ON THE DECOMPOSABILITY OF A SYZYGY OF THE RESIDUE FIELD

Ryo Takahashi

1. Introduction

Throughout the present paper, we assume that all rings are commutative noetherian local rings and all modules are finitely generated modules.

Dutta [10] proved the following theorem in his research into the homological conjectures:

Theorem 1.1 (Dutta). Let (R, 𝔪, k) be a local ring. Suppose that the nth syzygy module of k has a non-zero direct summand of finite projective dimension for some n ≥ 0. Then R is regular.

Since G-dimension is similar to projective dimension, this theorem naturally leads us to the following question:

Question 1.2. Let (R, 𝔪, k) be a local ring. Suppose that the nth syzygy module of k has a non-zero direct summand of finite G-dimension for some n ≥ 0. Then is R Gorenstein?

It is obviously seen from the indecomposability of k that this question is true if n = 0. Hence this question is worth considering just in the case where n ≥ 1.

We are able to answer in this paper that the above question is true if n ≤ 2. Furthermore, we can even determine the structure of a ring satisfying the assumption of the above question for n = 1, 2.

2. Main results

For a local ring R, we denote by mod R the category of finitely generated R-modules. First of all, we recall the definition of G-dimension.

Definition 2.1. (1) We denote by G(R) the full subcategory of mod R consisting of all R-modules M satisfying the following three conditions:

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

The final version of this paper has been submitted for publication elsewhere.

(2) Let M be an R-module. If n is a non-negative integer such that there is an exact sequence

0 → G_{n} → G_{n−1} → ⋯ → G_{1} → G_{0} → M → 0

of R-modules with G_{i} ∈ G(R) for every i, then we say that M has G-dimension 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 = ∞.

For properties of G-dimension, we refer to [3] or [9].

Proposition 2.2. Let (R, 𝔪, k) be a local ring. Suppose that there is a direct sum decomposition 𝔪 = I ⊕ J where I, J are non-zero ideals of R. Let M be a non-free indecomposable module in G(R). Then there exist elements x, y ∈ 𝔪 such that

(1) I = (x) and J = (y),

(2) (0 : x) = (y) and (0 : y) = (x),

(3) M is isomorphic to either (x) or (y). Hence the minimal free resolution of k is as follows:

⋯ \underrightarrow{( \begin{matrix}y & 0 \\ 0 & x \end{matrix} )} R^{2} \underrightarrow{( \begin{matrix}x & 0 \\ 0 & y \end{matrix} )} R^{2} \underrightarrow{( \begin{matrix}y & 0 \\ 0 & x \end{matrix} )} R^{2} ⟶^{( x y)} R ⟶ k ⟶ 0.

Proof. The modules M^{∗} and ΩM are also non-free indecomposable modules in G(R). There are isomorphisms

M^{∗} ≅ Hom_{R}(M, 𝔪)
= Hom_{R}(M, I ⊕ J)
≅ Hom_{R}(M, I) ⊕ Hom_{R}(M, J).

The indecomposability of M^{∗} implies that either Hom_{R}(M, I) = 0 or Hom_{R}(M, J) = 0. We may assume that

(2.2.1)  Hom_{R}(M, J) = 0.

There is an exact sequence

(2.2.2)  0 → ΩM → R^{n} → M → 0.

Dualizing this by J, we obtain another exact sequence

Hom_{R}(M, J) → J^{n} → Hom_{R}(ΩM, J).

We have Hom_{R}(ΩM, J) ≠ 0 by (2.2.1). Applying the above argument to the module ΩM yields

(2.2.3)  Hom_{R}(ΩM, I) = 0.

Also, dualizing (2.2.2) by I, we get an exact sequence

0 → Hom_{R}(M, I) → I^{n} → Hom_{R}(ΩM, I),

and hence M^{∗} ≅ Hom_{R}(M, I) ≅ I^{n}. The indecomposability of M^{∗} implies that n = 1 (i.e. M is cyclic), and M^{∗} ≅ I.

