ON MODULES OF G-DIMENSION ZERO OVER NON-GORENSTEIN LOCAL RINGS

RYO TAKAHASHI

1. Introduction

Throughout this note, we assume that all rings are commutative and noetherian, and that all modules are finitely generated.

G-dimension was defined by Auslander [1] and was deeply studied by Auslander and Bridger [2]. This is a homological invariant for modules analogous to projective dimension.

A Cohen-Macaulay local ring is called to be of finite Cohen-Macaulay representation type if there are only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules. Such a ring has been well researched for a long time. In several cases, all the isomorphism classes of indecomposable maximal Cohen-Macaulay modules over such a ring have already been classified completely. (See [14] for the details.)

Over a Gorenstein local ring, a module has G-dimension zero if and only if it is a maximal Cohen-Macaulay module. (Recall that a Gorenstein ring is always Cohen-Macaulay.) Hence a Gorenstein local ring has only finitely many isomorphism classes of indecomposable modules of G-dimension zero if and only if it is of finite Cohen-Macaulay representation type. Thus we are interested in a non-Gorenstein local ring which has only finitely many isomorphism classes of indecomposable modules of G-dimension zero, especially interested in determining all the isomorphism classes of indecomposable modules of G-dimension zero over such a ring.

Let R be such a ring. Our guess is that the only isomorphism class of indecomposable module of G-dimension zero is the isomorphism class of R. In other words, we guess that the following holds:

Conjecture 1.1. Let R be a non-Gorenstein local ring. Suppose that there exists a nonfree R-module of G-dimension zero. Then there exist infinitely many isomorphism classes of indecomposable R-modules of G-dimension zero.

Indeed, over a certain artinian local ring having a non-free module of G-dimension zero, Yoshino [16] actually constructed a family of modules of G-dimension zero with continuous parameters.

For a ring R, let us denote by modR the category of all finitely generated R-modules, and by G(R) the full subcategory of modR consisting of all R-modules of G-dimension zero. The main result of this note is the following theorem, which extends [16, Theorem 6.1].

Theorem 1.2. Let R be a henselian non-Gorenstein local ring of depth zero. Suppose that there exists a non-free R-module in G(R). Then the residue class field of R does not admit a G(R)-precover as an R-module. In particular, the category G(R) is not contravariantly finite in modR.

The detailed version [8] of this note has been submitted for publication elsewhere.

It is easily seen that this theorem gives a positive answer for Conjecture 1.1 if the depth of the local ring R is zero:

Corollary 1.3. Let R be a henselian non-Gorenstein local ring of depth zero. Suppose that there exists a non-free R-module of G-dimension zero. Then there exist infinitely many isomorphism classes of indecomposable R-modules of G-dimension zero.

We should remark that the above corollary especially asserts that Conjecture 1.1 holds if R is artinian.

2. Outline of the proof of Theorem 1.2

Throughout this section, R is always a commutative noetherian local ring with residue class field k. All R-modules considered in this section are finitely generated.

We define (−)^{∗} to be the dual functor Hom_{R}(−, R) from modR to itself, and denote by Ω_{R}^{n}M the nth syzygy module of an R-module M. To begin with, we recall the definition of G-dimension.

Definition 2.1. Let M be an R-module.

(1) If the following conditions hold, then we say that M has G-dimension zero, and write G-dim_{R}M = 0.

i) The natural homomorphism M → M^{∗∗} is isomorphic.

ii) Ext_{R}^{i}(M, R) = 0 for every i > 0. iii) Ext_{R}^{i}(M^{∗}, R) = 0 for every i > 0.

(2) If G-dim_{R}(Ω_{R}^{n}M) = 0 for a non-negative integer n, then we say that M has Gdimension at most n, and write G-dim_{R}M ≤ n. If such an integer n does not exist, then we say that M has infinite G-dimension, and write G-dim_{R}M = ∞.

As follows, G-dimension possesses many properties similar to those of projective dimension. For the proofs, we refer to [2], [6], [7], and [13].

Proposition 2.2. (1) The following conditions are equivalent.

i) R is Gorenstein.

ii) G-dim_{R}M < ∞ for any R-module M.

iii) G-dim_{R}k < ∞.

(2) Let M be a non-zero R-module with G-dim_{R}M < ∞. Then G-dim_{R}M = depthR − depth_{R}M.

(3) Let 0 → L → M → N → 0 be a short exact sequence of R-modules. If two of L, M, N have finite G-dimension, then so does the third.

(4) Let M be an R-module, and n a non-negative integer. Then G-dim_{R}(Ω_{R}^{n}M) = sup{G-dim_{R}M − n, 0}.

(5) Let M, N be R-modules. Then G-dim_{R}(M ⊕ N) = sup{G-dim_{R}M, G-dim_{R}N}.

Here we introduce the notion of a cover of a module.

Definition 2.3. Let X be a full subcategory of modR, and let ϕ : X → M be a homomorphism from X ∈ X to M ∈ modR.

