $i$ a commutative diagram of $R$-modules and $R$-homomorphisms

[TABLE]

By inductive assumption, $\zeta^{n}(\Gamma_{m}(M))$ is epimorphism and $\zeta^{i}(\Gamma_{m}(M)$ is isomorphism for all $i>n$. So $\zeta\text{-}{\mathrm{dim}}_{R}M\leq n$ if and only if $\zeta\text{-}{\mathrm{dim}}_{R}(M/\Gamma_{m}(M))\leq n$. We can therefore assume, in the inductive step that there exists an element $r\in R$ which is a non-zero divisor on $M$. The exact sequence

[TABLE]

induces for any integer $i$, a commutative diagram of $R$-modules and $R$-homomorphisms

[TABLE]

Since ${\mathrm{dim}}M/rM<{\mathrm{dim}}M$, by inductive assumption, $\zeta\text{-}{\mathrm{dim}}_{R}(M/rM)\leq n$. So $\zeta^{n}(M/rM)$ is epimorphism and $\zeta^{i}(M/rM)$ is isomorphism for all $i>n$. But $r\in\mathfrak{m}$ and so each element of $\widetilde{\text{Ext}}_{R}^{i}(M,k)$ is annihilated by multiplication by $r$. This fact in conjunction with the latter diagram, will implies that $\zeta^{n}(M)$ is epimorphism and $\zeta^{i}(M)$ is isomorphism for all $i>n$.

$(iii)\Rightarrow(ii)$. If $R$ is regular, the result is clear, because $\mathrm{pd}_{R}k$ is finite and so $\widetilde{\text{Ext}}_{R}^{i}(k,k)=0$ for all integer $i$. So let $R$ is non-regular. Then by [M2, Theorem 6], $\zeta^{i}(k)$ is monomorphism for all integer $i$. This follows the result. ∎

Following result shows that $\zeta\text{-}{\mathrm{dim}}_{R}M$ is a refinement of $G$-dimension. We preface it by recalling the structure of Tate cohomology, introduced through complete resolutions, that has been the subject of several recent expositions, in particular by Buchweitz [B] and Cornick and Kropholler [CK]. LetM be a finite $R$-module of finite $G$-dimension. Choose a complete resolution $T\xrightarrow{v}P\xrightarrow{\pi}M$ of M (see for instance [AM, Sec.5]). Then for each $R$-module $N$ and for each $n\in{\mathbb{Z}}$, Tate cohomology group is defined by the equality

[TABLE]

Theorem 2.4**.**

For any finitely generated $R$-module $M$, there is an inequality

[TABLE]

with equality, when G-${\mathrm{dim}}_{R}M$ is finite.

Proof.

Without loss of generality, we may assume that G-${\mathrm{dim}}_{R}M=g$ is finite. With this assumption, by [M1, §2], for any integer $i$, there is a natural isomorphism of cohomology functors $\widehat{\text{Ext}}_{R}^{i}(M,k)\cong\widetilde{\text{Ext}}_{R}^{i}(M,k)$, compatible with the maps coming from $\text{Ext}_{R}^{i}(M,k)$. If $g=0$, using the definition of Tate cohomology it is easily seen that $\widehat{\text{Ext}}_{R}^{0}(M,k)\cong\text{Hom}_{R}(M,k)/N$, for suitable $R$-module $N$. So $\zeta^{0}(M)$ is always epimorphism. Moreover it follows from [AM, 5.2(2)] that $\zeta^{i}(M)$ is isomorphism for all $i>0$. Hence $\zeta\text{-}{\mathrm{dim}}_{R}M=0$. Now let $g>0$. It follows from [AM, 5.2(2)] that $\zeta^{i}(M)$ is isomorphism for all $i>g$ and follows from [AM, 7.1] that $\zeta^{g}(M)$ is epimorphism. So $\zeta\text{-}{\mathrm{dim}}_{R}M\leq\text{G-}{\mathrm{dim}}_{R}M$. For equality, consider a Gorenstein resolution of $M$, say