We also have

M ≅ M^{∗∗}
≅ Hom_{R}(M^{∗}, 𝔪)
≅ Hom_{R}(M^{∗}, I) ⊕ Hom_{R}(M^{∗}, J).

Note that Hom_{R}(M^{∗}, I) is isomorphic to Hom_{R}(I, I), which contains the identity map of I. Hence Hom_{R}(M^{∗}, I) ≠ 0 and therefore

Hom_{R}(M^{∗}, J) = 0.

Applying the above argument to the module M^{∗}, we see that M^{∗} is also cyclic and M ≅ M^{∗∗} ≅ I. Thus, we have shown that M ≅ M^{∗} ≅ I and these modules are cyclic. Noting (2.2.3) and applying the above argument to the module ΩM, we see that ΩM ≅ (ΩM)^{∗} ≅ J and these modules are cyclic.

Now, writing I = (x) and J = (y), we can prove (x) = (0 : y) and (0 : x) = (y). Thus we obtain the minimal free resolutions of (x) and (y):

\{ \begin{matrix}⋯ & →^{y} & R & →^{x} & R & →^{y} & R & → & (x) & → & 0, \\ ⋯ & →^{x} & R & →^{y} & R & →^{x} & R & → & (y) & → & 0. \end{matrix}

Taking the direct sum of these exact sequence, we get

⋯ \underrightarrow{( \begin{matrix}y & 0 \\ 0 & x \end{matrix} )} R^{2} \underrightarrow{( \begin{matrix}x & 0 \\ 0 & y \end{matrix} )} R^{2} \underrightarrow{( \begin{matrix}y & 0 \\ 0 & x \end{matrix} )} R^{2} ⟶ 𝔪 ⟶ 0.

Joining this to the natural exact sequence 0 → 𝔪 → R → k → 0 constructs the minimal free resolution of k in the assertion. □

We denote by edim R the embedding dimension of a local ring R. When a homomorphic image of a regular local ring is given, we can choose a minimal presentation of the ring in the following sense:

Proposition 2.3. Let R be a homomorphic image of a regular local ring. Then there exist a regular local ring (S, 𝔫) and an ideal I of S contained in 𝔫^{2} such that R ≅ S/I.

Here we introduce a famous result due to Tate [17, Theorem 6]. See also [5, Remarks 8.1.1(3)].

Lemma 2.4 (Tate). Let (S, 𝔫, k) be a regular local ring, I an ideal of S contained in 𝔫^{2}, and R = S/I a residue class ring. Suppose that the complexity of k over R is at most one. (In other words, the set of all the Betti numbers of the R-module k is bounded.) Then I is a principal ideal.

We denote by β_{i}^{R}(M) the ith Betti number of a module M over a local ring R. Handling the above results, we can determine the structure of a local ring with decomposable maximal ideal having a non-free module of G-dimension zero, as follows:

Theorem 2.5. Let (S, 𝔫, k) be a regular local ring, I an ideal of S contained in 𝔫^{2}, and R = S/I a residue class ring. Suppose that there exists a non-free R-module in G(R). Then the following conditions are equivalent:

(1) The maximal ideal of R is decomposable;

(2) dim S = 2 and I = (xy) for some regular system of parameter x, y of S.

Proof. Let 𝔪 = 𝔫/I be the maximal ideal of R.

(2) ⇒ (1): It is easy to see that 𝔪 = xR ⊕ yR and that xR, yR are non-zero.

(1) ⇒ (2): First of all, note from the condition (1) that R is not an integral domain, hence is not a regular local ring.

Proposition 2.2 says that 𝔪 = xR⊕yR for some x, y ∈ 𝔫, and that β_{i}^{R}(k) = 2 for every i ≥ 2. It follows from Lemma 2.4 that I is a principal ideal. Hence R is a hypersurface.

