THE LEADING IDEAL OF A COMPLETE INTERSECTION OF HEIGHT TWO, PART II

SHIRO GOTO, WILLIAM HEINZER, AND MEE-KYOUNG KIM

Abstract. Let (S, n) be a regular 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. As in [GHK], we examine the leading form ideal I^{∗} of I in the associated graded ring G = gr_{n}(S). If gr_{𝔪}(R) is Cohen-Macaulay, we describe precisely the Hilbert series H(gr_{𝔪}(R), λ) in terms of the degrees of homogeneous generators of I^{∗} and of their successive GCD’s. If D = GCD(f^{∗}, g^{∗}) is a prime element of gr_{n}(S) that is regular on gr_{n}(S)/( \frac{f^{∗}}{D} , \frac{g^{∗}}{D}), we prove that I^{∗} is 3-generated and a perfect ideal. If ht_{gr_{𝔫}(S)}(f^{∗}, g^{∗}, h^{∗}) = 2, where h ∈ I is such that h^{∗} is of minimal degree in I^{∗} \ (f^{∗}, g^{∗}) gr_{n}(S), we prove I^{∗} is 3-generated and a perfect ideal of gr_{n}(S), so gr_{𝔪}(R) = gr_{n}(S)/I^{∗} is a Cohen-Macaulay ring. We give several examples to illustrate our theorems.

1. Introduction

This paper examines generators of the defining ideal of the tangent cone of a

complete intersection of codimension two. We fix the following notation.

Setting 1.1. Let (S, 𝔫) be a regular local ring of dimension s ≥ 2 and let I = (f, g)

be an ideal in S generated by a regular sequence f, g of length two. For simplicity

we assume that the residue class field k = S/ 𝔫 is infinite. We put 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 Rees algebras of 𝔫 and 𝔪 respectively, where t is an indeterminate. We

put

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

For each 0 ≠ h ∈ S let o(h) = sup\{i ∈ ℤ | h ∈ 𝔫^{i}\} and put h^{∗} = \overline{ht^{n}}, where n = o(h)

and \overline{ht^{n}} denotes the image of ht^{n} in G. The canonical map S → R induces the

Date: February 5, 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 Research Foundation (R04-2003-000-10113-0)2003.

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

epimorphism φ : G → gr_{𝔪}(R) of the associated graded rings. We put

I^{∗} = Ker (G →^{φ} gr_{𝔪}(R)).

Then the homogeneous components {[I^{∗}]_{i}}_{i∈ℤ} of the leading form ideal I^{∗} of I are given by

[I^{∗}]_{i} = \{\overline{ht^{i}} | h ∈ I ∩ 𝔫^{i}\}

for each i ∈ ℤ. We throughout assume that a = o(f) ≤ b = o(g) and that f^{∗} ∤ g^{∗} in G. The latter part of the condition is equivalent to saying that f^{∗}, g^{∗} form a part of a minimal homogeneous system of generators of I^{∗}.

The original motivation for our work comes from a paper of S. C. Kothari [K]. Kothari answers several questions raised by Abyhankar concerning the local Hilbert function of a pair of plane curves. Let ℓ_{S}(∗) denote length over S. In the case where dim S = 2, Kothari proves that 0 ≤ dim_{k}[gr_{𝔪}(R)]_{i} − dim_{k}[gr_{𝔪}(R)]_{i+1} ≤ 1 for all i ≥ a and that ℓ_{S}(R) ≥ ab; moreover, one has the equality ℓ_{S}(R) = ab if and only if f^{∗}, g^{∗} are coprime in G, that is, f^{∗}, g^{∗} form a G-regular sequence.

We have subsequently learned from an informative referee report of other work in this area. Indeed, F. Macaulay in a 1904 paper [M] employs a different method to determine the same necessary condition as Kothari on the Hilbert function of a pair of plane curves. Using his inverse systems, Macaulay establishes the structure of the Hilbert function H(A) of a complete intersection quotient A = k[[x, y]]/(f, g) to be of the form H = (1, 2, …, a, t_{a}, …, t_{j}, 0), (1) where a ≥ t_{a} ≥ t_{a+1} ≥ ⋯ ≥ t_{j} = 1 and |t_{i} − t_{i+1}| ≤ 1 for all i. Thus the Hilbert function H after an initial rising segment breaks up into platforms and regular flights of descending stairs, each step of height one. The structure of H(A) is studied from the point of view of parametrizations by J. Briançon [Br] and by A. Iarrobino [Ia1] and [Ia2]. These authors prove that every sequence satisfying the conditions in Equation 1 is realizable as the Hilbert function H(A) of some Gorenstein Artin algebra of the form A = k[[x, y]]/(f, g).

Let v(H) = 2 + #{platforms}. Iarrobino [Ia1], [Ia2] proves that I^{∗} needs two initial generators f^{∗}, g^{∗} and requires a new generator following each platform, and that v(H) is the minimum possible number of generators of a graded ideal defining a standard algebra with Hilbert function H. In [Ia1, Theorem 2.2.A], Iarrobino characterizes those graded ideals corresponding to I^{∗} for which I is a complete

THE LEADING IDEAL OF A COMPLETE INTERSECTION 3

intersection of height two. He proves they are exactly the graded ideals with v(H) generators. The referee has pointed out that our results in Theorem 1.2 and Theorem 1.3 can be deduced from these results of Iarrobino. While acknowledging the priority of these results of Iarrobino, we hope that our different approach is still of some interest.

Theorem 1.2. Let notation be as in Setting 1.1 and assume that dim S = 2 and n = μ_{G}(I^{∗}). Then I^{∗} contains a homogeneous system {ξ_{i}}_{1≤i≤n} of generators that satisfy the following three conditions.

(1) ξ_{1} = f^{∗} and ξ_{2} = g^{∗}.

(2) degξ_{i} + 2 ≤ degξ_{i+1} for all 2 ≤ i ≤ n − 1.

(3) ht_{G}(ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) = 1.

Let {ξ_{i}}_{1≤i≤n} be a homogeneous system of generators of I^{∗} satisfying conditions (1) and (2) in Theorem 1.2. We prove that the ideals

{(ξ_{j} | 1 ≤ j ≤ i)G}_{1≤i≤n}

of G are independent of the particular choice of the family {ξ_{i}}_{1≤i≤n} and are uniquely determined by I. Moreover, if D_{i} = GCD(ξ_{j} | 1 ≤ j ≤ i) and d_{i} = degD_{i}, then one has the strictly descending sequence

a = d_{1} > d_{2} > ⋯ > d_{n−1} > d_{n} = 0

and \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) for all 1 ≤ i ≤ n − 1 (Lemma 3.2 ). Let c_{i} = deg ξ_{i}

and let

H(gr_{𝔪}(R), λ) = ∑_{i=0}^{∞} dim_{k}[gr_{𝔪}(R)]_{i}λ^{i}

denote the Hilbert series of gr_{𝔪}(R). We explicitly describe H(gr_{𝔪}(R), λ) and the difference ℓ_{S}(R) − ab in terms of c_{i} and d_{i}, sharpening results proved by Kothari in [K].

Theorem 1.3. Let notation be as in Setting 1.1 and assume that dim S = 2 and n = μ_{G}(I^{∗}). The following assertions hold true.

(1) H(gr_{𝔪}(R), λ) = ^{\frac{∑_{i=2}^{n} λ^{d_{i}}(1−λ^{d_{i−1}−d_{i}})(1−λ^{c_{i}−d_{i}})}{(1−λ)^{2}}}.

(2) ℓ_{S}(R) = ∑_{i=2}^{n}(d_{i−1} − d_{i})(c_{i} − d_{i}) = ab + ∑_{i=2}^{n−1} d_{i}·[(c_{i+1} − c_{i}) − (d_{i−1} − d_{i})].

(3) c_{i+1} − c_{i} > d_{i−1} − d_{i} > 0 for all 2 ≤ i ≤ n − 1.

(4) [K, Corollary1] ℓ_{S}(R) = ab if and only if n = 2, i.e., f^{∗}, g^{∗} is a G-regular sequence.

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

Remark 1.4. In the case where dim S = s > 2, it is still true that ht_{G}(f^{∗}, g^{∗}) > 1 implies f^{∗}, g^{∗} is a G-regular sequence, and therefore I^{∗} = (f^{∗}, g^{∗})G also in this case. Thus we assume that ht_{G}(f^{∗}, g^{∗}) = 1 and put D_{2} = GCD(f^{∗}, g^{∗}) and d_{2} = degD_{2}. Let f^{∗} = D_{2}ξ and g^{∗} = D_{2}η. Notice that ξ, η is a regular sequence in G. We have b ≥ a > d_{2} > 0, and μ_{G}(I^{∗}) = n ≥ 3. There exists a minimal homogeneous system {ξ_{1}, ξ_{2}, …, ξ_{n}} of generators of I^{∗} such that ξ_{1} = f^{∗} and ξ_{2} = g^{∗}, and c_{i} := deg ξ_{i} ≤ deg ξ_{i+1} := c_{i+1} for each i ≤ n − 1. However, the ideal I^{∗} may fail to be perfect, and it is possible to have D_{3} := GCD(ξ_{1}, ξ_{2}, ξ_{3}) = D_{2} as is illustrated in [GHK, Example 1.6]. We prove in [GHK, Theorem 1.2] that I^{∗} is perfect if n = 3. We also prove in [GHK] that ξ_{3} = h^{∗}, where h has the form h = αf + βg ∈ I with o(α) = b − d_{2}, and o(β) = a − d_{2}, and that c_{3} := o(h) > a + b − d_{2}. Moreover, if q = σf + τg is such that q^{∗} ∉ (f^{∗}, g^{∗})G and (o)(σ) = b − d_{2}, then o(q) = o(h) and (f^{∗}, g^{∗}, h^{∗})G = (f^{∗}, g^{∗}, q^{∗})G. Thus the ideal (ξ_{1}, ξ_{2}, ξ_{3})G is independent of the choice of ξ_{3}. In the case where n ≥ 4, we also prove that c_{4} ≥ c_{3} + 2 [GHK, Proposition 2.4]. However, examples shown to us by Craig Huneke and Lance Bryant show that it is possible to have c_{i+1} = c_{i} for i ≥ 4. This resolves a question mentioned in [GHK, Discussion 2.5]).

If gr_{𝔪}(R) is a Cohen-Macaulay ring, we prove in Section 4 by passing to the factor ring of G modulo a suitable linear system of parameters for gr_{𝔪}(R) that it is possible to reduce the problems to the case where dim S = 2 and obtain results corresponding to those proved in Section 3 about the Hilbert series H(gr_{𝔪}(R), λ). In particular, if I^{∗} is perfect, then c_{i+1} > c_{i} + 1 for each i with 2 ≤ i ≤ n − 1.

With notation as in Setting 1.1, let e_{𝔪}^{0}(R) denotes the multiplicity of R with respect to 𝔪. Using Theorem 1.2, we prove in Section 4:

Theorem 1.5. Assume notation as in Setting 1.1 and Remark 1.4, and let D := D_{2}, d := d_{2} and c := c_{3}. If ht_{G}(f^{∗}, g^{∗}, h^{∗}) = 2, then the following assertions hold true.

(1) I^{∗} = (f^{∗}, g^{∗}, h^{∗}).

(2) gr_{𝔪}(R) is a Cohen-Macaulay ring.

(3) H(gr_{𝔪}(R), λ) = \frac{(1−λ^{c})(1−λ^{d})+λ^{d}(1−λ^{a−d})(1−λ^{b−d})}{(1−λ)^{dim S}}.

(4) e_{𝔪}^{0}(R) = ab + d·[(c + d) − (a + b)].

Let M = [gr_{𝔪}(R)]_{+} and let H_{M}^{s−2}(gr_{𝔪}(R)) denote the s − 2 \underline{th} local cohomology module of gr_{𝔪}(R) with respect to M. Recall that

a(gr_{𝔪}(R)) = max{i ∈ ℤ | [H_{M}^{s−2}(gr_{𝔪}(R))]_{i} ≠ (0)}

THE LEADING IDEAL OF A COMPLETE INTERSECTION 5

is the a-invariant of gr_{𝔪}(R). Using this notation and setting Q = (X_{1}, …, X_{s−2})G, where X_{1}, …, X_{s} are suitably chosen homogeneous elements of degree one in G such that G = k[X_{1}, …, X_{s}], and using the formula

a(gr_{𝔪}(R)/Qgr_{𝔪}(R)) = a(gr_{𝔪}(R)) + (s − 2)

of [GW, Remark (3.1.6)], we establish the following result in Section 4.

Theorem 1.6. Assume notation as in Setting 1.1 and Remark 1.4. If gr_{𝔪}(R) is a Cohen-Macaulay ring and n = μ_{G}(I^{∗}), then the following assertions hold true.

(1) H(gr_{𝔪}(R), λ) = \frac{∑_{i=2}^{n} λ^{d_{i}}(1 − λ^{d_{i−1}−d_{i}})(1 − λ^{c_{i}−d_{i}})}{(1 − λ)^{s}}.

