THE LEADING IDEAL OF A COMPLETE INTERSECTION OF HEIGHT TWO

SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

Abstract. Let (S, 𝔫) be a Noetherian local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Assume that the associated graded ring gr_{𝔫}(S) of S with respect to 𝔫 is a UFD. We examine generators of the leading form ideal I^{∗} of I in gr_{𝔫}(S) and prove that I^{∗} is a perfect ideal of gr_{𝔫}(S), if I^{∗} is 3-generated. Thus, in this case, letting R = S/I and 𝔪 = 𝔫/I, if gr_{𝔫}(S) is Cohen-Macaulay, then gr_{𝔪}(R) = gr_{𝔫}(S)/I^{∗} is Cohen-Macaulay. As an application, we prove that if (R, 𝔪) is a one-dimensional Gorenstein local ring of embedding dimension 3, then gr_{𝔪}(R) is Cohen-Macaulay if the reduction number of 𝔪 is at most 4.

1. Introduction

Setting 1.1. Let (S, 𝔫) be a Noetherian local ring and let I = (f, g) be an ideal in

S generated by a regular sequence f, g of length two. Let R = S/I and 𝔪 = 𝔫 /I.

Let

R′(𝔫) = ∑_{i∈ℤ}𝔫^{i} t^{i} ⊆ S[t, t^{−1}] and R′(𝔪) = ∑_{i∈ℤ}𝔪^{i} t^{i} ⊆ R[t, t^{−1}]

denote the extended Rees algebras of 𝔫 and 𝔪 respectively, where t is an indetermi-

nate. Let

gr_{𝔫}(S) = R′(𝔫)/t^{−1}R′(𝔫) and gr_{𝔪}(R) = R′(𝔪)/t^{−1}R′(𝔪).

Then the canonical map S → R induces the homomorphism φ : gr_{𝔫}(S) → gr_{𝔪}(R)

of the associated graded rings. We put

I^{∗} = Ker (gr_{𝔪}(S) →^{φ} gr_{𝔪}(R)).

Then the ideal I^{∗} is generated by the initial forms of elements of I and gr_{𝔪}(R) ≅

gr_{𝔫}(S)/I^{∗}. We assume that G = gr_{𝔫}(S) is a UFD. Hence ht_{G} I^{∗} = grade_{G} I^{∗} = 2.

Date: January 27, 2007. 1991 Mathematics Subject Classification. Primary: 13A30, 13C05; Secondary: 13E05, 13H15. Key words and phrases. ideal of initial forms, associated graded ring, multiplicity, reduction

number, Gorenstein ring, Cohen-Macaulay ring.

Shiro Goto is supported by the Grant-in-Aid for Scientific Researches in Japan (C(2), No.1364044).

Mee-Kyoung Kim is supported by the Korea Science and Engineering Foundation (R04-2003-000-10113-0).

2 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

We are interested in determining generators for I^{∗} and thereby obtaining conditions in order that gr_{𝔪}(R) be Cohen-Macaulay. The goal of the paper is to prove Theorem 1.2, the proof of which is given in Section 2.

Theorem 1.2. Assume notation as in Setting 1.1, so, in particular, gr_{𝔫}(S) is a UFD. If I^{∗} is 3-generated, then I^{∗} is a perfect ideal of gr_{𝔫}(S). Therefore if gr_{𝔫}(S) is Cohen-Macaulay, then gr_{𝔪}(R) = gr_{𝔫}(S)/I^{∗} is Cohen-Macaulay.

As an immediate corollary to Theorem 1.2, we have

Corollary 1.3. With notation as in Setting 1.1, if (S, 𝔫) is a regular local ring and I^{∗} is 3-generated, then gr_{𝔪}(R) is Cohen-Macaulay.

In Section 3 we discuss some consequences of Theorem 1.2.

Notation 1.4. Let G = gr_{𝔫}(S). For each f ∈ S let o(f) = sup{i ∈ ℤ | f ∈ 𝔫^{i}}, the order of f. We put

f^{∗} = \{ \begin{matrix}\overline{ft^{i}} & if f ≠ 0 and i = o(f), \\ 0 & if f = 0 \end{matrix}

and call it the initial form of f, where \overline{ft^{i}} denotes the image in G of ft^{i} ∈ 𝔫^{i} t^{i} in R′(𝔫). Then for all f, g ∈ S we have

o(fg) = o(f) + o(g), (fg)^{∗} = f^{∗}g^{∗},

o(f + g) ≥ min{o(f), o(g)}, and

o(f + g) = min{o(f), o(g)} if o(f) ≠ o(g).