We write I = (f) for some f ∈ 𝔫^{2}. Since 𝔪 is decomposable, the local ring R is not artinian. (Over an artinian Gorenstein local ring, the intersection of non-zero ideals is also non-zero; cf. [8, Exercise 3.2.15].) Hence we have 0 < dim R < edim R = 2, which says that dim R = 1 and dim S = 2.

Note that 𝔫 = (x, y, f). Because edim S = dim S = 2, one of the elements x, y, f belongs to the ideal generated by the other two elements. Noting that the images of elements x, y in 𝔪 form a minimal system of generators of 𝔪, we see that f ∈ (x, y), and hence x, y is a regular system of parameters of S.

On the other hand, noting xR ∩ yR = 0, we get xy ∈ I = (f). Write xy = cf for some c ∈ S. Since the associated graded ring gr_{𝔫}(S) is a polynomial ring over k in two variables \overline{x}, \overline{y} ∈ 𝔫/𝔫^{2}, we especially have \overline{x}\overline{y} ≠ 0 in 𝔫^{2}/𝔫^{3}, namely, xy ∉ 𝔫^{3}. It follows that c ∉ 𝔫 because f ∈ 𝔫^{2}. Therefore the element c is a unit of S, and thus I = (xy). □

Using Theorem 2.5 and Cohen’s structure theorem, we obtain the following corollary.

Corollary 2.6. Let (R, 𝔪) be a complete local ring. The following conditions are equivalent:

(1) There is a non-free module in G(R), and 𝔪 is decomposable;

(2) R is Gorenstein, and 𝔪 is decomposable;

(3) There are a complete regular local ring S of dimension two and a regular system of parameters x, y of S such that R ≅ S/(xy).

The finiteness of G-dimension is independent of completion. Thus, Corollary 2.6 not only gives birth to a generalization of [15, Proposition 2.3] but also guarantees that Question 1.2 is true if n = 1.

As far as here, we have observed a local ring whose maximal ideal is decomposable. From here to the end of this paper, we will observe a local ring such that the second syzygy module of the residue class field is decomposable. We begin with the following theorem, which implies that Question 1.2 is true if n = 2.

Theorem 2.7. Let (R, 𝔪, k) be a local ring. Suppose that 𝔪 is indecomposable and that Ω_{R}^{2}k has a non-zero proper direct summand of finite G-dimension. Then R is a Gorenstein ring of dimension two.

Proof. Replacing R with its 𝔪-adic completion, we may assume that R is a complete local ring. In particular, note that R is Henselian.

We have Ω_{R}^{2}k = M ⊕N for some non-zero R-modules M and N with G-dim_{R}M < ∞. There is an exact sequence

0 ⟶ M ⊕ N ⟶^{(f,g)} R^{e} ⟶ 𝔪 ⟶ 0

of R-modules, where e = edim R. Setting A = Coker f and B = Coker g, we get exact sequences

(2.7.1)  \begin{cases}0 → M →^{f} R^{e} →^{α} A → 0, \\ 0 → N →^{g} R^{e} →^{β} B → 0. \end{cases}

It is easily observed that there are exact sequences

(2.7.2)  0 ⟶ R^{e} \underrightarrow{\binom{α}{β}} A ⊕ B ⟶ 𝔪 ⟶ 0

and

(2.7.3)  \begin{cases}0 ⟶ M ⟶^{βf} B ⟶ 𝔪 ⟶ 0, \\ 0 ⟶ N ⟶^{αg} A ⟶ 𝔪 ⟶ 0. \end{cases}

Using (2.7.1), (2.7.2) and (2.7.3), we can prove that Ext_{R}^{2}(k, R) ≠ 0. (Hence depth R ≤ 2.)