(2) e_{𝔪}^{0}(R) = ab + ∑_{i=2}^{n−1} d_{i}·[(c_{i+1} − c_{i}) − (d_{i−1} − d_{i})] with

c_{i+1} − c_{i} > d_{i−1} − d_{i} > 0

for all 2 ≤ i ≤ n − 1.

(3) e_{𝔪}^{0}(R) ≤ a·[c_{n} + d_{n−1} − a], where the equality holds true if and only if n = 2.

(4) a(gr_{𝔪}(R)) = c_{n} + d_{n−1} − s.

Sections 5 is devoted to some examples, which illustrate our theorems. Let H = ⟨n_{1}, n_{2}, n_{3}⟩ be a Gorenstein numerical semigroup generated by the three integers n_{1}, n_{2}, n_{3}, where 0 < n_{1} < n_{2} < n_{3} and GCD(n_{1}, n_{2}, n_{3}) = 1. Let S = k[[X_{1}, X_{2}, X_{3}]] and T = k[[t]] be formal power series rings over a field k. We denote by φ : S → T the k-algebra map defined by φ(X_{i}) = t^{n_{i}} for i = 1, 2, 3. Let I = Ker φ, R = k[[t^{n_{1}}, t^{n_{2}}, t^{n_{3}}]], 𝔫 = (X_{1}, X_{2}, X_{3})S, and 𝔪 = (t^{n_{1}}, t^{n_{2}}, t^{n_{3}})R. Then, as was essentially shown in [H2] and [RV], gr_{𝔪}(R) is a Cohen-Macaulay ring if and only if I^{∗} is 3-generated. We shall recover this result in our context. In Example 5.5, we present a family of examples due to Takahumi Shibuta that demonstrates that for I = Ker φ as above, there is no bound on the number of elements needed to generate I^{∗}.

2. Preliminaries

Throughout this section, let notation be as in Setting 1.1, assume that dim S = 2 and let 𝔫 = (x, y).

Lemma 2.1. Let h ∈ S with m = o(h) and assume that x^{∗} ∤ h^{∗}. Then h = εy^{m}+xφ for some ε ∈ U(S) and φ ∈ 𝔫^{m−1}.

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

Proof. Let \overline{S} = S/(x) and denote by \overline{∗} the image in \overline{S}. Let ℓ = o(\overline{h}). Then ℓ ≥ m and \overline{h} = \overline{ε} · \overline{y}^{ℓ} for some ε ∈ U(S). We write h = εy^{ℓ} + xφ with φ ∈ S. Then φ ∈ 𝔫^{m−1}, because (x) ∩ 𝔫^{m} = x 𝔫^{m−1}. Hence ℓ = m, as x^{∗} ∤ h^{∗}. □

Lemma 2.2. There exist elements x, y, u, and g_{1} ∈ S satisfying the following conditions.

(1) 𝔫 = (x, y) and x^{∗} ∤ f^{∗}.

(2) u ∈ U(S), o(g_{1}) = b − 1, and g = uy^{b−a}f + xg_{1}

Proof. Let 𝔫 = (x, y). Then, since k = S/ 𝔫 is infinite, we have x^{∗} + cy^{∗} ∤ f^{∗} and x^{∗} + cy^{∗} ∤ g^{∗} for some c ∈ k. Let c ≡ α mod 𝔫 (α ∈ S) and z = x + αy. Then 𝔫 = (z, y). Because z^{∗} ∤ f^{∗} and z^{∗} ∤ g^{∗}, by Lemma 2.1, we have

f = εy^{a} + zξ and g = τy^{b} + zη

for some ε, τ ∈ U(S), ξ ∈ 𝔫^{a−1}, and η ∈ 𝔫^{b−1}. Let g_{1} = η − uy^{b−a}ξ where u = τε^{−1}. Then g = uy^{b−a}f + zg_{1} and o(g_{1}) = b − 1, because g_{1} ∈ 𝔫^{b−1} and f^{∗} ∤ g^{∗}. Replacing x with z, we get the required elements x, y, u, and g_{1} ∈ S as claimed. □

In what follows let x, y, u, and g_{1} ∈ S be elements which satisfy conditions (1) and (2) in Lemma 2.2. We put I_{1} = (f, g_{1}), X = x^{∗}, and Y = y^{∗}.

Proposition 2.3. The following assertions hold true.

(1) I = (f, xg_{1}) and I :_{S} x = I_{1}.

(2) (f^{∗}, g^{∗}) = (f^{∗}, Xg_{1}^{∗}) whence f^{∗} ∤ g_{1}^{∗}.

(3) f^{∗}, X is a G-regular sequence.

(4) I = 𝔫, if b = 1.

(5) ([K]) Suppose that b > 1. Then I_{1} is a parameter ideal in S and I^{∗} = (f^{∗}) + XI_{1}^{∗}. Hence I^{∗} :_{G} X = I_{1}^{∗}.

Proof. (1) Since g = uy^{b−a}f + xg_{1}, we get I = (f, xg_{1}), whence xI_{1} ⊆ I. Let φ ∈ I :_{S} x and write xφ = αf + β(xg_{1}) (α, β ∈ S). Then x(φ − βg_{1}) ∈ (f) so that φ − βg_{1} ∈ (f), because f, x is a regular sequence in S (recall that x ∤ f). Hence φ ∈ (f, g_{1}) = I_{1} and thus I :_{S} x = I_{1}.

(2) Recall that g^{∗} = u^{∗}Y^{b−a}f^{∗} + Xg_{1}^{∗}.

(3) This is clear, since X ∤ f^{∗}.

(4) We have a = 1, since a ≤ b. Hence o(g_{1}) = 0 and o(f mod (x)) = 1 (cf. Proof of Lemma 2.1), so that we have I = (f, xg_{1}) = (f, x) = 𝔫.

THE LEADING IDEAL OF A COMPLETE INTERSECTION 7

(5) Since b > 1, we get I ⊆ I_{1} ⊊ S. Hence I_{1} is a parameter ideal of S. Let i ≥ a−1 be an integer. Then, thanks to Proof of [K, Lemma], we see that for every k -basis W_{1}, W_{2}, ⋯, W_{r} of [I_{1}^{∗}]_{i}, the elements Y^{i+1−a}f^{∗}, XW_{1}, XW_{2}, ⋯, XW_{r} form a k-basis of [I^{∗}]_{i+1}. Consequently, [I^{∗}]_{i+1} ⊆ (f^{∗}) + XI_{1}^{∗} ⊆ I^{∗} (recall that xI_{1} ⊆ I), whence I^{∗} = (f^{∗}) + XI_{1}^{∗}, because [I^{∗}]_{i} = (0) for i ≤ a − 1. As f^{∗}, X is a G-regular sequence, we have the equality I^{∗} :_{G} X = I_{1}^{∗} similarly as in the proof of assertion (1). □

Corollary 2.4. Suppose that b > 1. Then H(G/I^{∗}, λ) = ∑_{i=0}^{a−1} λ^{i} + λ· H(G/I_{1}^{∗}, λ).

Proof. Notice that (X, f^{∗})/I^{∗} = (X, f^{∗})/[(f^{∗}) + XI_{1}^{∗}] ≅ (X)/XI_{1}^{∗} ≅ (G/I_{1}^{∗})(−1), because (X) ∩ (f^{∗}) = (Xf^{∗}) and f^{∗} ∈ I_{1}^{∗}. Then we get the exact sequence

0 → (G/I_{1}^{∗})(−1) → G/I^{∗} → G/(X, f^{∗}) → 0

of graded G-modules, so that

H(G/I^{∗}, λ) = H(G/(X, f^{∗}), λ) + λ· H(G/I_{1}^{∗}, λ)
= ∑_{i=0}^{a−1} λ^{i} + λ· H(G/I_{1}^{∗}, λ)

as claimed. □

The following fact plays a key role in our argument.

Corollary 2.5. Suppose that b > 1. Let n = μ_{G}(I^{∗}) and ℓ = μ_{G}(I_{1}^{∗}).

(1) Suppose that a < b. Then n = ℓ and, for every homogeneous system {η_{i}}_{1≤i≤n} of generators of I_{1}^{∗} with η_{1} = f^{∗} and η_{2} = g_{1}^{∗}, we have I^{∗} = (f^{∗}, g^{∗}) + (Xη_{i} | 3 ≤ i ≤ n).

(2) Suppose that a = b and g_{1}^{∗} ∤ f^{∗}. Then n = ℓ and, for every homogeneous system {η_{i}}_{1≤i≤n} of generators of I_{1}^{∗} with η_{1} = g_{1}^{∗} and η_{2} = f^{∗}, we have I^{∗} = (f^{∗}, g^{∗}) + (Xη_{i} | 3 ≤ i ≤ n).

(3) Suppose that a = b but g_{1}^{∗} | f^{∗}. Then n = ℓ + 1. Choose f_{1} ∈ S so that o(f_{1}) > a, I_{1} = (g_{1}, f_{1}), and g_{1}^{∗} ∤ f_{1}^{∗}. Then, for every homogeneous system {η_{i}}_{1≤i≤n−1} of generators of I_{1}^{∗} with η_{1} = g_{1}^{∗} and η_{2} = f_{1}^{∗}, we have I^{∗} = (f^{∗}, g^{∗}) + (Xη_{i} | 2 ≤ i ≤ n − 1).

Proof. (1) By Proposition 2.3 (2) we have f^{∗} ∤ g_{1}^{∗}. Let {η_{i}}_{1≤i≤ℓ} be a homogeneous system of generators of I_{1}^{∗} with η_{1} = f^{∗} and η_{2} = g_{1}^{∗}. Then, because I^{∗} = (f^{∗})+XI_{1}^{∗} and (f^{∗}, g^{∗}) = (f^{∗}, Xg_{1}^{∗}) (cf. Proposition 2.3, (2) and (5)), we have I^{∗} = (f^{∗}, Xη_{2})+

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

(Xη_{i} | 3 ≤ i ≤ ℓ). To see that n = ℓ, we shall check that f^{∗}, Xη_{2}, Xη_{3}, ⋯, Xη_{ℓ} is a minimal system of generators of I^{∗}. Since f^{∗} ∉ (X), it suffices to show that Xη_{i} ∉ (f^{∗}) + (Xη_{2}, ⋯, Xη_{i−1}, Xη_{i+1}, ⋯, Xη_{ℓ}) for any 2 ≤ i ≤ ℓ. Assume the contrary and write Xη_{i} = f^{∗}φ + ∑_{2≤j≤ℓ,j≠i} Xη_{j}φ_{j} with φ, φ_{j} ∈ G. Then X[η_{i} − ∑_{2≤j≤ℓ,j≠i} η_{j}φ_{j}] ∈ (f^{∗}). Because f^{∗}, X form a G-regular sequence, we get η_{i} ∈ (f^{∗}) + (η_{2}, ⋯, η_{i−1}, η_{i+1}, ⋯ η_{ℓ}), which is impossible (recall that f^{∗} = η_{1}, η_{2}, ⋯, η_{ℓ} is a minimal system of generators of I_{1}^{∗}). Thus n = ℓ.

(2) Let {η_{i}}_{1≤i≤ℓ} be a homogeneous system of generators of I_{1}^{∗} with η_{1} = g_{1}^{∗} and η_{2} = f^{∗}. Then I^{∗} = (f^{∗}, Xη_{1}) + (Xη_{i} | 3 ≤ i ≤ ℓ). For the same reason as in the proof of assertion (1), f^{∗}, Xη_{1}, Xη_{3}, ⋯, Xη_{ℓ} is a minimal system of generators of I^{∗} and we get n = ℓ.

(3) Let {η_{i}}_{1≤i≤ℓ} be a homogeneous system of generators of I_{1}^{∗} such that η_{1} = g_{1}^{∗} and η_{2} = f_{1}^{∗}. Then I^{∗} = (f^{∗}, Xη_{1}) + (Xη_{i} | 2 ≤ i ≤ ℓ). We want to show that f^{∗}, Xη_{1}, Xη_{2}, ⋯, Xη_{ℓ} is a minimal system of generators of I^{∗}. Let 1 ≤ i ≤ ℓ and assume that Xη_{i} ∈ (f^{∗}) + (Xη_{1}, ⋯, Xη_{i−1}, Xη_{i+1}, ⋯ Xη_{ℓ}). Then X[η_{i} − ∑_{1≤j≤ℓ,j≠i} η_{j}φ_{j}] ∈ (f^{∗}) for some φ_{j} ∈ G, so that we have η_{i} ∈ (f^{∗}) + (η_{1}, ⋯, η_{i−1}, η_{i+1}, η_{ℓ}). If i = 1, then η_{1} = g_{1}^{∗} ∈ (f^{∗}) + (η_{2}, η_{3}, ⋯, η_{ℓ}). Since deg f^{∗} = a > deg g_{1}^{∗} = a − 1, this forces η_{1} ∈ (η_{2}, η_{3}, ⋯, η_{ℓ}), which is impossible. Hence i > 1. Then, because η_{1} | f^{∗}, we have η_{i} ∈ (η_{1}, ⋯, η_{i−1}, η_{i+1}, ⋯, η_{ℓ}), which is absurd. Thus f^{∗}, Xη_{1}, Xη_{2}, ⋯, Xη_{ℓ} constitute a minimal system of generators of I^{∗} and so n = ℓ + 1. □

We close this section with the following.