With this notation the following two simple examples illustrate the situation we are considering. In both examples we let S = k[[x, y, z]] be the formal power series ring in the three variables x, y, z over a field k.

Example 1.5. Let R = k[[w^{5}, w^{6}, w^{9}]] be the subring of the formal power series ring k[[w]] and define the homomorphism ϕ : S → R of k-algebras by ϕ(x) = w^{5}, ϕ(y) = w^{6}, and ϕ(z) = w^{9}. Then the ideal I = Ker ϕ is generated by f = z^{2} − y^{3} and g = zy − x^{3}, whence R is a complete intersection of dimension one. We have gr_{𝔫}(S) = k[x^{∗}, y^{∗}, z^{∗}], f^{∗} = z^{∗2}, and g^{∗} = z^{∗}y^{∗}. Let h = yf − zg = zx^{3} − y^{4}. Then h^{∗} = z^{∗}x^{∗3} − y^{∗4}. Let

J = (f^{∗}, g^{∗}, h^{∗}) = (z^{∗2}, z^{∗}y^{∗}, z^{∗}x^{∗3} − y^{∗4}) ⊆ I^{∗}.

Then the Hilbert series of the graded ring gr_{𝔫}(S)/J is

\frac{1 + 2t + t^{2} + t^{3}}{1 − t} = 1 + 3t + 4t^{2} + 5t^{3} + 5t^{4} + ⋯ + 5t^{n} + ⋯

and these values are the same as those in the Hilbert series of gr_{𝔪}(R) = gr_{𝔫}(S)/I^{∗}, so that J = I^{∗}. The reduction number of 𝔪 = (w^{5}, w^{6}, w^{9}) with respect to the principal reduction (w^{5}) is 3 and the relation type of gr_{𝔪}(R) is 4. The ideal I^{∗} has grade 2 and is generated by the 2 × 2 minors of the following matrix

[\begin{matrix}y^{∗} & z^{∗} & 0 \\ −x^{∗3} & −y^{∗3} & z^{∗} \end{matrix}] .

Hence, by the theorem of Hilbert-Burch [BH, Theorem 1.4.17], I^{∗} is a perfect ideal and gr_{𝔫}(S)/I^{∗} = gr_{𝔪}(R) is a Cohen-Macaulay ring.

Example 1.6. Let R = k[[w^{6}, w^{7}, w^{15}]] be the subring of the formal power series ring k[[w]] and consider the homomorphism ϕ : S → R of k-algebras defined by ϕ(x) = w^{6}, ϕ(y) = w^{7}, and ϕ(z) = w^{15}. Then I = Ker ϕ is generated by f = z^{2} −x^{5} and g = zx − y^{3}, whence R is a complete intersection of dimension one. We have gr_{𝔫}(S) = k[x^{∗}, y^{∗}, z^{∗}], f^{∗} = z^{∗2}, and g^{∗} = z^{∗}x^{∗}. Let h = xf − zg = zy^{3} − x^{6}. Then h^{∗} = z^{∗}y^{∗3} and (f^{∗}, g^{∗}, h^{∗}) = (z^{∗2}, z^{∗}x^{∗}, z^{∗}y^{∗3}) ⊊ I^{∗}. The inclusion is strict, since ht_{gr_{𝔫}(S)} I^{∗} = 2 and z^{∗} is a common factor of f^{∗}, g^{∗}, and h^{∗}. We have o(f) = o(g) = 2 and o(h) = 4. Let h_{1} = xh − y^{3}g = y^{6} − z^{7} ∈ I. Then h_{1}^{∗} = y^{∗6}. We put

J = (z^{∗2}, z^{∗}x^{∗}, z^{∗}y^{∗3}, y^{∗6}) ⊆ I^{∗}.