Fix a non-free indecomposable module X ∈ G(R). Applying the functor Hom_{R}(X, −) to (2.7.2) gives an exact sequence

0 → (X^{∗})^{e} → Hom_{R}(X, A) ⊕ Hom_{R}(X, B) → Hom_{R}(X, 𝔪) → 0

and an isomorphism

(2.7.4)  Ext_{R}^{1}(X, A) ⊕ Ext_{R}^{1}(X, B) ≅ Ext_{R}^{1}(X, 𝔪).

We have (X^{∗})^{e} ∈ G(R) and Hom_{R}(X, 𝔪) ∈ G(R), hence

Hom_{R}(X, A) ∈ G(R).

Take the first syzygy module of X; we have an exact sequence

0 → ΩX → R^{n} → X → 0.

Dualizing this sequence by A, we obtain an exact sequence

0 → Hom_{R}(X, A) → A^{n} → Hom_{R}(ΩX, A) → Ext_{R}^{1}(X, A) → 0.

Divide this into two short exact sequences

(2.7.5)  \begin{cases}0 → Hom_{R}(X, A) → A^{n} → C → 0, \\ 0 → C → Hom_{R}(ΩX, A) → Ext_{R}^{1}(X, A) → 0 \end{cases}

of R-modules. Since ΩX is also a non-free indecomposable module in G(R), applying the above argument to ΩX instead of X shows that the module Hom_{R}(ΩX, A) also belongs to G(R). We have G-dim_{R}(A^{n}) < ∞ by the first sequence in (2.7.1). Hence it follows from (2.7.5) that G-dim_{R}C < ∞, and

(2.7.6)  G-dim_{R}(Ext_{R}^{1}(X, A)) < ∞.

On the other hand, applying the functor Hom_{R}(X, −) to the natural exact sequence 0 → 𝔪 → R → k → 0, we get an exact sequence

0 → Hom_{R}(X, 𝔪) → X^{∗} → Hom_{R}(X, k) → Ext_{R}^{1}(X, 𝔪) → 0.

There is an isomorphism Hom_{R}(X, k) ≅ Ext_{R}^{1}(X, 𝔪), hence Ext_{R}^{1}(X, 𝔪) is a k-vector space. Since Ext_{R}^{1}(X, A) is contained in Ext_{R}^{1}(X, 𝔪) by (2.7.4),

(2.7.7)  Ext_{R}^{1}(X, A) is a k-vector space.

Using (2.7.6) and (2.7.7), we can prove that the local ring R is Gorenstein.

Since the only number i such that Ext_{R}^{i}(k, R) ≠ 0 is the Krull dimension of R if R is Gorenstein, it follows from the above two claims that R is a Gorenstein local ring of dimension two, which completes the proof of the theorem. □

The above theorem interests us in the observation of a Gorenstein local ring of dimension two such that the second syzygy module of the residue class field is decomposable. We introduce here a related result due to Yoshino and Kawamoto.

A homomorphic image of a convergent power series ring over a field k is called an analytic ring over k. Any complete local ring containing a field is an analytic ring over its coefficient field, and it is known that any analytic local ring is Henselian; see [14, Chapter VII]. Yoshino and Kawamoto observed the decomposability of the fundamental module of an analytic normal domain.

Theorem 2.8 (Yoshino-Kawamoto). Let R be an analytic normal local domain of dimension two. Suppose that the residue class field of R is algebraically closed and has characteristic zero. Then the following conditions are equivalent:

(1) The fundamental module of R is decomposable;

(2) R is an invariant subring of a regular local ring by a cyclic group. (In other words, R is a cyclic quotient singularity.)

For the details of this theorem, see [21, Theorem (2.1)] or [19, Theorem (11.12)]. With the notation of the above theorem, suppose in addition that R is a complete

Gorenstein ring such that Ω_{R}^{2}k is decomposable. Then R satisfies the condition (1) in the above theorem. Hence the proof of the above theorem shows that R is of finite Cohen-Macaulay representation type (i.e. there exist only finitely many nonisomorphic maximal Cohen-Macaulay R-modules); see [21] or [19]. Therefore it follows from a theorem of Herzog [12] that R is a hypersurface. Thus the local ring R is a rational double point of type (A_{n}) for some n ≥ 1 by [21, Proposition (4.1)], namely, R is isomorphic to

k[[X, Y, Z]]/(XY − Z^{n+1}).