Proposition 2.6. Let P = k[X, Y] be the polynomial ring in two variables X, Y over a field k. Let J be a graded ideal of P with μ_{P} (J) = n and \sqrt{J} = (X, Y). Let {ξ_{i}}_{1≤i≤n}be a homogeneous system of generators of J and set D_{i} = GCD(ξ_{1}, ξ_{2}, ⋯, ξ_{i}) for 1 ≤ i ≤ n. If deg D_{i} > deg D_{i+1} and \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) for all 1 ≤ i ≤ n − 1, then the Hilbert series H(P/J, λ) = ∑_{i=0}^{∞} dim_{k}[P/J]_{i}λ^{i} of P/J is given by the formula

H(P/J, λ) = \frac{∑_{i=2}^{n} λ^{deg D_{i}}(1 − λ^{deg D_{i−1}−deg D_{i}})(1 − λ^{deg ξ_{i}−deg D_{i}})}{(1 − λ)^{2}}.

In particular,

H(G/(X^{3i}Y^{m−i−1} | 0 ≤ i ≤ m − 1), λ) = \frac{∑_{i=2}^{m} λ^{m−i}(1 − λ^{3(i−1)})}{1 − λ}

for all 2 ≤ m ∈ ℤ.

THE LEADING IDEAL OF A COMPLETE INTERSECTION 9

Proof. If n = 2, then ξ_{1}, ξ_{2} is a P -regular sequence and we get H(P/J, λ) = \frac{(1−λ^{deg ξ_{1}})(1−λ^{deg ξ_{2}})}{(1−λ)^{2}}. Suppose that n > 2 and that our assertion holds true for n − 1. Let D = D_{n−1}. Then J ⊆ (D, ξ_{n}) and D, ξ_{n} form a P -regular sequence (recall that GCD(D, ξ_{n}) = 1). We write ξ_{i} = Dη_{i} (1 ≤ i ≤ n − 1) and put K = (η_{i} | 1 ≤ i ≤ n − 1). Then ξ_{n} = \frac{ξ_{n}}{D_{n}} ∈ (\frac{ξ_{1}}{D} , \frac{ξ_{2}}{D} , ⋯, \frac{ξ_{n−1}}{D}) = K and so μ_{P} (K) = n − 1, since J = DK + (ξ_{n}). Let E_{i} = GCD(η_{1}, η_{2}, ⋯, η_{i}). Then D_{i} = DE_{i} so that we have deg E_{i} > deg E_{i+1} and \frac{η_{i+1}}{E_{i+1}} = \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) = (\frac{η_{1}}{E_{i}} , \frac{η_{2}}{E_{i}} , ⋯, \frac{η_{i}}{E_{i}}) for all 1 ≤ i ≤ n − 2. Therefore, thanks to the exact sequence

0 → (P/K)(− deg D) → P/J → P/(D, ξ_{n}) → 0

of graded P -modules (recall that (D, ξ_{n})/J = (D, ξ_{n})/[DK + (ξ_{n})] ≅ (D)/[DK + (D) ∩ (ξ_{n})] = (D)/DK ≅ (P/K)(− deg D), since (D) ∩ (ξ_{n}) = (Dξ_{n}) and ξ_{n} ∈ (K)) and the hypothesis of induction on n, we get

H(P/J, λ) = H(P/(D, ξ_{n}), λ) + λ^{deg D}· H(P/K, λ)
= \frac{(1 − λ^{deg D})(1 − λ^{deg ξ_{n}})}{(1 − λ)^{2}}
+ \frac{λ^{deg D}· ∑_{i=2}^{n−1} λ^{deg E_{i}}(1 − λ^{deg E_{i−1}−deg E_{i}})(1 − λ^{deg η_{i}−deg E_{i}})}{(1 − λ)^{2}}
= \frac{∑_{i=2}^{n} λ^{deg D_{i}}(1 − λ^{deg D_{i−1}−deg D_{i}})(1 − λ^{deg ξ_{i}−deg D_{i}})}{(1 − λ)^{2}}

as claimed.

For the last assertion, let ξ_{i} = X^{3(i−1)}Y^{m−i} for 1 ≤ i ≤ m. Then D_{i} = Y^{m−i} and

\frac{ξ_{i}}{D_{i}} = X^{3(i−1)} for all 1 ≤ i ≤ m. Hence

H(G/(X^{3i}Y^{m−i−1} | 0 ≤ i ≤ m − 1), λ) = \frac{∑_{i=2}^{m} λ^{m−i}(1 − λ)(1 − λ^{3(i−1)})}{(1 − λ)^{2}}
= \frac{∑_{i=2}^{m} λ^{m−i}(1 − λ^{3(i−1)})}{1 − λ}.
□

3. Proof of Theorems 1.2 and 1.3

Proof of Theorem 1.2. Assume that Theorem 1.2 fails to hold and choose the ideal I so that a = o(I) := sup{i ∈ ℤ | I ⊆ 𝔫^{i}} is as small as possible among the counterexamples. We furthermore choose our ideal I so that b = o(g) is the smallest among the counterexamples I with o(I) = a. Then n > 2, whence b > 1 (Proposition 2.3 (4)). Choose elements x, y, u, and g_{1} ∈ S so that conditions (1)

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

and (2) in Lemma 2.2 are satisfied and put I_{1} = (f, g_{1}). We then have the following three cases: (i) a < b, (ii) a = b and g_{1}^{∗} ∤ f^{∗}, and (iii) a = b but g_{1}^{∗} | f^{∗}.

Suppose that case (i) occurs. Then μ_{G}(I_{1}^{∗}) = n (cf. Corollary 2.5). Since o(I_{1}) = a but o(g_{1}) = b − 1, we may choose a minimal homogeneous system {η_{i}}_{1≤i≤n} of generators of I_{1}^{∗} so that

(1) η_{1} = f^{∗} and η_{2} = g_{1}^{∗},

(2) deg η_{i} + 2 ≤ deg η_{i+1} for all 2 ≤ i ≤ n − 1, and

(3) ht_{G}(η_{1}, η_{2}, ⋯, η_{n−1}) = 1. Then, thanks to Corollary 2.5 (1), we get I^{∗} = (f^{∗}, g^{∗}) + (Xη_{3}, ⋯, Xη_{n}). Letting ξ_{1} = f^{∗}, ξ_{2} = g^{∗}, and ξ_{i} = Xη_{i} (3 ≤ i ≤ n), we certainly have conditions (1) and (2) in Theorem 1.2, because deg g_{1}^{∗} = b − 1 ≤ deg η_{3} − 2. Since (ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) = (f^{∗}, g^{∗})+(ξ_{3}, ⋯, ξ_{n−1}) = (f^{∗}, Xg_{1}^{∗})+(Xη_{3}, ⋯, Xη_{n−1}) ⊆ (η_{1}, η_{2})+(η_{3}, ⋯, η_{n−1}), we get ht_{G}(ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) = 1. Thus case (i) cannot occur.

Suppose case (ii) occurs. Then μ_{G}(I_{1}^{∗}) = n. Since o(I_{1}) = a − 1, we may choose a minimal homogeneous system {η_{i}}_{1≤i≤n} of generators of I_{1}^{∗} so that

(1) η_{1} = g_{1}^{∗} and η_{2} = f^{∗},

(2) deg η_{i} + 2 ≤ deg η_{i+1} for all 2 ≤ i ≤ n − 1, and

(3) ht_{G}(η_{1}, η_{2}, ⋯, η_{n−1}) = 1. Then I^{∗} = (f^{∗}, g^{∗})+(Xη_{3}, ⋯, Xη_{n}) by Corollary 2.5 (2). Let ξ_{1} = f^{∗}, ξ_{2} = g^{∗}, and ξ_{i} = Xη_{i} (3 ≤ i ≤ n). Then deg ξ_{2} = b = a and deg ξ_{3} = deg η_{3} + 1 ≥ deg η_{2} + 3 = a + 3, so that conditions (1) and (2) in Theorem 1.2 are safely satisfied for the family {ξ_{i}}_{1≤i≤n}. Since (ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) = (f^{∗}, g^{∗}) + (ξ_{3}, ⋯, ξ_{n−1}) = (f^{∗}, Xg_{1}^{∗}) + (ξ_{3}, ⋯, ξ_{n−1}) ⊆ (η_{1}, η_{2}, ⋯, η_{n−1}), we also have condition (3) in Theorem 1.2 to be satisfied. Hence case (ii) cannot occur.

Thus we have case (iii). Hence μ_{G}(I_{1}^{∗}) = n − 1. We choose f_{1} ∈ S so that o(f_{1}) = a_{1} > a, I_{1} = (g_{1}, f_{1}), and g_{1}^{∗} ∤ f_{1}^{∗}. Because o(I_{1}) = a − 1 < a = o(I), we may choose a minimal homogeneous system {η_{i}}_{1≤i≤n−1} of generators of I_{1}^{∗} so that

(1) η_{1} = g_{1}^{∗} and η_{2} = f_{1}^{∗},

(2) deg η_{i} + 2 ≤ deg η_{i+1} for all 2 ≤ i ≤ n − 2, and

(3) ht_{G}(η_{1}, η_{2}, ⋯, η_{n−2}) = 1. Then I^{∗} = (f^{∗}, g^{∗}) + (Xη_{2}, Xη_{3}, ⋯, Xη_{n−1}). Let ξ_{1} = f^{∗}, ξ_{2} = g^{∗}, and ξ_{i} = Xη_{i−1} for 3 ≤ i ≤ n. Because deg η_{2} = a_{1} > a, we have deg ξ_{3} ≥ a + 2, so that conditions (1) and (2) in Theorem 1.2 are satisfied for the family {ξ_{i}}_{1≤i≤n}. Since

THE LEADING IDEAL OF A COMPLETE INTERSECTION 11

(f^{∗}, g^{∗}) + (ξ_{3}, ⋯, ξ_{n−1}) = (f^{∗}, Xg_{1}^{∗}) + (ξ_{3}, ⋯, ξ_{n−1}) ⊆ (g_{1}^{∗}) + (η_{2}, ⋯, η_{n−2}) = (η_{1}, η_{2}, ⋯, η_{n−2}) (recall that g_{1}^{∗} | f^{∗}), we also have condition (3). This is absurd and thus Theorem 1.2 holds true.

Discussion 3.1. Let ξ_{1}, ξ_{2}, ⋯, ξ_{n} be a homogeneous system of generators for I^{∗} which satisfies conditions (1) and (2) in Theorem 1.2. Let c_{i} = deg ξ_{i} for 1 ≤ i ≤ n, and let G_{+} = ∑_{i>0} G_{i}. We then have {c_{1}, c_{2}, ⋯, c_{n}} = {i ∈ ℤ | [I^{∗}/G_{+}·I^{∗}]_{i} ≠ (0)}, whence the degree sequence (c_{1}, c_{2}, ⋯, c_{n}) is independent of the choice of {ξ_{i}}_{1≤i≤n}. Because ξ_{1} = f^{∗}, ξ_{2} = g^{∗}, and c_{1} = a ≤ c_{2} = b < c_{3} < ⋯ < c_{n}, the ideals (ξ_{1}, ξ_{2}, ⋯, ξ_{i}) (1 ≤ i ≤ n)G also do not depend on the choice of {ξ_{i}}_{1≤i≤n}. We put D_{i} = GCD(ξ_{1}, ξ_{2}, ⋯, ξ_{i}) and d_{i} = deg D_{i} (1 ≤ i ≤ n). (Hence D_{1} = ξ_{1} and D_{n} = 1.) Since the ideal (ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) is independent of the choice of {ξ_{i}}_{1≤i≤n}, we have condition (3) in Theorem 1.2 that ht_{G}(ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) = 1 is always satisfied for every homogeneous system of generators {ξ_{i}}_{1≤i≤n} of I^{∗} which satisfies conditions (1) and (2) of Theorem 1.2. Similarly, the fact whether \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) or not does not depend on the particular choice of a homogeneous system {ξ_{i}}_{1≤i≤n} of generators of I^{∗} which satisfies conditions (1) and (2) in Theorem 1.2.

Lemma 3.2. With notation as in Discussion 3.1, the following assertions hold true. (1) d_{i} > d_{i+1} and \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) for all 1 ≤ i ≤ n − 1.

(2) c_{i+1} − c_{i} > d_{i−1} − d_{i} > 0 for all 2 ≤ i ≤ n − 1.

(3) c_{n} + d_{n−1} ≥ d_{i−1} + c_{i} − d_{i} for all 2 ≤ i ≤ n.

(4) c_{n} + d_{n−1} ≥ a + b.

Proof. Assume that Lemma 3.2 is false and choose an ideal I so that a = o(I) = sup{i ∈ ℤ | I ⊆ 𝔫^{i}} is as small as possible among the counterexamples. We furthermore choose the ideal I so that b = o(g) is the smallest among the counterexamples I with o(I) = a. Then b > 1, since n > 2. Let x, y, u, and g_{1} ∈ S be elements which satisfy conditions (1) and (2) in Lemma 2.2. We put I_{1} = (f, g_{1}). Then we have the following three cases: (i) a < b, (ii) a = b and g_{1}^{∗} ∤ f^{∗}, and (iii) a = b but g_{1}^{∗} | f^{∗}. For case (i) we have f^{∗} ∤ g_{1}^{∗} and for case (iii) we have some f_{1} ∈ S with o(f_{1}) = a_{1} > a such that I_{1} = (g_{1}, f_{1}) and g_{1}^{∗} ∤ f_{1}^{∗}. In any case, because the value a or the value b for I_{1} is less than that for I, Lemma 3.2 holds true for the ideal I_{1}. In what follows, we shall establish a contradiction by showing (i),(ii), and (iii) cannot occur.

