Cohen-Macaulay Fiber Cones

Clare D’Cruz, K. N. Raghavan and J. K. Verma Dedicated to the memory of Professor Wolfgang Vogel

1 Introduction

Let (R, m) be a local ring and let I be an ideal of R. The fiber cone of I is the graded ring F (I) := ⊕_{n≥0}I^{n}/mI^{n}. In this paper we will give a necessary and sufficient condition, in terms of Hilbert series of F (I), for it to be Cohen-Macaulay (CM). We apply this criterion to a variety of examples.

Cohen-Macaulay fiber cones have been studied by K. Shah [Sh] and T. Cortadellas and S. Zarzuela [CZ]. Shah characterized Cohen-Macaulay fiber cones terms of their Hilbert function. To recall his result let us introduce some notation and terminology. An ideal J ⊆ I is called a reduction of I if JI^{n} = I^{n+1} for some n. We say J is a minimal reduction of I if whenever K ⊆ I and K is a reduction of I, then K = J. These concepts were introduced by Northcott and Rees in [NR]. They showed that when R/m is infinite, any minimal reduction of I is minimally generated by dim F (I) elements. The Krull dimension of the fiber cone F (I) is an important invariant of I and it is called the analytic spread of I. We denote it by a(I). It is known [R2] that ht I ≤ a(I) ≤ min(μ(I), dim R) where μ(I) = dim I/mI. Let ℓ(M) denote the length of a module M. We can now state the necessary condition found by Shah for F (I) to be CM.

Theorem 1.1 (Shah) Let (R, m) be a local ring and let I be an ideal in R of analytic spread a. Let J be a minimal reduction of I. If F (I) is CM, then for all n ≥ 0

μ(I^{n}) = ∑_{i≥0} [μ(I^{i}) − ℓ (\frac{JI^{i−1}}{JI^{i−1} ∩ mI^{i}} )] \binom{n + a − i − 1}{a − 1}. (∗)

K. Shah conjectured that the converse is also true. Once we rewrite the equation (*) in terms of Hilbert series of F (I) it becomes clear why the above theorem and it’s converse are easy to see. We will use this theorem and it’s converse quite frequently. Since we do not find any reference for the Theorem 1.2 below we will provide a proof for the sake of completeness. We will denote the multiplicity of F (I) by e(F (I)). We know that if F (I) is CM then e(F (I)) ≥ μ(I) − a + 1 [A]. When equality holds, we say that F (I) has minimal multiplicity. The reduction number of I with respect to a minimal reduction J of I is defined as

r_{J}(I) = min{n ≥ 0 | JI^{n} = I^{n+1}}.

The reduction number of I denoted by r(I) is defined to be the minimum of r_{J}(I) where J varies over all minimal reductions of I.

Theorem 1.2 Let (R, m) be a local ring and let I be any ideal of R. Let J be any minimal reduction of I. Then the following are equivalent:

1. F (I) is CM.

2. H(F (I), t) = \frac{1}{(1 − t)^{a}} ∑_{i=0}^{r} ℓ (\frac{I^{i}}{JI^{i−1} + mI^{i}} ) t^{i} where r = r_{J}(I) and a = dim F (I).

3. e(F (I)) = ∑_{n=0}^{r} ℓ (\frac{I^{n}}{JI^{n−1} + mI^{n}} ).

Proof: (1) ⇒ (2). Suppose that F (I) is CM. Since JF (I) is generated by a homogeneous system of parameters, it is generated by a regular sequence. Thus

H(F (I), t) = \frac{1}{(1 − t)^{a}} H (\frac{F (I)}{JF (I)}, t) .

Notice that

JF (I) = ⨁_{i=1}^{r} (\frac{JI^{i−1} + mI^{i}}{mI^{i}} ) ⨁ [ ⨁_{i=r+1}^{∞} (\frac{I^{i}}{mI^{i}} )] .

Hence

\frac{F (I)}{JF (I)} = \frac{R}{m} ⨁ [⨁_{i=1}^{r} (\frac{I^{i}}{JI^{i−1} + mI^{i}} )] .

This gives

H (\frac{F (I)}{JF (I)}, t) = ∑_{i=0}^{r} ℓ (\frac{I^{i}}{JI^{i−1} + mI^{i}} ) t^{i}.

which gives (2).

(2) ⇒ (3). Put t = 1 in the numerator of H(F (I), t).

(3) ⇒ (1). Let M denote the maximal homogeneous ideal of F (I). It is enough to show that F (I)_{M} is CM by [MR]. Since F (I) is homogeneous graded ring, the associated graded ring gr(MF (I)_{M}, F (I)_{M}) ≅ F (I). Thus e(F (I)) = e(F (I)_{M}). It is also clear that ℓ(F (I)/JF (I)) = ℓ(F (I)_{M}/JF (I)_{M}). Thus F (I)_{M} is CM and hence F (I) is so.

Remark 1.3 Let (R, m) be a local ring and let I be any ideal of R. If F (I) is Cohen-Macaulay , then by theorem 1.2, for any minimal reduction J of I, r_{J}(I) is the degree of the numerator of the Hilbert series of F (I) in the reduced form. Hence the reduction number r_{J}(I) is independent of the choice of the minimal reduction J of I. This observation is also proved, by different means, by Huckaba and Marley in [HM].