(1) We call ϕ an X -precover (or a right X -approximation) of M if for any homomorphism ϕ′ : X′ → M with X′ ∈ X there exists a homomorphism f : X′ → X such that ϕ′ = ϕf.

(2) Assume that ϕ is an X -precover of M. We call ϕ an X -cover (or a right minimal X - approximation) of M if any endomorphism f of X with ϕ = ϕf is an automorphism.

A full subcategory X of modR is said to be contravariantly finite in modR if every M ∈ modR has an X -precover.

The following proposition helps us to see whether a given precover is a cover or not.

Proposition 2.4. [15, Lemma (2.2)] Let X be a full subcategory of modR, and let

0 → N →^{ψ} X →^{ϕ} M

be an exact sequence in modR where ϕ is an X -precover. Suppose that R is henselian. Then the following conditions are equivalent.

i) ϕ is not an X -cover.

ii) There exists a non-zero submodule L of N such that ψ(L) is a direct summand of X.

Let X be a full subcategory of modR. We say that X is closed under direct summands provided that for any object M of X and any direct summand N of M we have N is also an object of X. Note by Proposition 2.2.5 that the category G(R) is closed under direct summands. Using the above proposition, we can easily prove that the existence of an X -precover in fact implies the existence of an X -cover if X is a full subcategory of modR which is closed under direct summands. Hence we have the following.

Corollary 2.5. Let X be a full subcategory of modR which is closed under direct summands. Suppose that R is henselian. Then an R-module admits an X -cover if and only if it admits an X -precover.

We say that a full subcategory X of modR is closed under extensions if for any short exact sequence

0 → L → M → N → 0

in modR with L, N ∈ X, we have M is also an object of X. Note by 2.2.2 and 2.2.3 that the category G(R) is closed under extensions. The lemma below is so-called Wakamatsu’s Lemma, which plays an important role in the notion of a cover. For the proof, see [11] or [12, Lemma 2.1.1].

Lemma 2.6 (Wakamatsu). Let X be a full subcategory of modR which is closed under extensions, and let

0 → N → X →^{ϕ} M

be an exact sequence in modR where ϕ is an X -cover of M. Then Ext_{R}^{1}(X′, N) = 0 for any X′ ∈ X.

Now, we shall give the outline of the proof of our theorem.

Outline of the proof of Theorem 1.2. Let R be a henselian non-Gorenstein local ring of depth zero such that there exists a non-free module in G(R). Suppose that the residue field k of R has a G(R)-precover as an R-module. We want to derive contradiction. By Corollary 2.5, there exists a short exact sequence

0 → L →^{θ} Z →^{π} k → 0

of R-modules such that π is a G(R)-cover. Dualizing this sequence, we obtain an exact sequence

0 → k^{∗} →^{π^{∗}} Z^{∗} →^{θ^{∗}} L^{∗}.

Set C = Im(θ^{∗}), and let α : Z^{∗} → C be the surjection induced by θ^{∗} and β : C → L^{∗} be the natural embedding.

We shall show that the homomorphism α : Z^{∗} → C is a G(R)-cover of C. Fix X ∈ G(R). To prove that any homomorphism X → C is factored as X → Z^{∗} →^{α} C, we may assume that X is non-free and indecomposable. Applying the functor Hom_{R}(X, −) to the above exact sequence, we get an exact sequence

0 ⟶ Hom_{R}(X, k^{∗}) ⟶^{Hom_{R}(X,π^{∗})} Hom_{R}(X, Z^{∗}) ⟶^{Hom_{R}(X,θ^{∗})} Hom_{R}(X, L^{∗}).

We can prove that the homomorphism Hom_{R}(X, θ^{∗}) is a split epimorphism. Since Hom_{R}(X, θ^{∗}) = Hom_{R}(X, β) · Hom_{R}(X, α) and Hom_{R}(X, β) is an injection, the homomorphism Hom_{R}(X, β) is an isomorphism. Therefore Hom_{R}(X, α) is a split epimorphism, and hence it is especially a surjection. This means that the homomorphism α : Z^{∗} → C is a G(R)-precover of C. Assume that α is not a G(R)-cover. Then Proposition 2.4 shows that k^{∗} and Z^{∗} have some common non-zero summand. Since k^{∗} is a k-vector space, the R-module Z^{∗} has a summand isomorphic to the R-module k, and hence k ∈ G(R) by Proposition 2.2.5. It follows from Proposition 2.2.1 that R is Gorenstein, which contradicts the assumption of the theorem. Therefore α must be a G(R)-cover of C.

Thus we can apply Lemma 2.6, and get Ext_{R}^{1}(Y, k^{∗}) = 0 for every Y ∈ G(R). Since the local ring R has depth zero, that is to say, k^{∗} is a non-zero k-vector space, every module in G(R) is free, which is contrary to the assumption of our theorem. This contradiction proves our theorem.

References

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Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan

  E-mail address: takahasi@math.okayama-u.ac.jp