Then the Hilbert series of gr_{𝔫}(S)/J is given by

\frac{1 + 2t + t^{2} + t^{3} + t^{5}}{1 − t} = 1 + 3t + 4t^{2} + 5t^{3} + 5t^{4} + 6t^{5} + ⋯ + 6t^{n} + ⋯

and these values are the same as those in the Hilbert series of gr_{𝔪}(R) = gr_{𝔫}(S)/I^{∗}, so that J = I^{∗}. The reduction number of 𝔪 = (w^{6}, w^{7}, w^{15}) with respect to the principal reduction (w^{6}) is 5 and the relation type of gr_{𝔪}(R) is 6. The ring gr_{𝔪}(R) is not Cohen-Macaulay. This is implied by the gap in the numerator of the Hilbert series, and can be deduced also from the fact that the ideal I^{∗} has radical (y^{∗}, z^{∗}) and the ideal I^{∗} : z^{∗} is primary with \sqrt{I^{∗} : z^{∗}} = (x^{∗}, y^{∗}, z^{∗}).

4 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

2. Proof of Theorem 1.2

The purpose of this section is to prove Theorem 1.2. We assume notation as in Setting 1.1. Let G = gr_{𝔫}(S) and J = I^{∗}. We choose f, g ∈ S so that I = (f, g) with a = o(f) ≤ b = o(g). Without loss of generality we may assume that f^{∗} ∉ NJ and g^{∗} ∉ NJ + (f^{∗}), where N = G_{+}. Hence the elements f^{∗}, g^{∗} form part of a minimal system of homogeneous generators of J. Notice that if ht_{G}(f^{∗}, g^{∗}) = 2, then the sequence f^{∗}, g^{∗} is G-regular whence J = (f^{∗}, g^{∗}). In what follows we assume that

ht_{G}(f^{∗}, g^{∗}) = 1.

Let D = GCD(f^{∗}, g^{∗}) and write f^{∗} = ξD, g^{∗} = ηD, where D, ξ, η are homogeneous elements of G with degree d > 0, a − d, and b − d, respectively. Then {ξ, η} is a G-regular sequence.

We begin with Lemma 2.1 which gives some information about homogeneous elements of J that are not in the ideal (f^{∗}, g^{∗}).

Lemma 2.1. Let α, β ∈ S and h = αf + βg. Assume that h^{∗} ∉ (f^{∗}, g^{∗}). Then (1) o(αf) = o(βg) < o(h).

(2) o(α) + a = o(β) + b, o(α) ≥ b − d, and o(β) ≥ a − d.

(3) α^{∗}ξ + β^{∗}η = 0.

Proof. We have o(h) ≥ min{o(αf), o(βg)}. If o(αf) < o(βg), then o(h) = o(αf) and h^{∗} = α^{∗}f^{∗} ∈ (f^{∗}), which is impossible. We similarly have o(αf) = o(βg). Hence o(h) > o(αf) = o(βg), because h^{∗} ∉ (f^{∗}, g^{∗}). Thus α^{∗}f^{∗} +β^{∗}g^{∗} = (α^{∗}ξ +β^{∗}η)D = 0 whence α^{∗}ξ + β^{∗}η = 0. Therefore, since the sequence ξ, η is G-regular, we get α^{∗} = −φη and β^{∗} = φξ for some homogeneous element φ of G. Thus o(α) = deg φ+(b−d) and o(β) = deg φ+(a−d), so that o(α)+a = o(β)+b, o(α) ≥ b−d, and o(β) ≥ a−d, as was claimed. □

The existence of a third generator of the leading ideal J of a certain form is guaranteed by Proposition 2.2.

Proposition 2.2. Assume that the local ring S is 𝔫-adically complete. Then there exist elements α, β of S such that o(α) = b − d, o(β) = a − d, and (αf + βg)^{∗} ∉ (f^{∗}, g^{∗}).