We now describe the contents of later sections briefly. In section two we define ideals of minimal mixed multiplicity in CM local rings and calculate the Hilbert series of their fiber cone. It turns out that for such ideals the fiber cone is CM if and only if r(I) ≤ 1. By using this result we construct zero dimensional ideals in a two dimensional regular local ring whose fiber cones are not CM. Section four is about fiber cones of ideals generated by quadratic sequences.

The first author wishes to thank the National Board of Higher Mathematics for financial support and the Indian Institute of Technology, Bombay, where the research on this paper was carried out. The second named author wishes to acknowledge financial help in the form a grant from the Department of Science and Technology of the Government of India to the SPIC Mathematical Institute for purchase of computing equipment on which calculations using Macaulay could be performed. The last named author wishes to acknowledge the hospitality of the SPIC Mathematical Institute during a visit on which part of the work on this paper was done.

2 Fiber cones of ideals with minimal mixed multiplicity

Let (R, m) be a local ring. Define the function H : IN^{g} → IN for g m-primary ideals I_{1}, …, I_{g} as follows:

H(r_{1}, …, r_{g}) = ℓ (\frac{R}{I_{1}^{r_{1}} ⋯ I_{g}^{r_{g}}} ) .

It was proved in [T] that H(r_{1}, …, r_{g}) is given by a polynomial P (r_{1}, …, r_{g}) with rational coefficients for all large r_{1}, …, r_{g}. The total degree of P is d and the terms of degree d in P are captured in the expression

∑_{i_{1}+···+i_{g} = d} e(I_{1}^{[i_{1}]}| ⋯ |I_{g}^{[i_{g}]})\binom{r_{1} + i_{1}}{i_{1}} ⋯ \binom{r_{g} + i_{g}}{i_{g}}

for certain positive integers e(I_{1}^{[i_{1}]}| ⋯ |I_{g}^{[i_{g}]}). These positive integers are called mixed multiplicities of I_{1}, …, I_{g}. When g = 2 we adopt the notation

e_{i}(I_{1}|I_{2}) = e(I_{1}^{[d−i]}|I_{2}^{[i]}).

Rees [R1] showed that e_{0}(I_{1}|I_{2}) = e(I_{1}) and e_{d}(I_{1}|I_{2}) = e(I_{2}). Other mixed multiplicities can also be interpreted as multiplicities of certain parameter ideals. Teissier and Risler [T] provided an interpretation of mixed multiplicities in terms of superficial elements. Rees [R3] introduced joint reductions to study them. A set of elements x_{1}, …, x_{d} is called a joint reduction of I_{1}, …, I_{d} if x_{i} ∈ I_{i} for i = 1, …, d and there exists a positive integer n so that

[∑_{j=1}^{d} x_{j} I_{1} ⋯ Î_{j} ⋯ I_{d}] (I_{1} ⋯ I_{d})^{n−1} = (I_{1} ⋯ I_{d})^{n}.

Rees proved that if R/m is infinite, then joint reductions exist and

e(I_{1}^{[i_{1}]}| ⋯ |I_{g}^{[i_{g}]}) = e(x_{1}, …, x_{d})

where (x_{1}, …, x_{d}) is any joint reduction of the set of ideals I_{1}, …, I_{1}, …, I_{g}, …, I_{g} where I_{j} is repeated i_{j} times for j = 1, …, g..

Abhyankar [A] showed that for a CM local ring (R, m), μ(m) − dim R + 1 ≤ e(R). We extend this to m-primary ideals.

Lemma 2.1 Let (R, m) be a CM local ring of dimension d. Let I be an m-primary ideal. Then

e_{d−1}(m|I) ≥ μ(I) − d + 1.

Proof: By passing to the ring R(X) = R[X]_{mR[X]} we may assume that R/m is infinite. Let (x, a_{1}, …, a_{d−1}) be a joint reduction of m, I, …, I where I is repeated d − 1 times. Consider the map

f : \frac{R}{I} ⨁ (\frac{R}{m})^{d−1} ⟶ \frac{(x, a_{1}, …, a_{d−1})}{xI + (a_{1}, …, a_{d−1})m}

defined as f(a′, b′_{1}, …, b′_{d−1}) = (xa + a_{1}b_{1} + ⋯ + a_{d−1}b_{d−1})′ where primes denote the residue classes. Let K = Kernel f. Then

ℓ(K) + ℓ (\frac{R}{xI + (a_{1}, …, a_{d−1})m}) − e_{d−1}(m|I) = ℓ (\frac{R}{I}) + d − 1.

Hence

e_{d−1}(m|I) = ℓ(K) + ℓ (\frac{R}{Im}) − ℓ (\frac{R}{I}) − d + 1 − ℓ (\frac{R}{Im}) + ℓ (\frac{R}{xI + (a_{1}, …, a_{d−1})m})
= ℓ(K) + μ(I) − d + 1 + ℓ (\frac{Im}{xI + (a_{1}, …, a_{d−1})m}) .