Suppose that case (i) occurs. Then μ_{G}(I_{1}^{∗}) = n. Let {η_{i}}_{1≤i≤n} be a homogeneous system of generators of I_{1}^{∗} such that η_{1} = f^{∗}, η_{2} = g_{1}^{∗}, and deg η_{i} + 2 ≤ deg η_{i+1}

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

for 2 ≤ i ≤ n − 1. Then Lemma 3.2 holds true for the family {η_{i}}_{1≤i≤n} and by Corollary 2.5 we have

I^{∗} = (f^{∗}, g^{∗}) + (Xη_{3}, ⋯, Xη_{n}).

Let ξ_{1} = f^{∗}, ξ_{2} = g^{∗}, and ξ_{i} = Xη_{i} (3 ≤ i ≤ n). Then the homogeneous system {ξ_{i}}_{1≤i≤n} of generators of I^{∗} satisfies conditions (1) and (2) in Theorem 1.2. We put c′_{i} = deg η_{i}, D′_{i} = GCD(η_{1}, η_{2}, ⋯, η_{i}), and d′_{i} = deg D′_{i}. Then, because (ξ_{1}, ξ_{2}) = (f^{∗}, g^{∗}) = (f^{∗}, Xg_{1}^{∗}) = (η_{1}, Xη_{2}) and X ∤ f^{∗}, we have D_{i} = D′_{i} for all 1 ≤ i ≤ n, while c′_{1} = a = c_{1} and c′_{i} = c_{i} − 1 for all 2 ≤ i ≤ n. Consequently, assertions (2), (3), (4), and the former part of assertion (1) in Lemma 3.2 are safely deduced from those on the ideal I_{1}. Let us check that \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}). Since D_{1} = ξ_{1}, we may assume i ≥ 2. First of all, recall that \frac{η_{i+1}}{D_{i+1}} ∈ (\frac{η_{1}}{D_{i}} , \frac{η_{2}}{D_{i}} , ⋯, \frac{η_{i}}{D_{i}}) and we have \frac{Xη_{i+1}}{D_{i+1}} ∈ (\frac{η_{1}}{D_{i}} , \frac{Xη_{2}}{D_{i}} , ⋯, \frac{Xη_{i}}{D_{i}}) = (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}), because (η_{1}, Xη_{2}, ⋯, Xη_{i}) = (ξ_{1}, ξ_{2}, ⋯, ξ_{i}). Hence \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) as ξ_{i+1} = Xη_{i+1}. Thus case (i) does not occur.

Suppose case (ii). Then μ_{G}(I_{1}^{∗}) = n. Let {η_{i}}_{1≤i≤n} be a homogeneous system of generators of I_{1}^{∗} such that η_{1} = g_{1}^{∗}, η_{2} = f^{∗}, and deg η_{i} + 2 ≤ deg η_{i+1} for all 2 ≤ i ≤ n − 1. Then I^{∗} = (f^{∗}, g^{∗}) + (Xη_{3}, ⋯, Xη_{n}) by Corollary 2.5. Let ξ_{1} = f^{∗}, ξ_{2} = g^{∗}, and ξ_{i} = Xη_{i} for 3 ≤ i ≤ n. Then {ξ_{i}}_{1≤i≤n} is a homogeneous system of generators of I^{∗} which satisfies conditions (1) and (2) in Theorem 1.2. We put c′_{i} = deg η_{i}, D′_{i} = GCD(η_{1}, η_{2}, ⋯, η_{i}), and d′_{i} = deg D′_{i} for each 1 ≤ i ≤ n. Then c′_{1} = a − 1 = c_{1} − 1, c′_{2} = a = c_{2}, and c′_{i} = c_{i} − 1 for 3 ≤ i ≤ n. Because X ∤ f^{∗} and (ξ_{1}, ξ_{2}) = (f^{∗}, g^{∗}) = (η_{2}, Xη_{1}), we get D′_{1} = η_{1} = g_{1}^{∗} and D′_{i} = D_{i} for 2 ≤ i ≤ n. Hence d′_{1} = a − 1 = d_{1} − 1 and d′_{i} = d_{i} for all 2 ≤ i ≤ n. Consequently, it is direct to check that assertions (2), (3), (4), and the former part of assertion (1) hold true for the ideal I. Let us show \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) for all 1 ≤ i ≤ n − 1. We may assume i ≥ 2. Because \frac{η_{i+1}}{D_{i+1}} ∈ (\frac{η_{1}}{D_{i}} , \frac{η_{2}}{D_{i}} , ⋯, \frac{η_{i}}{D_{i}}), we have \frac{Xη_{i+1}}{D_{i+1}} ∈ (\frac{Xη_{1}}{D_{i}} , \frac{Xη_{2}}{D_{i}} , ⋯, \frac{Xη_{i}}{D_{i}}) ⊆ (\frac{ξ_{1}}{D_{i}}) + (\frac{Xη_{j}}{D_{i}} |1 ≤ j ≤ i, j ≠ 2) = (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) (use the fact (ξ_{1}, ξ_{2}) = (ξ_{1}, Xη_{1})). Hence \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}) as ξ_{i+1} = Xη_{i+1}. Thus case (ii) does not occur.

Now we consider case (iii). We have μ_{G}(I_{1}^{∗}) = n−1. Let f_{1} ∈ S such that o(f_{1}) = a_{1} > a, I_{1} = (g_{1}, f_{1}), and g_{1}^{∗} ∤ f_{1}^{∗}. Choose a homogeneous system {η_{i}}_{1≤i≤n−1} of generators for I_{1}^{∗} so that η_{1} = g_{1}^{∗}, η_{2} = f_{1}^{∗}, and deg η_{i} + 2 ≤ deg η_{i+1} for all 2 ≤ i ≤ n−2. Then I^{∗} = (f^{∗}, g^{∗})+(Xη_{2}, ⋯, Xη_{n−1}). We put ξ_{1} = f^{∗}, ξ_{2} = g^{∗}, and ξ_{i} = Xη_{i−1} for 3 ≤ i ≤ n−1. Then the homogeneous system {ξ_{i}}_{1≤i≤n} of generators

THE LEADING IDEAL OF A COMPLETE INTERSECTION 13

of I^{∗} satisfies conditions (1) and (2) in Theorem 1.2. Let D′_{i} = GCD(η_{1}, η_{2}, ⋯, η_{i}), d′_{i} = deg D′_{i}, and c′_{i} = deg η_{i} for each 1 ≤ i ≤ n − 1. Then D′_{i} = D_{i+1} for 1 ≤ i ≤ n − 1 (recall that g_{1}^{∗} | f^{∗} and (ξ_{1}, ξ_{2}) = (f^{∗}, g^{∗}) = (f^{∗}, Xg_{1}^{∗}) = (f^{∗}, Xη_{1})). Hence c′_{1} = a − 1 = c_{1} − 1, c′_{i} = c_{i+1} − 1 for 2 ≤ i ≤ n − 1, and d′_{i} = d_{i+1} for 1 ≤ i ≤ n − 1. Consequently, assertions (2), (3), (4), and the former part of assertion (1) hold true (use the fact that c_{3} = a_{1} + 1 ≥ a + 2, d′_{1} = a − 1, and c_{n} ≥ a + 2). Let us check the latter part of assertion (1). We may assume i ≥ 2. Then, since \frac{η_{i}}{D′_{i}} ∈ (\frac{η_{1}}{D′_{i−1}} , \frac{η_{2}}{D′_{i−1}} , ⋯, \frac{η_{i−1}}{D′_{i−1}} ), we get \frac{Xη_{i}}{D_{i+1}} ∈ (\frac{f^{∗}}{D_{i}} , \frac{Xη_{1}}{D_{i}} , ⋯, \frac{Xη_{i−1}}{D_{i}}). Hence \frac{ξ_{i+1}}{D_{i+1}} ∈ (\frac{ξ_{1}}{D_{i}} , \frac{ξ_{2}}{D_{i}} , ⋯, \frac{ξ_{i}}{D_{i}}), because ξ_{i+1} = Xη_{i} and (f^{∗}, Xη_{1}) = (ξ_{1}, ξ_{2}). Thus even case (iii) cannot occur. We conclude that Lemma 3.2 holds true. □

Proof of Theorem 1.3. Items (1) and (3) follow from Proposition 2.6 and Lemma 3.2.

For items (2) and (4), since

H(gr_{𝔪}(R), λ) = ∑_{i=2}^{n} λ^{d_{i}}( ∑_{j=0}^{d_{i−1}−d_{i}−1} λ^{j})( ∑_{j=0}^{c_{i}−d_{i}−1} λ^{j})

and ℓ_{S}(R) = dim_{k} gr_{𝔪}(R), we readily get ℓ_{S}(R) = ∑_{i=2}^{n}(d_{i−1} − d_{i})(c_{i} − d_{i}) = c_{1}c_{2} + ∑_{i=2}^{n−1} d_{i}·[(c_{i+1} − c_{i}) − (d_{i−1} − d_{i})] = ab + ∑_{i=2}^{n−1} d_{i}·[(c_{i+1} − c_{i}) − (d_{i−1} − d_{i})]. We have ℓ_{S}(R) = ab if and only if n = 2, because (c_{i+1} − c_{i}) − (d_{i−1} − d_{i}) > 0 for all 2 ≤ i ≤ n − 1 by Lemma 3.2 (2). Since I^{∗} = (f^{∗}, g^{∗}) if and only if n = 2, we have ℓ_{S}(R) = ab if and only if f^{∗}, g^{∗} form a regular sequence in G.

Corollary 3.3. Assume notation as in Theorem 1.3, and let a(gr_{𝔪}(R)) = max{i ∈ ℤ | [gr_{𝔪}(R)]_{i} ≠ (0)}. The following assertions hold true.

(1) a(gr_{𝔪}(R)) = c_{n} + d_{n−1} − 2.

(2) ℓ_{S}(R) ≤ a·[c_{n} + d_{n−1} − a].

(3) ℓ_{S}(R) = a·[c_{n} + d_{n−1} − a] if and only if n = 2.

Proof. Since a(gr_{𝔪}(R)) = deg H(gr_{𝔪}(R), λ), thanks to Theorem 1.3 (1), we have a(gr_{𝔪}(R)) = max{d_{i} +(d_{i−1} −d_{i} −1)+(c_{i} −d_{i} −1) | 2 ≤ i ≤ n}. Hence a(gr_{𝔪}(R)) = c_{n} + d_{n−1} − 2 by Lemma 3.2 (3). Because d_{1} − 1 = a − 1 ≥ d_{i} for all 2 ≤ i ≤ n − 1 and c_{n} + d_{n−1} ≥ a + b by Lemma 3.2 (1), (4), we get by Theorem 1.3 (2) that

ℓ_{S}(R) ≤ ab + (a − 1)· ∑_{i=2}^{n−1}[(c_{i+1} − c_{i}) − (d_{i−1} − d_{i})]
= ab + (a − 1)·[c_{n} + d_{n−1} − (a + b)]
= a·[c_{n} + d_{n−1} − a] − [c_{n} + d_{n−1} − (a + b)]
≤ a·[c_{n} + d_{n−1} − a].

If the equality ℓ_{S}(R) = a·[c_{n} + d_{n−1} − a] holds true, then c_{n} + d_{n−1} − (a + b) = 0, so that ℓ_{S}(S/I) = a·[c_{n} + d_{n−1} − a] = ab, whence n = 2. Since c_{2} = b and d_{1} = a, we certainly have ℓ_{S}(R) = a(c_{n} + d_{n−1} − a) if n = 2. This completes the proof of Corollary 3.3. □

Suppose that ht_{G}(f^{∗}, g^{∗}) = 1. Let D = GCD(f^{∗}, g^{∗}) and d = deg D. We write f^{∗} = Dξ and g^{∗} = Dη with ξ, η ∈ G. Then b ≥ a > d > 0 and by [GHK, Proposition 2.2] we may choose h = αf + βg with α, β ∈ S so that o(α) = b − d, o(β) = a − d, and h^{∗} ∉ (f^{∗}, g^{∗}). We call such an element h^{∗} the third generator of I^{∗}. We put c = o(h). With this notation we have the following.

Corollary 3.4. Suppose that ht_{G}(f^{∗}, g^{∗}) = 1 and ht_{G}(f^{∗}, g^{∗}, h^{∗}) = 2. Then the following assertions hold true.

(1) I^{∗} = (f^{∗}, g^{∗}, h^{∗}).

(2) H(gr_{𝔪}(R), λ) = \frac{(1−λ^{c})(1−λ^{d})+λ^{d}(1−λ^{a−d})(1−λ^{b−d})}{(1−λ)^{2}}.