THE LEADING IDEAL OF A COMPLETE INTERSECTION 5

Proof. Assume the contrary. Let f_{0}, g_{0} ∈ S with o(f_{0}) = a − d and o(g_{0}) = b − d such that ξ = f_{0}^{∗} and η = g_{0}^{∗}. We are going to construct two sequences {f_{i}}_{i=0,1,2,...} and {g_{i}}_{i=0,1,2,...} of elements in S which satisfy the following conditions: Let h_{i} = (− ∑_{k=0}^{i} g_{k})f + (∑_{k=0}^{i} f_{k})g for each i ≥ 0. Then

(1) h_{i} ≠ 0,

(2) o(h_{i}) < o(h_{i+1}),

(3) o(h_{i}) − b ≤ o(f_{i+1}) and o(h_{i}) − a ≤ o(g_{i+1}) for all i ≥ 0. To construct the sequences, firstly we put h_{0} = (−g_{0})f +f_{0}g. Then o(f_{0}) = a−d and o(g_{0}) = b−d. We notice h_{0} ≠ 0, because b−d = o(g_{0}) < o(g) = b (recall that f, g is a regular sequence). Hence h_{0}^{∗} ∈ (f^{∗}, g^{∗}) by our assumption. We write h_{0}^{∗} = f^{∗}φ+g^{∗}ψ with φ ∈ G_{o(h_{0})−a} and ψ ∈ G_{o(h_{0})−b}. Let φ = \overline{g_{1}t^{o(h_{0})−a}} and ψ = \overline{(−f_{1})t^{o(h_{0})−b}} with g_{1} ∈ 𝔫^{o(h_{0})−a} and f_{1} ∈ 𝔫^{o(h_{0})−b}. Then h_{0} = g_{1}f +(−f_{1})g +h_{1} for some h_{1} ∈ 𝔫^{o(h_{0})+1}; hence

h_{1} = [−(g_{0} + g_{1})]f + (f_{0} + f_{1})g,

where o(f_{1}) ≥ o(h_{0}) − b, o(g_{1}) ≥ o(h_{0}) − a, and o(h_{1}) > o(h_{0}). Because

\overline{h_{0}t^{a+b−d}} = \overline{(−g_{0})t^{b−d}} · \overline{ft^{a}} + \overline{f_{0}t^{a−d}} · \overline{gt^{b}}
= (−ηf^{∗}) + ξg^{∗}
= (−η · ξD) + ξ(ηD)
= 0,

we get o(h_{0}) > a+b−d, so that o(f_{1}) ≥ o(h_{0})−b > a−d and o(g_{1}) ≥ o(h_{0}) > b−d. Thus o(g_{0} + g_{1}) = o(g_{0}) = b − d < b and o(f_{0} + f_{1}) = o(f_{0}) = a − d < a, whence h_{1} = [−(g_{0} + g_{1})]f + (f_{0} + f_{1})g ≠ 0. Repeating this procedure, we get the required sequences {f_{i}}_{i=0,1,2,...} and {g_{i}}_{i=0,1,2,...} of elements in S.

Now let α = − ∑_{k=0}^{∞} g_{k} and β = ∑_{k=0}^{∞} f_{k}. We then have

αf + βg = ∑_{k=0}^{∞}[(−g_{k})f + f_{k}g]
= lim_{i→∞}[(− ∑_{k=0}^{i} g_{k})f + (∑_{k=0}^{i} f_{k})g]
= lim_{i→∞} h_{i}
= 0,

whence β ∈ (f), which is impossible because o(β) < a (recall that β = f_{0} +∑_{k=1}^{∞} f_{k}, o(f_{0}) = a−d, and o(f_{k}) ≥ o(h_{0})−b > a−d for all k ≥ 1). Thus (αf +βg)^{∗} ∉ (f^{∗}, g^{∗}) for some elements α, β of S with o(α) = b − d and o(β) = a − d. □

6 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

Remark 2.3. Let α, β ∈ S with o(α) = b−d and assume that (αf +βg)^{∗} ∉ (f^{∗}, g^{∗}). Then α^{∗} = −ūη and β^{∗} = ūξ for some unit u in S. Hence α^{∗}, β^{∗} form a G-regular sequence.

Proof. With the same notation as in the proof of Lemma 2.1 we have 0 ≠ φ ∈ G_{0} = S/𝔫. Letting φ = ū with a unit u in S, we readily get α^{∗} = −ūη and β^{∗} = ūξ. □

Let n = μ_{G}(J) and k = S/𝔫. In Proposition 2.4 (3) we prove the uniqueness of the order of o(αf + βg) for the elements α and β in S given by Proposition 2.2 and the uniqueness of the ideal (f^{∗}, g^{∗}, h^{∗}) as well, where h = αf + βg.