[TABLE]

with all the $P_{i}$’s finitely generated and projective and with $G_{0}$ of Gorenstein dimension zero. But then we can also assume that each $P_{i}\rightarrow P_{i-1}$ gives a projective cover of the image of $P_{i}$ in $P_{i-1}$. So in particular this means that $P_{g}\subset\mathfrak{m}P_{g-1}$. But this implies that $\text{Hom}_{R}(P_{g-1},k)\rightarrow\text{Hom}_{R}(P_{g},k)$ is the zero map. So if $P_{g}\neq 0$ we see that $\text{Ext}_{\mathcal{G}}^{g}(M,k)\neq 0$. Here we are tacitly assuming $g\geq 2$. If $g=1$, then the resolution of $M$ looks like $0\rightarrow F\rightarrow G\rightarrow M\rightarrow 0$. Here $F$ is a finitely generated free $R$-module. We can assume that $F\subset\mathfrak{m}G$, for if not one can use Nakayama’s lemma to get a copy of $R$ in $F$ a direct summand of $G$. But then we can go modulo this copy of $R$. So repeating if necessary, we finally get that $F\subset\mathfrak{m}G$. But then if $F\neq 0$, by the same type argument as above we get that $\text{Ext}_{\mathcal{G}}^{1}(M,k)\neq 0$. Hence by [AM, 7.1], either $\zeta^{g}(M)$ is not injective or $\zeta\text{-}{\mathrm{dim}}^{g-1}(M)$ is not epimorphism. So $\zeta\text{-}{\mathrm{dim}}_{R}M\geq g$. This completes the proof.

∎

Now we are in position to put all our results together to present a characterization for Gorenstein rings in terms of $\zeta$-dimension. We need the following proposition.

Proposition 2.5**.**

Let $M$ be a finitely generated $R$-module of finite $\zeta$-dimension, say $n$. Then for any finitely generated $R$-module $N$, $\zeta^{n}(M,N)$ is epimorphism and $\zeta^{i}(M,N)$ is isomorphism for all $i>n$.

Proof.

Let $N$ be a finitely generated $R$-module. By following the same type argument as we have used for the proof of Theorem 1.3, we may assume inductively that ${\mathrm{dim}}N>0$, the result holds for all finitely generated modules of dimension less than ${\mathrm{dim}}N$ and also there exists a non-zerodivisor $r$ on $N$. So we have a short exact sequence $0\rightarrow N\xrightarrow{r}N\rightarrow N/rN\rightarrow 0$. This induces, for any integer $i$, a commutative diagram of $R$-modules and $R$-homomorphisms

[TABLE]

Since for any $i\geq n$, $\zeta^{i}(M,N/rN)$ is epimorphism and $\zeta^{i+1}(M,N/rN)$ is isomorphism, by a diagram chasing one can see that the multiplication map by $r$ restricted to $\text{Ker}\zeta^{i+1}(M,N)$ is epimorphism. So using Nakayama’s Lemma, for any $i\geq n$, we get that the map $\zeta^{i+1}(M,N)$ is monomorphism. Now consider the commutative diagram

[TABLE]

Since for any integer $i\geq n$, $\zeta^{i}(M,N/rN)$ is epimorphism and $\zeta^{i+1}(M,N)$ is monomorphism, the restriction of the multiplication map $r$ to $\text{Coker}~{}\zeta^{i}(M,N)$ is surjective, and so by Nakayama’s Lemma, $\text{Coker}~{}\zeta^{i}(M,N)=0$. Therefore $\zeta^{i}(M,N)$ is epimorphism for all $i\geq n$. This completes the proof. ∎

Theorem 2.6**.**

The following conditions are equivalent:

• i)

$R$* is Gorenstein.*

• ii)

$\zeta\text{-}{\mathrm{dim}}_{R}M<\infty$, for all finitely generated $R$-module $M$.

• iii)

$\zeta\text{-}{\mathrm{dim}}_{R}k<\infty$.

Proof.

$(i)\Rightarrow(ii)$. Since $R$ is Gorenstein, by [AB], G-${\mathrm{dim}}_{R}M<\infty$, for all finitely generated $R$-module $M$. So by Theorem 1.4, $\zeta\text{-}{\mathrm{dim}}_{R}M<\infty$.

$(ii)\Rightarrow(iii)$. This trivially holds.

$(iii)\Rightarrow(i)$. Let $\zeta\text{-}{\mathrm{dim}}_{R}k<\infty$. By Proposition 1.5, $\zeta^{i}(k,R)$ is isomorphism for all integer $i>\zeta\text{-}{\mathrm{dim}}_{R}k$. But $\widetilde{\text{Ext}}_{R}^{i}(k,R)=0$ for all $i$. This implies that $\text{Ext}_{R}^{i}(k,R)=0$ for all $i$ large enough. So $R$ is Gorenstein.