(3) e_{𝔪}^{0}(R) = ab + d·[(c + d) − (a + b)].

Proof. For each 3 ≤ i ≤ n, let ξ_{i} = \overline{h_{i}t^{c_{i}}} where h_{i} ∈ I with o(h_{i}) = c_{i}. We write h_{i} = α_{i}f+β_{i}g with α_{i}, β_{i} ∈ S. Then o(α_{i}) ≥ b−d and o(h_{i}) ≥ o(h)+[o(α_{i})−(b−d)] ≥ o(h) (cf. [GHK, Proposition 2.4 (1)]). Let h^{∗} = ∑_{i=1}^{n} ξ_{i}φ_{i} with φ_{i} ∈ G_{c−c_{i}}. Then, since h^{∗} ∉ (f^{∗}, g^{∗}) = (ξ_{1}, ξ_{2}), we have c − c_{i} ≥ 0 for some 3 ≤ i ≤ n. Hence c ≥ c_{i} ≥ c_{3}, so that c = c_{3}, because c_{3} ≥ c+[o(α_{3})−(b−d)] ≥ c. We furthermore have o(α_{3}) = b−d, whence, thanks to [GHK, Proposition 2.4 (3)], we get (f^{∗}, g^{∗}, ξ_{3}) = (f^{∗}, g^{∗}, h^{∗}). Thus n = 3 by Theorem 1.2 (3), because ht_{G}(f^{∗}, g^{∗}, h^{∗}) = 2 by our assumption, so that I^{∗} = (ξ_{1}, ξ_{2}, ξ_{3}) = (f^{∗}, g^{∗}, h^{∗}) as claimed. Assertions (2) and (3) now readily follow from Theorem 1.3 (1) and (2). □

THE LEADING IDEAL OF A COMPLETE INTERSECTION 15

Remark 3.5. With notation as in Setting 1.1, it follows from Part (1) of Lemma 3.2 that there exists a strictly descending chain

(\frac{ξ_{1}}{D_{2}} , \frac{ξ_{2}}{D_{2}} )G ⊃ (\frac{ξ_{1}}{D_{3}} , \frac{ξ_{2}}{D_{3}} , \frac{ξ_{3}}{D_{3}} )G ⊃ ⋯ ⊃ (\frac{ξ_{1}}{D_{n−1}} , \frac{ξ_{2}}{D_{n−1}} , ⋯ \frac{ξ_{n−1}}{D_{n−1}} )G ⊃ I^{∗}

of height-two ideals of G. In particular, I^{∗} is contained in the ideal (\frac{ξ_{1}}{D_{2}} , \frac{ξ_{2}}{D_{2}} )G. This behavior fails to hold in general in the higher dimensional case. The leading ideal of a complete intersection of height two in a three-dimensional regular local ring may fail to have this property as is demonstrated by Example 1.6 of [GHK].

4. Proof of Theorems 1.5 and 1.6

The goal of this section is to prove Theorems 1.5, and 1.6 and deduce several consequences of these theorems. We use the following lemma.

Lemma 4.1. Assume notation as in Setting 1.1. Let 0 ≠ h ∈ 𝔫 and m = o(h). Let X_{1}, X_{2}, ⋯, X_{s−1} ∈ G be a linear system of parameters for the graded ring G/(h^{∗}) and write X_{i} = x_{i}^{∗} with x_{i} ∈ 𝔫. Then x_{1}, x_{2}, ⋯, x_{s−1} is a part of a regular system of parameters of S and for all 1 ≤ ℓ ≤ s − 1, we have o(\overline{h}) = m, where \overline{h} denotes the image of h in \overline{S} = S/(x_{1}, x_{2}, ⋯, x_{ℓ}).

Proof. Since X_{1}, X_{2}, ⋯, X_{s−1} are algebraically independent over k, the elements x_{1}, x_{2}, ⋯, x_{s−1} form a part of a regular system of parameters in S. If

h ∈ 𝔫^{m+1} +(x_{1}, x_{2}, ⋯, x_{ℓ}),

then since (x_{1}, x_{2}, ⋯, x_{ℓ}) ∩ 𝔫^{m} = (x_{1}, x_{2}, ⋯, x_{ℓ}) 𝔫^{m−1}, we get

h ∈ 𝔫^{m+1} +(x_{1}, x_{2}, ⋯, x_{ℓ}) 𝔫^{m−1} .

Thus h^{∗} ∈ (X_{1}, X_{2}, ⋯, X_{ℓ}), which is impossible, because X_{1}, X_{2}, ⋯, X_{ℓ}, h^{∗} forms a regular sequence in G. Hence o(\overline{h}) = m as claimed.

Proof of Theorem 1.5. By Corollary 3.4, we may assume that dim S = s > 2. Choose X_{1}, X_{2}, ⋯, X_{s−1} ∈ G_{1} so that X_{1}, X_{2}, ⋯, X_{s−1} is a homogeneous system of parameters for the graded rings G/(f^{∗}), G/(g^{∗}), G/(h^{∗}), G/(α^{∗}), G/(β^{∗}), and G/(D) and X_{1}, X_{2}, ⋯, X_{s−2} is a homogeneous system of parameters for the graded rings G/(f^{∗}, g^{∗}, h^{∗}), G/(ξ, η), and gr_{𝔪}(R). For each i with 1 ≤ i ≤ s − 1, choose x_{i} ∈ 𝔫 such that x_{i}^{∗} = X_{i}. Then x_{1}, x_{2}, ⋯, x_{s−1} form a part of a regular system of parameters for S. Let 𝔮 = (x_{1}, x_{2}, ⋯, x_{s−2})S. We put \overline{S} = S/𝔮, \overline{𝔫} = 𝔫 /𝔮, and \overline{I} = (\overline{f}, \overline{g}), where overline denotes image in \overline{S}. Notice that 𝔮R is a minimal reduction

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

of 𝔪. Thus I + 𝔮 is a parameter ideal for S and \overline{I} = (\overline{f}, \overline{g})\overline{S} is a parameter ideal in the regular local ring \overline{S} of dimension 2. Lemma 4.1 implies that o(\overline{f}) = a, o(\overline{g}) = b and o(\overline{h}) = c.

Let Q = (X_{1}, X_{2}, …, X_{s−2})G. We prove that the following diagram is commutative:

S ⟶ S/𝔮 := \overline{S}
φ_{1}↓  ^{φ_{2}}↓
G := gr_{n}(S) ⟶ G/Q := G̃ ≅ gr\overline{𝔫}(\overline{S}).

Here φ_{1} and φ_{2} denote the canonical maps associating an element with its leading form in the associated graded ring, and the identification G̃ ≅ gr\overline{n}(\overline{S}) is because Q is the leading ideal in G of the ideal 𝔮 of S. We denote with a tilde the image in G/Q of elements and ideals of G. Since X_{1}, X_{2}, ⋯, X_{s−2}, ξ, η is a homogeneous system of parameters in G, ξ̃, η̃ is a homogeneous system of parameters in G/Q. Thus GCD(ξ̃, η̃) = 1, and D̃ = GCD(f̃^{∗}, g̃^{∗}). Since o(f) = o(\overline{f}), we have f̃^{∗} = \overline{f}^{∗}. Similarly, g̃^{∗} = \overline{g}^{∗} and h̃^{∗} = \overline{h}^{∗}. We have Ĩ^{∗} ⊆ \overline{I}^{∗}. Moreover, Ĩ^{∗} = \overline{I}^{∗} if and only if X_{1}, …, X_{s−2} is a regular sequence on G/I^{∗}. Thus Ĩ^{∗} = \overline{I}^{∗} if and only if I^{∗} is a perfect ideal of G.

We furthermore have the following.

Claim 4.2. The following assertions hold true.

(1) \overline{f}^{∗} ∤ \overline{g}^{∗} in gr\overline{𝔫}(\overline{S}).

(2) o(\overline{α}) = b − d, o(\overline{β}) = a − d, and o(\overline{h}) = c.

(3) \overline{h}^{∗} ∉ (\overline{f}^{∗}, \overline{g}^{∗}). Thus \overline{h}^{∗} is the third generator of \overline{I}^{∗} in gr\overline{𝔫}(\overline{S}).

Proof of Claim 4.2. (1) Suppose that \overline{f}^{∗} | \overline{g}^{∗}. Then, via the identification G/Q = gr\overline{𝔫}(\overline{S}), we have g^{∗} ∈ (f^{∗}) + Q. Let us write g^{∗} = f^{∗}φ + ∑_{i=1}^{s−2} X_{i}φ_{i} with φ, φ_{i} ∈ G. Then, since f^{∗} = Dξ and g^{∗} = Dη, we have D(η − ξφ) ∈ Q. Hence η − ξφ ∈ Q, because X_{1}, X_{2}, ⋯, X_{s−2}, D is a regular sequence in G. Thus η ∈ Q + (ξ), which is impossible, because X_{1}, X_{2}, ⋯, X_{s−2}, ξ, η is a G-regular sequence. Hence \overline{f}^{∗} ∤ \overline{g}^{∗}.

(2) See Lemma 4.1

(3) We have h^{∗} ∈ (ξ, η) ([GHK, Remark 2.3]; recall that h ∈ (α, β)). Write h^{∗} = ξφ + ηψ with φ, ψ ∈ G. Then

(f^{∗}, g^{∗}, h^{∗}) = I_{2} ( \begin{matrix}φ & −ψ & D \\ ξ & η & 0 \end{matrix} ) ,

THE LEADING IDEAL OF A COMPLETE INTERSECTION 17

so that (f^{∗}, g^{∗}, h^{∗}) is a perfect ideal with μ_{G}(f^{∗}, g^{∗}, h^{∗}) = 3, since ht_{G}(f^{∗}, g^{∗}, h^{∗}) = 2. Therefore G/(f^{∗}, g^{∗}, h^{∗}) is a Cohen-Macaulay ring, whence X_{1}, X_{2}, ⋯, X_{s−2} form a regular sequence in G/(f^{∗}, g^{∗}, h^{∗}). Thus \overline{h}^{∗} ∉ (\overline{f}^{∗}, \overline{g}^{∗}), because μ_{gr\overline{𝔫}(\overline{S})}(\overline{f}^{∗}, \overline{g}^{∗}, \overline{h}^{∗}) = 3. □

Therefore \overline{I}^{∗} = (\overline{f}^{∗}, \overline{g}^{∗}, \overline{h}^{∗}) by Corollary 3.4, because \overline{h}^{∗} is the third generator of \overline{I}^{∗} in gr\overline{𝔫}(\overline{S}) with ht_{gr\overline{𝔫}(\overline{S})}(\overline{f}^{∗}, \overline{g}^{∗}, \overline{h}^{∗}) = 2. We now look at the estimation (∗):

ℓ_{R}(R/𝔮R) = ℓ\overline{S}(\overline{S}/\overline{I}) = dim_{k} gr\overline{𝔫}(\overline{S})/(\overline{f}^{∗}, \overline{g}^{∗}, \overline{h}^{∗})
= dim_{k} G/[Q + (f^{∗}, g^{∗}, h^{∗})]
≥ dim_{k} G/[Q + I^{∗}]
= dim_{k} gr_{𝔪}(R)/Qgr_{𝔪}(R)
≥ e^{0}_{Qgr_{𝔪}(R)}(gr_{𝔪}(R))
= e_{𝔪}^{0}(R)
= ℓ_{R}(R/𝔮R),

since 𝔮R is a minimal reduction of 𝔪. Thus gr_{𝔪}(R) = G/I^{∗} is Cohen-Macaulay, since dim_{k} gr_{𝔪}(R)/Qgr_{𝔪}(R) = e_{Qgr_{𝔪}(R)}^{0}(gr_{𝔪}(R)) (cf. estimation (∗)), and so the sequence X_{1}, X_{2}, ⋯, X_{s−2} is gr_{𝔪}(R)-regular. Hence I^{∗} = (f^{∗}, g^{∗}, h^{∗}), because Q + (f^{∗}, g^{∗}, h^{∗}) = Q + I^{∗} and Q ∩ I^{∗} = QI^{∗}. We furthermore have that

H(gr_{𝔪}(R), λ) = \frac{H(gr\overline{𝔫}(\overline{S})/(\overline{f}^{∗}, \overline{g}^{∗}, \overline{h}^{∗}), λ)}{(1 − λ)^{s−2}},

whence

H(gr_{𝔪}(R), λ) = \frac{(1 − λ^{c})(1 − λ^{d}) + λ^{d}(1 − λ^{a−d})(1 − λ^{b−d})}{(1 − λ)^{s}}

by Corollary 3.4. Thus e_{𝔪}^{0}(R) = ab + d·[(c + d) − (a + b)] as claimed. This completes the proof of Theorem 1.5.

Remark 4.3. Without the assumption in Theorem 1.5 that ht(f^{∗}, g^{∗}, h^{∗}) = 2, it is still possible to specialize via 𝔮 and Q to obtain f̃^{∗} = \overline{f}^{∗}, g̃^{∗} = \overline{g}^{∗} and D̃ = GCD(f̃^{∗}, g̃^{∗}). However, h̃^{∗} = \overline{h}^{∗} may fail to be a minimal generator of \overline{I}^{∗} as we demonstrate in Example 4.4.