Proposition 2.4. Let α, β, σ, τ ∈ S with o(α) = b − d. Let h = αf + βg and q = σf + τg. Assume that h^{∗} ∉ (f^{∗}, g^{∗}). Then the following assertions hold true.

(1) Assume that q^{∗} ∉ (f^{∗}, g^{∗}). Then o(q) ≥ o(h) + o(σ) − (b − d).

(2) Assume that q^{∗} ∉ (f^{∗}, g^{∗}, h^{∗}). Then o(q) > o(h) + o(σ) − (b − d).

(3) Assume that q^{∗} ∉ (f^{∗}, g^{∗}) and o(σ) = b − d. Then o(q) = o(h) and (f^{∗}, g^{∗}, q^{∗}) = (f^{∗}, g^{∗}, h^{∗}).

(4) The elements f^{∗}, g^{∗}, h^{∗} form a part of a minimal system of homogeneous generators of J.

(5) Assume that n ≥ 4 and I ⊆ 𝔫^{2}. Then writing J = ⨁ J_{n}, we have J ⊋ (J_{i} | 1 ≤ i ≤ 5)G.

Proof. Assume that q^{∗} ∉ (f^{∗}, g^{∗}) and let c = o(σ) − (b − d). Then σ^{∗}ξ + τ^{∗}η = 0 by Lemma 2.1. Choose a unit u in S so that α^{∗} = −ūη and β^{∗} = ūξ. Then, since σ^{∗}ξū + τ^{∗}ηū = 0, we get σ^{∗}β^{∗} = τ^{∗}α^{∗}. Hence σ^{∗} = α^{∗}δ^{∗} and τ^{∗} = β^{∗}δ^{∗} for some δ ∈ S with o(δ) = c, because α^{∗}, β^{∗} is a G-regular sequence. Thus σ = αδ + σ_{1} and τ = βδ + τ_{1} for some σ_{1}, τ_{1} ∈ S with o(σ_{1}) > o(σ) and o(τ_{1}) > o(τ); q = hδ + (σ_{1}f + τ_{1}g). (1)

Now let

Λ = \{o(σ′f + τ′g)| \begin{matrix}σ′, τ′ ∈ S such that \\ (σ′f + τ′g)^{∗} ∉ (f^{∗}, g^{∗}) and o(σ′) ≥ b − d + c \end{matrix} \} .

Then o(q) ∈ Λ. Let n = min Λ and put

Γ = \{o(σ′)| \begin{matrix}σ′ ∈ S for which there exists τ′ ∈ S such that \\ (σ′f + τ′g)^{∗} ∉ (f^{∗}, g^{∗}), o(σ′) ≥ b − d + c, and o(σ′f + τ′g) = n \end{matrix} \} .

THE LEADING IDEAL OF A COMPLETE INTERSECTION 7

Then Γ ≠ ∅ and γ < n − a for all γ ∈ Γ (cf. Lemma 2.1 (1)). Let γ = max Γ and choose σ′, τ′ ∈ S so that (σ′f + τ′g)^{∗} ∉ (f^{∗}, g^{∗}), γ = o(σ′) ≥ b − d + c, and o(σ′f + τ′g) = n. Let q′ = σ′f + τ′g. Then, because q′^{∗} ∉ (f^{∗}, g^{∗}), similarly as in equation (1) we have

q′ = hδ′ + (σ_{2}f + τ_{2}g)

for some δ′, σ_{2}, τ_{2} ∈ S with o(δ′) = o(σ′) − (b − d), o(σ_{2}) > o(σ′), and o(τ_{2}) > o(τ′). Let q′′ = σ_{2}f + τ_{2}g and assume that o(q′) < o(hδ′). We then have

n = o(q′) = o(q′′)  and  q′^{∗} = q′′^{∗},

whence q′′^{∗} ∉ (f^{∗}, g^{∗}). On the other hand, because o(σ_{2}) > o(σ′) ≥ b − d + c, we get o(σ_{2}) ∈ Γ, which is impossible (recall that o(σ′) = max Γ). Thus o(q′) ≥ o(hδ′) and so

o(q) ≥ n = o(q′) ≥ o(h) + o(δ′)
= o(h) + o(σ′) − (b − d)
≥ o(h) + [(b − d) + c] − (b − d)
= o(h) + c,

as was claimed. This proves assertion (1).