∎

Proposition 2.7**.**

Let $\mathfrak{p}$ be a prime ideal in $\textnormal{Spec}(R)$. Then for any finitely generated $R$-module $M$,

[TABLE]

Proof.

For proof just one should note that both functors $\text{Ext}_{R}^{*}$ and $\widetilde{\text{Ext}}_{R}^{*}$ are well behaved under localization. ∎

Now we aim to give a lower bound for $\zeta\text{-}{\mathrm{dim}}_{R}M$. There is a refinement of projective dimension of $M$ denoted by $\mathrm{apd}_{R}M$, defined by the formula

[TABLE]

It is proved in [AB, Theorem 4.13] that $\mathrm{apd}_{R}$ is also a refinement of G-dimension G-${\mathrm{dim}}_{R}$ . We shall show that $\mathrm{apd}_{R}M\leq\zeta\text{-}{\mathrm{dim}}_{R}M$, with equality where $\zeta\text{-}{\mathrm{dim}}_{R}M$ is finite.

Theorem 2.8**.**

Let $M$ be a finitely generated $R$-module. Then

[TABLE]

with equality if $\zeta\text{-}{\mathrm{dim}}_{R}M$ is finite.

Proof.

We may (and do) assume that $\zeta\text{-}{\mathrm{dim}}_{R}M=n$ is finite. So by Proposition 1.5, for any finitely generated $R$-module $N$, $\zeta^{n}(M,N)$ is epimorphism and $\zeta^{i}(M,N)$ is isomorphism for all $i>n$. Now let $N$ be an $R$-module of finite projective dimension. By Proposition 1.1, $\widetilde{\text{Ext}}_{R}^{i}(M,N)=0$ for all integer $i$. So we get $\text{Ext}_{R}^{i}(M,N)=0$ for all $i>n$. This implies that $\mathrm{apd}_{R}M\leq\zeta\text{-}{\mathrm{dim}}_{R}M$. Now let $\mathrm{apd}_{R}(M)=s$ and $s<n$. We seek for a contradiction. The short exact sequence $0\rightarrow\mathfrak{m}\rightarrow R\rightarrow k\rightarrow 0$ induces the following commutative diagram

[TABLE]

Since $s<n$, $\text{Ext}_{R}^{i}(M,R)=0$ for all $i\geq n$. So since $\zeta^{n+1}(M,\mathfrak{m})$ is isomorphism, we get $\zeta^{n}(M,k)$ is isomorphism and since $\zeta^{n}(M,\mathfrak{m})$ is epimorphism, we get $\zeta^{n-1}(M,k)$ is epimorphism. Therefore $\zeta\text{-}{\mathrm{dim}}_{R}M\leq n-1$. This is the desired contradiction. So $s=n$.

∎

Corollary 2.9**.**

For any finitely generated $R$-module $M$,

[TABLE]

with equality to the left of any finite ones.

Finally we show that the category of modules of finite $\zeta$-dimension has resolving property. Let $\mathcal{Z}$ (resp. $\mathcal{\tilde{Z}}$) denotes the full subcategory of $\mathcal{F}$, the category of finitely generated $R$-modules and $R$-homomorphisms, whose objects are modules of $\zeta$-dimension zero (resp. of finite $\zeta$-dimension).

Proposition 2.10**.**

The category $\mathcal{Z}$ is closed under extension and kernels of epimorphisms. Moreover $\mathcal{Z}$ contains $\mathcal{P}$, the category of finite projective $R$-modules.

Proof.

Let $0\rightarrow E\rightarrow L\rightarrow M\rightarrow 0$ be an exact sequence of $R$-modules with $M\in\mathcal{Z}$. We shall show that $L\in\mathcal{Z}$ if and only if $E\in\mathcal{Z}$. The above short exact sequence induces for any integer $i$, a commutative diagram of $R$-modules and $R$-homomorphisms

[TABLE]

It is now easy to deduce the first assertion, by a simple diagram chasing and also five lemma. The last assertion is elementary. ∎

Proposition 2.11**.**

The category $\mathcal{\tilde{Z}}$ is closed under extension, kernel of epimorphisms and cokernel of monomorphisms. Moreover $\mathcal{\tilde{Z}}\supseteq\mathcal{\tilde{G}}$, where $\mathcal{\tilde{G}}$ denotes the subcategory of $\mathcal{F}$ whose objects are finitely generated $R$-modules of finite $G$-dimension.

Proof.

Last assertion follows from Theorem 1.4. The other ones are easy to see. ∎