Example 4.4. Let S = k[[x, y, z]] be the formal power series ring in the three variables x, y, z over a field k, and let X, Y, Z denote the leading forms of x, y, z in G = gr_{n}(S) = k[X, Y, Z]. As in [GHK, Example 1.6], let I = (f, g), where f = z^{2} − x^{5} and g = zx − y^{3}. Thus R = S/I is a complete intersection of dimension

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

one. We have I^{∗} = (Z^{2}, ZX, ZY^{3}, Y^{6})G. We consider several choices for an element w ∈ n \ n^{2} and behavior of the specialization S → S/wS = \overline{S}. Since dim G/I^{∗} = 1 and I^{∗} is not a perfect ideal, one always has the strict inequality I^{∗}G̃ ⊊ (I\overline{S})^{∗}.

(1) Let w = x. Then \overline{S} = k[[y, z]], \overline{f} = z^{2} and \overline{g} = −y^{3}. We have

I^{∗}G̃ = (Z^{2}, ZY^{3}, Y^{6})G̃ ⊊ (I\overline{S})^{∗} = (Z^{2}, Y^{3})k[Y, Z]. The multiplicity of G/I^{∗} is 6 as is the multiplicity of gr\overline{n}(\overline{S})/(I\overline{S})^{∗}. The Hilbert series for G/I^{∗} is

H(G/I^{∗}, λ) = \frac{1 + 2λ + λ^{2} + λ^{3} + λ^{5}}{1 − λ} ,

while the Hilbert series for gr\overline{n}(\overline{S})/(I\overline{S})^{∗} is

H(gr\overline{n}(\overline{S})/(I\overline{S})^{∗}, λ) = \frac{1 + 2λ + 2λ^{2} + λ^{3}}{1 − λ} .

The multiplicity of G̃/I^{∗}G̃ is 9, and the Hilbert series for G̃/I^{∗}G̃ is

H(G̃/I^{∗}G̃, λ) = \frac{1 + 2λ + 2λ^{2} + 2λ^{3} + λ^{4} + λ^{5}}{1 − λ} .

(2) Let w = x − y and use this to eliminate x. Then \overline{S} = k[[y, z]], \overline{f} = z^{2} − y^{5}

and \overline{g} = zy − y^{3}. We have

I^{∗}G̃ = (Z^{2}, ZY, Y^{6})G̃ ⊊ (I\overline{S})^{∗} = (Z^{2}, ZY, Y^{5})k[Y, Z].

The multiplicity and Hilbert series of G/I^{∗} are as given in part (1). The multiplicity of gr\overline{n}(\overline{S})/(I\overline{S})^{∗} is 6, while the multiplicity of G̃/I^{∗}G̃ is 7. The Hilbert series of gr\overline{n}(\overline{S})/(I\overline{S})^{∗} is

H(gr\overline{n}(\overline{S})/(I\overline{S})^{∗}, λ) = \frac{1 + 2λ + λ^{2} + λ^{3} + λ^{4}}{1 − λ} ,

while the Hilbert series of G̃/I^{∗}G̃ is

H(G̃/I^{∗}G̃, λ) = \frac{1 + 2λ + λ^{2} + λ^{3} + λ^{4} + λ^{5}}{1 − λ} .

Example 4.5. Let S = k[[x, y, z, u]] be the formal power series ring in the four variables x, y, z, u over a field k, and let X, Y, Z, U denote the leading forms of x, y, z, u in G = gr_{n}(S) = k[X, Y, Z, U]. Let I = (f, g)S, where f = xy and g = xz + u^{3}. Thus R = S/I is a complete intersection of dimension two. It can be seen directly, and also is a consequence of Theorem 1.5, that I^{∗} = (XY, XZ, Y U^{3})G. Since I^{∗} is a perfect ideal and dim G/I^{∗} = 2, it is possible to choose Q = (X_{1}, X_{2})G, the leading form ideal of 𝔮 = (x_{1}, x_{2})S such that Ĩ^{∗} = \overline{I}^{∗}. We illustrate how to successively choose x_{1} and x_{2}.

THE LEADING IDEAL OF A COMPLETE INTERSECTION 19

(1) Let x_{1} = y − u and use this to eliminate u. Thus \overline{S} = k[[x, y, z]], \overline{f} = xy

and \overline{g} = xz + y^{3}. We have

I^{∗}G̃ = (XY, XZ, Y^{4})G̃ = (I\overline{S})^{∗} = (XY, XZ, Y^{4})k[X, Y, Z]. We now apply the process again:

(2) Let x_{2} = z − x and use this to eliminate z. Thus \overline{S} = k[[x, y]], \overline{f} = xy and

\overline{g} = x^{2} + y^{3}. We have

I^{∗}G̃ = (XY, X^{2}, Y^{4})G̃ = (I\overline{S})^{∗} = (XY, X^{2}, Y^{4})k[X, Y].

The numerator polynomial of the Hilbert series in each case is 1+2t+t^{2} +t^{3}.

We record the following corollary to Theorem 1.5.

Corollary 4.6. Assume notation as in Setting 1.1 and Remark 1.4. If D := D_{2} is a prime element of G that is regular on G/(ξ, η), then μ(I^{∗}) = 3 and I^{∗} is perfect.

Proof. It suffices to show that GCD(f^{∗}, g^{∗}, h^{∗}) = 1. If this fails, then

h^{∗} ∈ (D) ∩ (ξ, η) = (Dξ, Dη) = (f^{∗}, g^{∗}),

a contradiction to the assumption that h^{∗} is the third generator of I^{∗}. □

Example 4.7. Let S = k[[x, y, z]] be the formal power series ring in the three variables x, y, z over a field k. Let f = xy^{i} + z^{s} and g = xz^{j}, where s > i + 1 and i and j are positive. By Corollary 4.6, μ(I^{∗}) = 3 and I^{∗} is perfect.

We use Lemma 4.1 and Theorem 1.2 to establish in Theorem 4.8 conditions on the degrees of a minimal homogeneous system of generators for I^{∗} in the case where I^{∗} is perfect.

Theorem 4.8. Assume notation as in Setting 1.1 and Remark 1.4. If gr_{𝔪}(R) is a Cohen-Macaulay ring and n = μ_{G}(I^{∗}), then there exist homogeneous elements {ξ_{i}}_{1≤i≤n} of G such that

(1) I^{∗} = (ξ_{1}, ξ_{2}, ⋯, ξ_{n}),

(2) ξ_{1} = f^{∗} and ξ_{2} = g^{∗},

(3) deg ξ_{i} + 2 ≤ deg ξ_{i+1} for all 2 ≤ i ≤ n − 1, and

(4) ht_{G}(ξ_{1}, ξ_{2}, ⋯, ξ_{n−1}) = 1.

Proof. By Theorem 1.2, we may assume s > 2. If n = 2, then I^{∗} = (f^{∗}, g^{∗}) and there is nothing to prove. Assume n > 2 and let D = GCD(f^{∗}, g^{∗}). We write f^{∗} = Dξ

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

and g^{∗} = Dη; hence ξ, η is a G-regular sequence. We choose, similarly as in the proof of Theorem 1.5, the elements X_{1}, X_{2}, ⋯, X_{s−1} ∈ G_{1} so that {X_{i}}_{1≤i≤s−1} is a homogeneous system of parameters for the rings G/(f^{∗}), G/(g^{∗}), and G/(D) and {X_{i}}_{1≤i≤s−2} is a homogeneous system of parameters for the rings G/(ξ, η) and gr_{𝔪}(R). Let x_{i} ∈ 𝔫 with X_{i} = x_{i}^{∗}. We put 𝔮 = (x_{i} | 1 ≤ i ≤ s − 2), \overline{S} = S/𝔮, \overline{𝔫} = 𝔫 /𝔮, and \overline{I} = (\overline{f}, \overline{g}), where \overline{f} and \overline{g} respectively denote the images of f and g in \overline{S}. Then \overline{f}^{∗} ∤ \overline{g}^{∗} (cf. Proof of Claim 4.2 (1)). The sequence X_{1}, X_{2}, ⋯, X_{s−2} is regular in the ring gr_{𝔪}(R), because gr_{𝔪}(R) is Cohen-Macaulay. We identify

gr\overline{𝔫}(\overline{S}) = G/Q and \overline{I}^{∗} = [I^{∗} + Q]/Q,

where Q = (X_{i} | 1 ≤ i ≤ s − 2). Therefore, since μ_{gr\overline{𝔫} \overline{S}}(\overline{I}^{∗}) = μ_{G}(I^{∗}) = n, thanks to Theorem 1.2, the ideal \overline{I}^{∗} contains a homogeneous system \{η_{i}\}_{1≤i≤n} of generators which satisfies the conditions

(1) η_{1} = \overline{f}^{∗} and η_{2} = \overline{g}^{∗},

(2) deg η_{i} + 2 ≤ deg η_{i+1} for all 2 ≤ i ≤ n − 1, and

(3) ht_{gr\overline{𝔫}(\overline{S})}(η_{i} | 1 ≤ i ≤ n − 1) = 1. Thus, taking ξ_{i} ∈ I^{∗} to be a preimage of η_{i}, we readily get a homogeneous system {ξ_{i}}_{1≤i≤n} of generators of I^{∗} which satisfies conditions (2) and (3) in Theorem 4.8.

Let us check condition (4) is also satisfied. Assume the contrary and rechoose the system {X_{i}}_{1≤i≤s−1} so that {X_{i}}_{1≤i≤s−2} is also a homogeneous system of parameters for the ring G/(ξ_{i} | 1 ≤ i ≤ n − 1) of dimension s − 2. Let \overline{ξ_{i}} denote the image of ξ_{i} in G/Q. Then \{\overline{ξ_{i}}\}_{1≤i≤n} constitutes a minimal homogeneous system of generators of \overline{I}^{∗} = [I^{∗} + Q]/Q with deg \overline{ξ_{i}} ≤ deg \overline{ξ_{i+1}} for all 2 ≤ i ≤ n − 1. Consequently, even though we do not necessarily have η_{i} = \overline{ξ_{i}} (1 ≤ i ≤ n) for the second choice of \{X_{i}\}_{1≤i≤s−1}, we still have (η_{i} | 1 ≤ i ≤ n − 1) = (\overline{ξ_{i}} | 1 ≤ i ≤ n − 1), because the ideals \{(η_{j} | 1 ≤ j ≤ i)\}_{1≤i≤n} of gr\overline{𝔫}(\overline{S}) are independent of the choice of minimal homogeneous systems \{η_{i}\}_{1≤i≤n} of generators of \overline{I}^{∗} which satisfy the condition that η_{1} = \overline{f}^{∗}, η_{2} = \overline{g}^{∗}, and deg η_{i} + 2 ≤ deg η_{i+1} for all 2 ≤ i ≤ n − 1. This is however impossible, since ht_{gr\overline{𝔫}(\overline{S})}(η_{i} | 1 ≤ i ≤ n − 1) = 1 while dim G/[Q + (ξ_{i} | 1 ≤ i ≤ n − 1)] = 0. Thus ht_{G}(ξ_{i} | 1 ≤ i ≤ n − 1) = 1 as claimed. □

Proof of Theorem 1.6. Assume that gr_{𝔪}(R) is a Cohen-Macaulay ring and let {ξ_{i}}_{1≤i≤n} be a homogeneous system of generators of I^{∗} which satisfies conditions (2) and (3) in Theorem 4.8. Let X_{1}, X_{2}, ⋯, X_{s−2} ∈ G_{1} and write X_{i} = x_{i}^{∗} with x_{i} ∈ 𝔫. We put 𝔮 = (x_{i} | 1 ≤ i ≤ s − 2), \overline{S} = S/𝔮, \overline{𝔫} = 𝔫 /𝔮, and \overline{I} = (\overline{f}, \overline{g}), where \overline{f} and \overline{g}

THE LEADING IDEAL OF A COMPLETE INTERSECTION 21

respectively denote the images of f and g in \overline{S}. We put Q = (X_{i} | 1 ≤ i ≤ s − 2). Then, choosing {X_{i}}_{1≤i≤s−2} to be sufficiently general, we may assume that

(1) \{X_{i}\}_{1≤i≤s−2} is a homogeneous system of parameters for gr_{𝔪}(R), so that \overline{S} is a regular local ring of dimension 2 with the parameter ideal \overline{I}, and

(2) D̃_{i} = GCD(ξ̃_{1}, ξ̃_{2}, ⋯, ξ̃_{i}) for all 1 ≤ i ≤ n, where D̃_{i} and ξ̃_{i} respectively denote the image of D_{i} and ξ_{i} in G/Q = gr\overline{𝔫}(\overline{S}). Then the minimal homogeneous system \{ξ̃_{i}\}_{1≤i≤n} of generators of the ideal Ĩ^{∗} = \overline{I}^{∗} in G/Q = gr\overline{𝔫}(\overline{S}) satisfies conditions (1) and (2) in Theorem 1.2. We have

H(gr_{𝔪}(R), λ) = \frac{H(gr\overline{𝔫}(\overline{S})/\overline{I}^{∗}, λ)}{(1 − λ)^{s−2}},

because X_{1}, X_{2}, ⋯, X_{s−2} form a regular sequence in gr_{𝔪}(R). The assertions in Theorem 1.6 readily follow from this.