Now assume that q^{∗} ∉ (f^{∗}, g^{∗}, h^{∗}). Then o(q) ≥ o(h) + c by assertion (1), where c = o(σ) − (b − d). Assume o(q) = o(h) + c and write q = hδ + (σ_{1}f + τ_{1}g) for some δ, σ_{1}, τ_{1} ∈ S with o(δ) = c, o(σ_{1}) > o(σ), and o(τ_{1}) > o(τ) (cf. equation (1)). We put q_{1} = σ_{1}f + τ_{1}g. Then, because o(q) = o(hδ) ≥ min{o(hδ), o(q_{1})}, o(q_{1}) ≥ o(hδ). If o(q_{1}) > o(hδ), then we have q^{∗} = (hδ)^{∗} = h^{∗}δ^{∗} ∈ (f^{∗}, g^{∗}, h^{∗}), which is impossible. Hence o(q_{1}) = o(hδ) = o(q) so that q^{∗} = h^{∗}δ^{∗} + q_{1}^{∗} ∉ (f^{∗}, g^{∗}, h^{∗}). Consequently q_{1}^{∗} ∉ (f^{∗}, g^{∗}) and so we get by assertion (1) that

o(h) + c = o(hδ) = o(q_{1})
≥ o(h) + o(σ_{1}) − (b − d)
≥ o(h) + [o(σ) + 1] − (b − d)
= o(h) + c + 1,

which is absurd. Hence o(q) > o(h) + c. This proves assertion (2).

8 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

To show assertion (3), thanks to assertion (2), it is enough to check the equality o(q) = o(h). The inequality o(q) ≥ o(h) follows from assertion (1), whence o(h) = o(q) by symmetry.

We now prove assertions (4) and (5). Let V = J/NJ and choose homogeneous elements δ_{1}, δ_{2},..., δ_{n} of J so that their images δ̄_{1}, δ̄_{2},..., δ̄_{n} in V form a k-basis of V. We may assume δ_{1} = f^{∗}, δ_{2} = g^{∗}. Hence J = (f^{∗}, g^{∗}, δ_{3}, ..., δ_{n}). For each 3 ≤ i ≤ n let δ_{i} = q_{i}^{∗} with q_{i} ∈ I and write q_{i} = σ_{i}f + τ_{i}g for some σ_{i}, τ_{i} ∈ S. Then o(σ_{i}) ≥ b − d by Lemma 2.1. We have o(q_{i}) = o(h) and (f^{∗}, g^{∗}, q_{i}^{∗}) = (f^{∗}, g^{∗}, h^{∗}) (resp. o(q_{i}) > o(h)), if o(σ_{i}) = b − d (resp. if o(σ_{i}) > b − d) by assertion (3) (resp. assertion (1)). Hence o(q_{i}) ≥ o(h). We may assume o(q_{3}) ≤ o(q_{4}) ≤ ... ≤ o(q_{n}). Then, because h^{∗} ∈ (f^{∗}, g^{∗}, δ_{3}, δ_{4}, ..., δ_{n}) but h^{∗} ∉ (f^{∗}, g^{∗}), we get deg h^{∗} = o(h) ≥ deg δ_{3} = o(q_{3}) so that o(q_{3}) = o(h), whence (f^{∗}, g^{∗}, δ_{3}) = (f^{∗}, g^{∗}, h^{∗}) by assertion (3). Thus assertion (4) follows. Suppose that n ≥ 4. Then δ_{4} = q_{4}^{∗} ∉ (f^{∗}, g^{∗}, δ_{3}) = (f^{∗}, g^{∗}, h^{∗}). Therefore o(σ_{4}) > b − d. Hence by assertion (2) we have

deg δ_{4} = o(q_{4})
≥ o(h) + [o(σ_{4}) − (b − d) + 1] ≥ o(h) + 2
≥ (a + b − d) + 3  (by Lemma 2.1)
≥ b + 4 ≥ a + 4.