From a more general viewpoint, we can give a characterization as follows:

Theorem 2.9. Let (S, 𝔫, k) be a regular local ring, I an ideal of S contained in 𝔫^{2}, and R = S/I a residue class ring. Suppose that R is a Henselian Gorenstein ring of dimension two. Then the following conditions are equivalent:

(1) Ω_{R}^{2}k is decomposable;

(2) dim S = 3 and I = (xy − zf) for some regular system of parameters x, y, z of S and f ∈ 𝔫.

It is necessary to prepare three elementary lemmas to prove this theorem. The first one is both well-known and easy to check, and we omit the proof.

Lemma 2.10. Let (S, 𝔫, k) be a regular local ring of dimension three and R = S/(f) a hypersurface with f ∈ 𝔫^{2}. Then f = xf_{x} + yf_{y} + zf_{z} for some f_{x}, f_{y}, f_{z} ∈ 𝔫, and the minimal free resolution of k over R is as follows:

⋯ ⟶^{C} R^{4} ⟶^{D} R^{4} ⟶^{C} R^{4} ⟶^{D} R^{4} ⟶^{C} R^{4} ⟶^{B} R^{3} ⟶^{A} R ⟶ k ⟶ 0,

where

\begin{aligned}A & = & (^{x y z}) , & B & = & ( \begin{matrix}0 & −z & y & f_{x} \\ z & 0 & −x & f_{y} \\ −y & x & 0 & f_{z} \end{matrix} ) , \\ C & = & ( \begin{matrix}0 & −f_{z} & f_{y} & x \\ f_{z} & 0 & −f_{x} & y \\ −f_{y} & f_{x} & 0 & z \\ −x & −y & −z & 0 \end{matrix} ) , & D & = & ( \begin{matrix}0 & −z & y & f_{x} \\ z & 0 & −x & f_{y} \\ −y & x & 0 & f_{z} \\ −f_{x} & −f_{y} & −f_{z} & 0 \end{matrix} ) . \end{aligned}

Lemma 2.11. Let (R, 𝔪, k) be a local ring and x ∈ 𝔪 − 𝔪^{2} an R-regular element. Then we have a split exact sequence

0 → k →^{θ} 𝔪/x𝔪 →^{π} 𝔪/xR → 0,

where θ is defined by θ(\overline{a}) = \overline{xa} for \overline{a} ∈ R/𝔪 = k and π is the natural surjection.

Proof. Let x_{1}, x_{2}, …, x_{n} be a minimal system of generators of 𝔪 with x_{1} = x. Define a homomorphism ε : 𝔪/x𝔪 → k by ε(∑_{i=1}^{n} \overline{x_{i}a_{i}}) = \overline{a_{1}}. We easily see that the composite map εθ is the identity map of k, which means that θ is a split-monomorphism. □

Lemma 2.12. Let (R, 𝔪, k) be a Cohen-Macaulay local ring of dimension one. Then the following conditions are equivalent:

(1) R is a discrete valuation ring;

(2) 𝔪^{∗} is a cyclic R-module.

Proof. (1) ⇒ (2): This implication is obvious since the maximal ideal 𝔪 is a free R-module of rank one.

(2) ⇒ (1): We have 𝔪^{∗} ≅ R/I for some ideal I of R. Dualizing the natural exact sequence 0 → 𝔪 → R → k → 0, we obtain an exact sequence

Hom_{R}(k, R) → R → 𝔪^{∗}.