Question 4.9. With notation as in Setting 1.1 and Remark 1.4, if I^{∗} is perfect, does it follow that I^{∗} ⊆ (ξ_{1}/D_{2}, ξ_{2}/D_{2})G?

5. Examples with μ_{G}(I^{∗}) = 3 and with given μ_{G}(I^{∗})

Let 0 < n_{1} < n_{2} < n_{3} be integers such that GCD(n_{1}, n_{2}, n_{3}) = 1 and let S = k[[X_{1}, X_{2}, X_{3}]] and T = k[[t]] be the formal power series rings over a field k. We denote by φ : S → T the k-algebra map defined by φ(X_{i}) = t^{n_{i}} for i = 1, 2, 3. Let

I = Kerφ, R = k[[t^{n_{1}}, t^{n_{2}}, t^{n_{3}}]], 𝔫 = (X_{1}, X_{2}, X_{3})S, and 𝔪 = (t^{n_{1}}, t^{n_{2}}, t^{n_{3}})R.

We then have the following, which is essentially due to J. Herzog [H2] (see p.191– 192) and L. Robbiano and G. Valla [RV]. Let us include a brief proof in our context for the sake of completeness.

Theorem 5.1. Suppose that μ_{S}(I) = 2, namely, R is a Gorenstein ring. Then gr_{𝔪}(R) is a Cohen-Macaulay ring if and only if the leading form ideal I^{∗} of I is 3-generated.

Proof. See [GHK, Theorem 1.2] for the proof of the if part. Suppose now that gr_{𝔪}(R) is a Cohen-Macaulay ring. Let G = gr_{𝔫}(S), which we shall identify with the polynomial ring k[X_{1}, X_{2}, X_{3}] over k. We will show that μ_{G}(I^{∗}) ≤ 3. Since μ_{S}(I) = 2, as for the system of generators of I we distinguish the following four cases ([H1]):

(1) I = (X_{1}^{c_{1}} − X_{2}^{c_{2}}, X_{1}^{c_{1}} − X_{3}^{c_{3}}),

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

(2) I = (X_{2}^{c_{2}} − X_{3}^{c_{3}}, X_{1}^{c_{1}} − X_{2}^{s_{12}}X_{3}^{s_{13}}) (s_{12} > 0, s_{13} > 0),

(3) I = (X_{1}^{c_{1}} − X_{3}^{c_{3}}, X_{2}^{c_{2}} − X_{1}^{s_{21}}X_{3}^{s_{23}}) (s_{21} > 0, s_{23} > 0), and

(4) I = (X_{1}^{c_{1}} − X_{2}^{c_{2}}, X_{3}^{c_{3}} − X_{1}^{s_{31}}X_{2}^{s_{32}}) (s_{31} > 0, s_{32} > 0) where c_{i} = min{0 < c ∈ ℤ | 0 ≠ X_{i}^{c} −X_{1}^{α_{1}}X_{2}^{α_{2}}X_{3}^{α_{3}} ∈ I for some 0 ≤ α_{1}, α_{2}, α_{3} ∈ ℤ}. For cases (1), (3), and (4), the ideal I+(X_{1}) is generated by monomials in X_{1}, X_{2}, X_{3} and so, thanks to [H2, Theorem 1], we have μ_{G}(I^{∗}) = μ_{S}(I) = 2, once gr_{𝔪}(R) is a Cohen-Macaulay ring. We are now concentrated in case (2), where

I = (X_{2}^{c_{2}} − X_{3}^{c_{3}}, X_{1}^{c_{1}} − X_{2}^{s_{12}}X_{3}^{s_{13}})

for some integers s_{12} > 0 and s_{13} > 0. Then c_{1} = (n_{2}, n_{3}), n_{2} = c_{1}c_{3}, and n_{3} = c_{1}c_{2} ([H1]); hence c_{3} < c_{2}. We write s_{13} = c_{3}q + s′_{13} with integers q, s′_{13} such that 0 ≤ q, 0 ≤ s′_{13} < c_{3} and put s′_{12} = c_{2}q + s_{12}. Then

s′_{13} = 0 or c_{1} + c_{3} − s′_{13} ≥ c_{2} + s′_{12},

because gr_{𝔪}(R) is a Cohen-Macaulay ring (see [H2, p.192]). Let f = X_{2}^{c_{2}} − X_{3}^{c_{3}} and g = X_{1}^{c_{1}} − X_{2}^{s′_{12}}X_{3}^{s′_{13}}. Then I = (f, g), since g ≡ X_{1}^{c_{1}} − X_{2}^{s_{12}}X_{3}^{s_{13}} mod (f). If s′_{13} = 0, then g^{∗} = X_{1}^{c_{1}} if c_{1} < s′_{12}, g^{∗} = X_{1}^{c_{1}} − X_{2}^{s′_{12}} if c_{1} = s′_{12}, and g^{∗} = −X_{2}^{s′_{12}} if c_{1} > s′_{12}. Since f^{∗} = −X_{3}^{c_{3}} (recall that c_{3} < c_{2}), in any case the forms f^{∗}, g^{∗} constitute a regular sequence in G, so that we have I^{∗} = (f^{∗}, g^{∗}).

Assume that s′_{13} > 0. Then g^{∗} = −X_{2}^{s′_{12}}X_{3}^{s′_{13}}, since c_{1} − (s′_{12} + s′_{13}) ≥ c_{2} − c_{3} > 0. We put h := X_{2}^{s′_{12}}f + X_{3}^{c_{3}−s′_{13}}g = X_{1}^{c_{1}}X_{3}^{c_{3}−s′_{13}} − X_{2}^{c_{2}+s′_{12}}. Let J = (f^{∗}, g^{∗}, h^{∗}) ⊆ I^{∗}. Then

J = (X_{2}^{c_{2}+s′_{12}}, X_{2}^{s′_{12}}X_{3}^{s′_{13}}, X_{3}^{c_{3}}) = I_{2} (\begin{matrix}0 & X_{2}^{c_{2}} & X_{3}^{s′_{13}} \\ X_{2}^{s′_{12}} & X_{3}^{c_{3}−s′_{13}} & 0 \end{matrix})

(resp. J = (X_{1}^{c_{1}}X_{3}^{c_{3}−s′_{13}} − X_{2}^{c_{2}+s′_{12}}, X_{2}^{s′_{12}}X_{3}^{s′_{13}}, X_{3}^{c_{3}}) = I_{2} (\begin{matrix}X_{1}^{c_{1}} & X_{2}^{c_{2}} & X_{3}^{s′_{13}} \\ X_{2}^{s′_{12}} & X_{3}^{c_{3}−s′_{13}} & 0 \end{matrix}))

if c_{1} + c_{3} − s′_{13} > c_{2} + s′_{12} (resp. c_{1} + c_{3} − s′_{13} = c_{2} + s′_{12}). We now want to show I^{∗} = J. For this purpose we firstly look at the exact sequence

0 → I^{∗}/J → G/J → gr_{𝔪}(R) → 0.

Then, since a = t^{n_{1}} is a minimal reduction of the ideal 𝔪, the element X_{1} ∈ G acts on the Cohen-Macaulay ring gr_{𝔪}(R) as a non-zerodivisor, whence we have the exact sequence

0 → (I^{∗}/J)/X_{1}(I^{∗}/J) → G/[(X_{1}) + J] →^{ε} gr_{𝔪 /(a)}(R/(a)) → 0.

THE LEADING IDEAL OF A COMPLETE INTERSECTION 23

Therefore, to show I^{∗} = J, by Nakayama’s lemma it is enough to check that ε is an isomorphism, or equivalently, to check that

dim_{k} G/[(X_{1}) + J] ≤ dim_{k} gr_{𝔪 /(a)}(R/(a)).

We have

dim_{k} gr_{𝔪 /(a)}(R/(a)) = ℓ_{R}(R/(a)) = e_{𝔪}^{0}(R) = n_{1}

and n_{1} = c_{3}s′_{12} + c_{2}s′_{13} (recall that n_{1}c_{1} = n_{2}s′_{12} + n_{3}s′_{13}, n_{2} = c_{1}c_{3}, and n_{3} = c_{1}c_{2}). On the other hand, since

G/[(X_{1}) + J] ≅ k[X_{2}, X_{3}]/(X_{2}^{c_{2}+s′_{12}}, X_{2}^{s′_{12}}X_{3}^{s′_{13}}, X_{3}^{c_{3}}),

we readily get dim_{k} G/[(X_{1}) + J] ≤ c_{3}(c_{2} + s′_{12}) − c_{2}(c_{3} − s′_{13}) = c_{3}s′_{12} + c_{2}s′_{13} = n_{1}. Hence I^{∗} = J so that we have μ_{G}(I^{∗}) = 3 as claimed. □

Corollary 5.2 (to the proof). Assume that μ_{S}(I) = 2. Then μ_{G}(I^{∗}) = 3 if and only if there exist integers α, β ∈ ℤ such that 0 < α, 0 < β < c_{3}, c_{1} + c_{3} ≥ c_{2} + (α + β), and I = (X_{2}^{c_{2}} − X_{3}^{c_{3}}, X_{1}^{c_{1}} − X_{2}^{α}X_{3}^{β}). When this is the case, we have c_{1} = GCD(n_{2}, n_{3}), n_{2} = c_{1}c_{3}, n_{3} = c_{1}c_{2}, n_{1} = c_{3}α + c_{2}β, and the leading form ideal I^{∗} of I is given by

I^{∗} = I_{2} (\begin{matrix}0 & X_{2}^{c_{2}} & X_{3}^{β} \\ X_{2}^{α} & X_{3}^{c_{3}−β} & 0 \end{matrix}) (resp. I^{∗} = I_{2} (\begin{matrix}X_{1}^{c_{1}} & X_{2}^{c_{2}} & X_{3}^{β} \\ X_{2}^{α} & X_{3}^{c_{3}−β} & 0 \end{matrix}))

if c_{1} + c_{3} > c_{2} + (α + β) (resp. c_{1} + c_{3} = c_{2} + (α + β) ).

Remark 5.3. This result classifies Gorenstein numerical semigroups H = ⟨n_{1}, n_{2}, n_{3}⟩ generated by 3 integers n′_{i}s with 0 < n_{1} < n_{2} < n_{3} and GCD(n_{1}, n_{2}, n_{3}) = 1, for which the associated graded rings gr_{𝔪}(R) (R = k[[t^{n_{1}}, t^{n_{2}}, t^{n_{3}}]], k a field) are non-Gorenstein Cohen-Macaulay rings. In fact, firstly we choose integers c_{2}, c_{3} so that 2 ≤ c_{3} < c_{2} and GCD(c_{2}, c_{3}) = 1. Let α, β be integers such that 0 < α, 0 < β < c_{3} and put n_{1} = c_{3}α + c_{2}β. We choose an integer c_{1} so that c_{1} > \frac{n_{1}}{c_{3}} , GCD(n_{1}, c_{1}) = 1, and c_{1} + c_{3} ≥ c_{2} + (α + β). Lastly let n_{2} = c_{1}c_{3} and n_{3} = c_{1}c_{2}. Then for H = ⟨n_{1}, n_{2}, n_{3}⟩ we easily get the equality

I = (X_{2}^{c_{2}} − X_{3}^{c_{3}}, X_{1}^{c_{1}} − X_{2}^{α}X_{3}^{β})

and c_{i} = min{0 < c ∈ ℤ | 0 ≠ X_{i}^{c} − X_{1}^{α_{1}}X_{2}^{α_{2}}X_{3}^{α_{3}} ∈ I for some 0 ≤ α_{1}, α_{2}, α_{3} ∈ ℤ} as well for each i = 1, 2, 3. Hence by Corollary 5.2 the ring gr_{𝔪}(R) is a non-Gorenstein Cohen-Macaulay ring. Let f = X_{2}^{c_{2}} − X_{3}^{c_{3}}, g = X_{1}^{c_{1}} − X_{2}^{α}X_{3}^{β}, and h = X_{1}^{c_{1}}X_{3}^{c_{3}−β} − X_{2}^{c_{2}+α} (= X_{2}^{α}f + X_{3}^{c_{3}−β}g). Then, since c_{2} > c_{3} and c_{1} − (α + β) ≥ c_{2} − c_{3} > 0, we have f^{∗} = −X_{3}^{c_{3}} and g^{∗} = −X_{2}^{α}X_{3}^{β} whence GCD(f^{∗}, g^{∗}) = X_{3}^{β},

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

while h^{∗} = −X_{2}^{c_{2}+α} (resp. h^{∗} = X_{1}^{c_{1}}X_{3}^{c_{3}−β} − X_{2}^{c_{2}+α}) if c_{1} + c_{3} > c_{2} + (α + β) (resp. c_{1} + c_{3} = c_{2} + (α + β)), which is the third generator of I^{∗}. Hence

H(gr_{𝔪}(R), λ) = \frac{(1 − λ^{c_{2}+α})(1 − λ^{β}) + λ^{β}(1 − λ^{c_{3}−β})(1 − λ^{α})}{(1 − λ)^{3}}

and e_{𝔪}^{0}(R) = c_{3}(α + β) + β[{(c_{2} + α) + β} − {c_{3} + (α + β)}] = c_{3}α + c_{2}β = n_{1} by Theorem 1.5.