Consequently, deg δ_{4} = o(q_{4}) ≥ 6, if I ⊆ 𝔫^{2}. Hence J ⊋ (J_{i} | 1 ≤ i ≤ 5)G, which completes the proof of Proposition 2.4. □

We are now ready to prove Theorem 1.2 .

Proof of Theorem 1.2. We may assume that S is complete and ht_{G}(f^{∗}, g^{∗}) = 1. Hence μ_{G}(J) = 3. Choose α, β ∈ S so that o(α) = b − d and (αf + βg)^{∗} ∉ (f^{∗}, g^{∗}). Let h = αf + βg. Then J = (f^{∗}, g^{∗}, h^{∗}) by Proposition 2.4 (4). We furthermore have h^{∗} ∈ (α^{∗}, β^{∗}), because α^{∗}, β^{∗} is a G-regular sequence (cf. Remark 2.3) and h ∈ (α, β). Let h^{∗} = α^{∗}φ + β^{∗}ψ with φ, ψ ∈ G. Then, since α^{∗} = −ūη and β^{∗} = ūξ for some unit u in S, we see

J = I_{2} (\begin{matrix}ūφ & ūψ & D \\ ξ & η & 0 \end{matrix})

where D ∈ G is the element such that f^{∗} = ξD and g^{∗} = ηD. Thus J is a perfect ideal of G, because grade_{G} J = 2. □

THE LEADING IDEAL OF A COMPLETE INTERSECTION 9

Discussion 2.5. Assume notation as in Setting 1.1 and also assume that I ⊂ 𝔫^{2}. Let μ(I^{∗}) denote the minimal number of generators of I^{∗}. If μ(I^{∗}) = 3, then I^{∗} = (f^{∗}, g^{∗}, h_{0}^{∗})G, where h_{0} = αf + βg and o(α) = b − d. We have

2 ≤ deg f^{∗} ≤ deg g^{∗} < deg g^{∗} + 2 ≤ deg h_{0}^{∗},

so deg h_{0}^{∗} ≥ 4. If μ(I^{∗}) ≥ 4, then there exist homogeneous generators for I^{∗} so that

I^{∗} = (f^{∗}, g^{∗}, h_{0}^{∗}, h_{1}^{∗}, …, h_{r}^{∗})G,

where we have r = μ(I^{∗}) − 3, and

2 ≤ deg f^{∗} ≤ deg g^{∗} < deg g^{∗} + 2 ≤ deg h_{0}^{∗} < deg h_{0}^{∗} + 2 ≤ deg h_{1}^{∗} ≤ ⋯ ≤ deg h_{r}^{∗}.

The inequality deg h_{1}^{∗} ≥ deg h_{0}^{∗} + 2 is by Proposition 2.4 (2). In particular, if μ(I^{∗}) ≥ 4, then the relation type of gr_{𝔪}(R) is greater than or equal to 6.

It would be interesting to know whether deg h_{i}^{∗} + 2 ≤ deg h_{i+1}^{∗} holds for all i with 0 ≤ i < r, or, if this fails to hold in general, whether deg h_{i}^{∗} + 1 ≤ deg h_{i+1}^{∗}. An interesting result of Kothari [K] shows that if S is a 2-dimensional regular local ring containing a coefficient field, then deg h_{i}^{∗} + 1 ≤ deg h_{i+1}^{∗} for all i with 1 ≤ i < r.

3. Applications of the theorem

Let us give some consequences of Theorem 1.2. We begin with the following.

Corollary 3.1. Let (R, 𝔪) be a d-dimensional Gorenstein local ring. Assume that 𝔪 is minimally generated by d + 2 elements. Then gr_{𝔪}(R) is a Cohen-Macaulay ring, if the relation type of gr_{𝔪}(R) is less than or equal to 5.