Since Hom_{R}(k, R) = 0 by the assumption that R is Cohen-Macaulay, there is an injective homomorphism R → R/I. We easily observe that I = 0, equivalently, 𝔪^{∗} ≅ R. This implies the condition (1). □

Let R be a local ring and I an ideal of R. We recall that the grade of I is defined to be the infimum of the integers n such that Ext_{R}^{n}(R/I, R) ≠ 0, and is denoted by grade I. As is well-known, it coincides with the length of any maximal R-sequence in I. Now let us prove Theorem 2.9.

Proof of Theorem 2.9. (2) ⇒ (1): We have xy − zf = x · 0 + y · x + z · (−f). Lemma 2.10 gives a finite free presentation

R^{4} ⟶^{C} R^{4} ⟶ Ω_{R}^{2}k ⟶ 0

of the R-module Ω_{R}^{2}k, where C = ( \begin{matrix}0 & f & x & x \\ −f & 0 & 0 & y \\ −x & 0 & 0 & z \\ −x & −y & −z & 0 \end{matrix} ). Putting P = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & −1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) and Q =

( \begin{matrix}0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & −1 \\ 0 & 0 & 0 & 1 \end{matrix} ), we obtain

P CQ = (\begin{matrix}U & 0 \\ 0 & ^{t}U \end{matrix})

where U = ( \begin{matrix}x & f \\ z & y \end{matrix} ). It is easily seen that the matrices P, Q are invertible. Denoting by M (resp. N) the cokernel of the homomorphism defined by the matrix U (resp. ^{t}U), we get an isomorphism Ω_{R}^{2}k ≅ M ⊕ N.

(1) ⇒ (2): First of all, note that the local ring R is not regular. We denote by 𝔪 the maximal ideal 𝔫/I of R.

We can choose an element z ∈ 𝔫 − 𝔫^{2} whose image in 𝔪 is an R-regular element and that the module 𝔪/zR is decomposable. Put \overline{(−)} = (−) ⊗_{S} S/(z). Note that \overline{S} is also a regular local ring because z is a minimal generator of the maximal ideal 𝔫 of S (see the proof of Proposition 2.3). Since the maximal ideal 𝔪\overline{R} of \overline{R} is decomposable, we can apply Theorem 2.5 and see that dim \overline{S} = 2 and I\overline{S} = xy\overline{S} for some x, y ∈ 𝔫 whose images in \overline{S} form a regular system of parameter of \overline{S}. Hence \overline{R} = \overline{S}/xy\overline{S} is a hypersurface, in particular a complete intersection, of dimension one. Therefore R is a complete intersection of dimension two by [8, Theorem 2.3.4(a)]. Since S is a regular local ring of dimension three with regular system of parameter x, y, z, the ideal I is generated by an S-sequence by [8, Theorem 2.3.3(c)]. Noting ht I = dim S−dim R = 1, we see that I is a principal ideal. Write I = (l) for some l ∈ I. There is an element f ∈ S such that l = xy − zf. Assume that f ∉ 𝔫. Then f is a unit of S, and we see that zR ⊆ xyR. Hence 𝔪 = (x, y)R, and edim R = dim R = 2. This implies that R is regular, which is a contradiction. It follows that f ∈ 𝔫. □

Combining Theorem 2.7 with Theorem 2.9 gives birth to the following corollary. Compare it with Corollary 2.6.

Corollary 2.13. Let (R, 𝔪, k) be a complete local ring. Suppose that 𝔪 is indecomposable. Then the following conditions are equivalent:

(1) Ω_{R}^{2}k has a non-zero proper direct summand of finite G-dimension;

(2) R is Gorenstein, and Ω_{R}^{2}k is decomposable;

(3) There are a complete regular local ring (S, 𝔫) of dimension three, a regular system of parameters x, y, z of S, and f ∈ 𝔫 such that R ≅ S/(xy − zf).

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Department of Mathematics, Faculty of Science, Okayama University, 1-1 Naka 3-chome Tsushima, Okayama, 700-8530, JAPAN

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