Let us note more concrete examples.

Example 5.4. (1) Let q ≥ 0 be an integer and put n_{1} = 6q + 5, n_{2} = 2(3q + 4), and n_{3} = 3(3q + 4). Then, letting c_{2} = 3, c_{3} = 2, α = 3q + 1, β = 1, and c_{1} = 3q + 4, by Corollary 5.2 and Remark 5.3 we get I^{∗} = I_{2} (\begin{matrix}0 & X_{2}^{3} & X_{3} \\ X_{2}^{3q+1} & X_{3} & 0 \end{matrix}). If we take q = 0, then n_{1} = 5, n_{2} = 8, n_{3} = 12.

(2) Similarly, let q ≥ 0 be an integer and put n_{1} = 6q + 5, n_{2} = 2(3q + 3), and n_{3} = 3(3q + 3). Then, letting c_{2} = 3, c_{3} = 2, α = 3q + 1, β = 1, and c_{1} = 3q + 3, by Corollary 5.2 and Remark 5.3 we get I^{∗} = I_{2} (\begin{matrix}X_{1}^{3q+3} & X_{2}^{3} & X_{3} \\ X_{2}^{3q+1} & X_{3} & 0 \end{matrix}). If we take q = 0, then n_{1} = 5, n_{2} = 6, n_{3} = 9, which is [GHK, Example 1.5].

We close this section with an example due to Takahumi Shibuta (Kyusyu University). His example shows that, unless gr_{𝔪}(R) is Cohen-Macaulay, we do not necessarily have the descending sequence

a = d_{1} > d_{2} > ⋯ > d_{n−1} > d_{n} = 0

of degrees of GCD’s of ξ′_{i}s even for a minimal homogeneous system {ξ_{i}}_{1≤i≤n} of generators of I^{∗} which satisfies the conditions in Theorem 4.8.

Example 5.5. Let 2 ≤ m ∈ ℤ and put n_{1} = 3m, n_{2} = 3m + 1, and n_{3} = 6m + 3. Then I = (X_{1}^{2m+1} − X_{3}^{m}, X_{2}^{3} − X_{1}X_{3}) in S and I^{∗} = (X_{1}X_{3}) + (X_{2}^{3i}X_{3}^{m−i} | 0 ≤ i ≤ m) in G = k[X_{1}, X_{2}, X_{3}] with μ_{G}(I^{∗}) = m + 2. Letting ξ_{1} = X_{1}X_{3} and ξ_{i} = X_{2}^{3(i−2)}X_{3}^{m−i+2} for 2 ≤ i ≤ m + 2, we see that the minimal homogeneous system {ξ_{i}}_{1≤i≤m+2} of generators of I^{∗} satisfies the conditions in Theorem 4.8, while GCD(ξ_{1}, ξ_{2}, ⋯, ξ_{i}) = X_{3} for 2 ≤ i ≤ m + 1. We have

H(gr_{𝔪}(R), λ) = \frac{∑_{i=2}^{m} λ^{m−i+1} − ∑_{i=2}^{m} λ^{m+2i−2} + ∑_{i=0}^{3m−1} λ^{i}}{1 − λ} .

Proof. It is routine to check that I = (X_{1}^{2m+1} − X_{3}^{m}, X_{2}^{3} − X_{1}X_{3})S. Hence we have X_{1}X_{3}, X_{3}^{m} ∈ I^{∗}. Let h_{i} = X_{1}^{2m+i+1} − X_{2}^{3i}X_{3}^{m−i} for 1 ≤ i ≤ m. We put J = (X_{1}X_{3}) + (X_{2}^{3i}X_{3}^{m−i} | 0 ≤ i ≤ m) in G. Then h_{i} ∈ I for all 1 ≤ i ≤ m,

THE LEADING IDEAL OF A COMPLETE INTERSECTION 25

whence J ⊆ I^{∗}. Let K = (X_{1}) + (X_{2}^{3i}X_{3}^{m−i−1} | 0 ≤ i ≤ m − 1) in G. Then \sqrt{K} = G_{+} = (X_{1}, X_{2}, X_{3}), J :_{G} X_{3} = K, and (X_{3})+J = (X_{2}^{3m}, X_{3}). Consequently, H_{N}^{0}(G/J) = (\overline{X_{3}}), where \overline{X_{3}} is the image of X_{3} in G/J and H_{N}^{0}(G/J) denotes the 0\underline{th} local cohomology module of G/J with respect to N = G_{+}. Hence

(0) :_{G/J} N = ∑_{i=1}^{m−1} k\overline{X_{2}^{3i−1}X_{3}^{m−i}},

because (\overline{X_{3}}) ≅ [G/K](−1) and the k-vector space (0) :_{G/K} N is spanned by the images of {X_{2}^{3i−1}X_{3}^{m−i−1}}_{1≤i≤m−1}.

Let \overline{θ} : G/[J + (X_{3})] → gr_{𝔪}(R)/H_{N}^{0}(gr_{𝔪}(R)) be the epimorphism induced from the canonical epimorphism G → gr_{𝔪}(R). Recall that X_{1} is a parameter for the ring gr_{𝔪}(R), since t^{3m} is a minimal reduction of 𝔪, so that X_{1} is a non-zerodivisor in the Cohen-Macaulay ring \overline{gr_{𝔪}(R)} = gr_{𝔪}(R)/H_{N}^{0}(gr_{𝔪}(R)). Hence \overline{θ} is an isomorphism, because dim_{k} G/[J + (X_{1}, X_{3})] = 3m and

dim_{k} gr_{𝔪}(R)/[H_{N}^{0}(gr_{𝔪}(R)) + X_{1}gr_{𝔪}(R)] = e_{X_{1}gr_{𝔪}(R)}^{0}(\overline{gr_{𝔪}(R)})
= e_{X_{1}gr_{𝔪}(R)}^{0}(gr_{𝔪}(R))
= e_{𝔪}^{0}(R)
= 3m.

Therefore, the kernel of the epimorphism θ : G/J → gr_{𝔪}(R) induced from the canonical epimorphism G → gr_{𝔪}(R) is contained in (\overline{X_{3}}) = H_{N}^{0}(G/J) and so, to see that θ is an isomorphism, it suffices to show that θ is injective on the socle

(0) :_{G/J} N = ∑_{i=1}^{m−1} k\overline{X_{2}^{3i−1}X_{3}^{m−i}}

of G/J, that is, it is enough to show θ(\overline{X_{2}^{3i−1}X_{3}^{m−i}}) ≠ 0 in gr_{𝔪}(R) for any 1 ≤ i ≤ m − 1, because the degrees of \overline{X_{2}^{3i−1}X_{3}^{m−i}} are distinct.

Let x = t^{n_{1}}, y = t^{n_{2}}, and z = t^{n_{3}}. We put U = k[x, y, z] in R. Hence U is a graded ring with deg x = n_{1}, deg y = n_{2}, and deg z = n_{3}. Let M = U_{+} = (x, y, z)U. We denote by U_{i} the homogeneous component of U of degree i. In what follows we will show that y^{3i−1}z^{m−i} ∉ 𝔪^{m+2i} for any 1 ≤ i ≤ m − 1. Assume that y^{3i−1}z^{m−i} ∈ 𝔪^{m+2i}, or equivalently, assume that y^{3i−1}z^{m−i} ∈ M^{m+2i} for some 1 ≤ i ≤ m − 1. Then we have the following.

Claim 5.6. y^{3i−1}z^{m−i} ∈ M^{m+2i+ℓ} for all 0 ≤ ℓ ≤ m − i.

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

P roof of Claim 5.6. When ℓ = 0, we have nothing to prove. Assume that 0 ≤ ℓ < m − i and that our assertion holds true for ℓ. We put δ = (3i − 1)n_{2} + (m − i)n_{3} = 6m^{2} + 3mi − 1. Then t^{δ} = y^{3i−1}z^{m−i} ∈ M^{m+2i+ℓ} = ∑_{α=0}^{m+2i+ℓ}(x, y)^{m+2i+ℓ−α}·z^{α} in U. Take 0 ≤ α ∈ ℤ and assume that m − i − ℓ ≤ α ≤ m + 2i + ℓ. Then

(x, y)^{m+2i+ℓ−α}·z^{α} = (x^{β}y^{γ}z^{α} | 0 ≤ β, γ ∈ ℤ such that β + γ = m + 2i + ℓ − α). We now choose 0 ≤ β, γ ∈ ℤ so that β+γ = m+2i+ℓ−α and put η = βn_{1}+γn_{2}+αn_{3}. Then

η ≥ (β + γ)n_{1} + αn_{3}
= 3(m + 2i + ℓ − α)m + α(6m + 3)
= 3m^{2} + 6mi + 3mℓ + 3mα + 3α
≥ 6m^{2} + 3mi + 3(m − i − ℓ) (since α ≥ m − i − ℓ)
≥ 6m^{2} + 3mi
> δ = 6m^{2} + 3mi − 1.

Consequently, t^{δ} ∈ ∑_{α=0}^{m−i−ℓ−1}(x, y)^{m+2i+ℓ−α}·z^{α}. We write t^{δ} = ∑_{α=0}^{m−i−ℓ−1} φ_{α}z^{α} with φ_{α} ∈ (x, y)^{m+2i+ℓ−α} such that φ_{α} ∈ U_{δ−αn_{3}}. Let us furthermore write φ_{α} = ∑_{β=0}^{m+2i+ℓ−α} w_{α,β}·x^{β}y^{m+2i+ℓ−α−β} with w_{α,β} ∈ U_{δ−αn_{3}−(βn_{1}+(m+2i+ℓ−α−β)n_{2}}. Then, if 0 ≤ α ≤ m − i − ℓ − 1, choosing 0 ≤ β, γ ∈ ℤ with β + γ = m + 2i + ℓ − α, we have

βn_{1} + γn_{2} + αn_{3} = 3mβ + (3m + 1)γ + αn_{3}
≤ (3m + 1)(β + γ) + α(6m + 3)
= 3m^{2} + m + 6mi + 2i + 3ℓm + ℓ + 3mα + 2α
≤ 6m^{2} + 3mi − ℓ − 2 (since α ≤ m − i − ℓ − 1)
< δ = 6m^{2} + 3mi − 1,

whence βn_{1} + γn_{2} < δ − αn_{3}. Consequently, w_{α,β} ∈ M for each α and β, so that φ_{α} ∈ M^{m+2i+ℓ−α+1} for all 0 ≤ α ≤ m − i − ℓ − 1, whence t^{δ} ∈ M^{m+2i+ℓ+1} as claimed. □

Therefore t^{δ} ∈ M^{2m+i}, which is however impossible, because

βn_{1} + γn_{2} + τn_{3} ≥ (β + γ + τ)n_{1} = (2m + i)·3m
= 6m^{2} + 3mi > δ

for all 0 ≤ β, γ, τ ∈ ℤ with β + γ + τ = 2m + i. Thus the epimorphism θ : G/J → gr_{𝔪}(R) is injective on the socle of G/J, so that θ is an isomorphism. Hence I^{∗} = J.

Because H_{N}^{0}(G/I^{∗}) = (\overline{X_{3}}) ≅ (G/K)(−1), thanks to the exact sequence

0 → (G/K)(−1) → G/I^{∗} → G/(X_{2}^{3m}, X_{3}) → 0

of graded G-modules, we have

H(G/I^{∗}, λ) = λ·H(G/K, λ) + \frac{1 − λ^{3m}}{(1 − λ)^{2}} .

Therefore

H(gr_{𝔪}(R), λ) = \frac{∑_{i=2}^{m} λ^{m−i+1}(1 − λ^{3(i−1)})}{1 − λ} + \frac{1 − λ^{3m}}{(1 − λ)^{2}}

= \frac{∑_{i=2}^{m} λ^{m−i+1} − ∑_{i=2}^{m} λ^{m+2i−2} + ∑_{i=0}^{3m−1} λ^{i}}{1 − λ}

by Proposition 2.6, since G/K = k[X_{2}, X_{3}]/(X_{2}^{3i}X_{3}^{m−i−1} | 0 ≤ i ≤ m − 1).

□

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[H2] J. Herzog, When is a regular sequence super regular?, Nagoya Math. J., 83 (1981), 183–195.

[Ia1] A. Iarrobino, Tangent cone of a Gorenstein singularity, in Proceedings of the conference in Algebraic Geometry, Berlin 1985, H. Kurke and M. Roczen, eds., Teubnertexte zur Math. 92, Teubner, Leipzig, 1986, 163-176.

[Ia2] A. Iarrobino, Associated Graded Algebra of a Gorenstein Artin Algebra, Memoirs Amer. Math. Soc., 107, num. 514, 1994.

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

[M] F. S. Macaulay, On a method of dealing with the intersection of plane curves, Trans. Amer. Math. Soc., 5 (1904), 385-410.

[RV] L. Robbiano and G. Valla, On the equations defining tangent cones, Math. Proc. Camb. Phil. Soc., 88 (1980), 281-297.

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@skku.edu