Proof. We may assume that (R, 𝔪) is complete. Hence, thanks to the structure theorem of Cohen ([BH, Theorem A.21]), we get R = S/I, where I is an ideal of a (d + 2)-dimensional regular local ring (S, 𝔫). Because R is a Gorenstein ring and dim R = d, the ideal I is generated by a regular sequence f, g of length 2. Let J = Ker (gr_{𝔫}(S) →^{φ} gr_{𝔪}(R)), where φ : gr_{n}(S) → gr_{m}(R) denotes the canonical map. We may assume that μ_{gr_{n}(S)}(J) ≥ 3. Then by Proposition 2.4 (5) the ideal J is 3-generated, because the relation type of gr_{𝔪}(R) is at most 5, whence by Theorem 1.2, gr_{𝔪}(R) is a Cohen-Macaulay ring since the polynomial ring gr_{𝔫}(S) is a UFD. □

10 SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

Corollary 3.2. Let (R, 𝔪) be a one-dimensional Gorenstein local ring and assume that 𝔪 is minimally generated by 3 elements. If the reduction number of 𝔪 is less than or equal to 4, then gr_{𝔪}(R) is a Cohen-Macaulay ring.

Proof. The result of Huckaba [H, Theorem 2.3] shows that in our setting the relation type of gr_{𝔪}(R) is at most one more than the reduction number of 𝔪. Hence by Corollary 3.1 the ring gr_{𝔪}(R) is Cohen-Macaulay. □

The example studied in Example 1.6 shows that Corollary 3.2 may fail if the reduction number of 𝔪 is 5. The following example is explored by Sally [S, Example 2.2] and shows that Corollary 3.1 may fail if we assume that R is a Cohen-Macaulay (rather than Gorenstein) ring.

Example 3.3. Let S = k[[x, y, z]] be the formal power series ring with three variables x, y, z over a field k . Let R = k[[w^{4}, w^{5}, w^{11}]] be the subring of the formal power series ring k[[w]] and consider the homomorphism ϕ : S → R of k-algebras defined by ϕ(x) = w^{4}, ϕ(y) = w^{5}, and ϕ(z) = w^{11}. Then I = Ker ϕ is generated by xz − y^{3}, yz − x^{4}, and z^{2} − x^{3}y^{2}. We have gr_{𝔫}(S) = k[x^{∗}, y^{∗}, z^{∗}],

I^{∗} = (z^{∗2}, z^{∗}y^{∗}, z^{∗}x^{∗}, y^{∗4}),

and the ring gr_{𝔪}(R) = gr_{𝔫}(S)/I^{∗} is not Cohen-Macaulay. The relation type of gr_{𝔪}(R) is 4 and the reduction number of 𝔪 is 3.

Corollary 3.4. Let (R, 𝔪) be a one-dimensional Gorenstein local ring and assume that 𝔪 is minimally generated by 3 elements. If the reduction number r of 𝔪 is less than or equal to 4, then gr_{𝔪}(R) is a Gorenstein ring if and only if J^{r} : 𝔪^{r} = 𝔪^{r}, where J is a reduction of 𝔪.

Proof. By Corollary 3.2, gr_{𝔪}(R) is Cohen-Macaulay. Therefore all the powers of 𝔪 are closed in the sense of Ratliff-Rush. Hence gr_{𝔪}(R) is a Gorenstein ring if and only if J^{r} : 𝔪^{r} = 𝔪^{r} (cf. [HKU, Corollary 4.8]). □

Acknowledgement. The main question considered in this paper arose during the collaboration of W. Heinzer and M-K Kim with Bernd Ulrich in [HKU]. We would like to thank him for his help in developing this material.

References

[BH] W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge University Press, Cambridge, 1993.

[HKU] W. Heinzer, Mee-Kyoung Kim and B. Ulrich, The Gorenstein and complete intersection properties of associated graded rings, J. Pure Appl. Algebra, to appear.

[H] S. Huckaba, Reduction numbers for ideals of higher analytic spread, Math. Proc. Camb. Phil. Soc. 102(1987), 49-57.

[K] S. Kothari, The local Hilbert function of a pair of plane curves, Proc. Amer. Math. Soc. 72 (1978), 439-442.

[S] J. D. Sally, Tangent cones at Gorenstein singularities, Compositio Math. 40 (1980), 167-175.

Department of Mathematics, School of Science and Technology Meiji University, 214-8571 Japan

   E-mail address: goto@math.meiji.ac.jp

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

   E-mail address: heinzer@math.purdue.edu

Department of Mathematics, Sungkyunkwan University Jangangu Suwon 440-746, Korea

   E-mail address: mkkim@math.skku.ac.kr