Hence

e_{d−1}(m|I) ≥ μ(I) − d + 1.

Definition 2.2 Let (R, m) be a CM local ring, I an m-primary ideal of R. We say that I has minimal mixed multiplicity if

e_{d−1}(m|I) = μ(I) − d + 1.

We now find the Hilbert series of F (I) when I is an m-primary ideal with minimal mixed multiplicity in a d-dimensional CM local ring. We begin with one dimensional local rings.

Proposition 2.3 Let (R, m) be a CM local ring of dimension one with μ(I) = e(R) := e for an m-primary ideal I of R. Then

H(F (I), t) = \frac{1 + (e − 1)t}{1 − t}.

Proof: By passing to the ring R(X) we may assume that R/m is infinite. Let xR be a minimal reduction of m. Then for any n ≥ 1,

e(R) = ℓ (\frac{R}{xR}) = ℓ (\frac{R}{xR}) + ℓ (\frac{xR}{xI^{n}} ) − ℓ (\frac{R}{I^{n}} ) = ℓ (\frac{I^{n}}{xI^{n}} ) .

Since ℓ(I/mI) = e(R) = ℓ(I/xI), it follows that mI = xI. Thus xI^{n} = mI^{n} which gives that μ(I^{n}) = e(R) for all n ≥ 1. Hence

H(F (I), t) = 1 + ∑_{n=1}^{∞} e t^{n} = 1 − e + \frac{e}{1 − t} = \frac{1 + (e − 1)t}{1 − t}.

Theorem 2.4 Let (R, m) CM local ring of dimension d ≥ 2. Let I be an m-primary ideal having minimal mixed multiplicity. Then

H(F (I), t) = \frac{1 + (μ(I) − d)t}{(1 − t)^{d}}

Proof: By passing to the ring R(X) we assume that R/m is infinite. Let x, a_{1}, …, a_{d−1} be a joint reduction of (m, I, …, I) where I is repeated d − 1 times. Then e_{d−1}(m|I) = ℓ(R/(x, a_{1}, …, a_{d−1})) and μ(I) = d − 1 + ℓ(R/(x, a_{1}, …, a_{d−1})). We will show that for all n ≥ 1,

μ(I^{n}) = \binom{n + d − 2}{d − 2} + \binom{n + d − 2}{d − 1}e_{d−1}(m|I).

Fix n. Put r = \binom{n+d−2}{d−2}. Let M_{1}, …, M_{r} denote all the monomials of degree n in a_{1}, …, a_{d−1}. Consider the map

f : \frac{R}{I^{n}} ⊕ (\frac{R}{m})^{r} ⟶ \frac{(x, (a_{1}, …, a_{d−1})^{n})}{xI^{n} + (a_{1}, …, a_{d−1})^{n}m}

given by f(y′, z′_{1}, …, z′_{r}) = (yx + ∑_{i=1}^{r} z_{i}M_{i})′ where primes denote the residue classes. If f(y′, z′_{1}, …, z′_{r}) = 0, then yx + ∑_{i=1}^{r} z_{i}M_{i} ∈ xI^{n} + (a_{1}, …, a_{d−1})^{n}m. Hence there exist b ∈ I^{n} and c_{1}, …, c_{r} ∈ m such that

yx + ∑_{i=1}^{r} z_{i}M_{i} = xb + ∑_{i=1}^{r} c_{i}M_{i}.  (1)

Thus x(y − b) = ∑_{i=1}^{r} M_{i}(c_{i} − z_{i}). Since (x, a_{1}, …, a_{d−1}) is a regular sequence, y − b ∈ (M_{1}, …, M_{r}) = (a_{1}, …, a_{d−1})^{n}. Hence y = b + ∑_{i=1}^{r} d_{i}M_{i} for some d_{1}, …, d_{r} ∈ R. Substituting for y in (1), we get ∑_{i=1}^{r} M_{i}(z_{i} − c_{i} + xd_{i}) = 0. Since a_{1}, …, a_{d−1} is a regular sequence, z_{i} − c_{i} + xd_{i} ∈ (a_{1}, …, a_{d−1}) for all i = 1, …, r. Hence z_{i} ∈ m for all i = 1, …, r. Thus f is an isomorphism which implies that

ℓ (\frac{R}{xI^{n} + (a_{1}, …, a_{d−1})^{n}m}) − ℓ (\frac{R}{(x, (a_{1}, …, a_{d−1})^{n})}) = ℓ (\frac{R}{I^{n}} ) + r.

Put n = 1 to get

ℓ (\frac{R}{xI + (a_{1}, …, a_{d−1})m}) − ℓ (\frac{R}{I}) = r + e_{d−1}(m|I) = μ(I) = ℓ (\frac{R}{mI} ) − ℓ (\frac{R}{I}) .

Hence Im = xI + (a_{1}, …, a_{d−1})m. Thus for all n ≥ 1, I^{n}m = xI^{n} + (a_{1}, …, a_{d−1})^{n}m by an easy induction on n. Hence for all n ≥ 1,

μ(I^{n}) = r + \binom{n + d − 2}{d − 1}e_{d−1}(m|I).

Therefore

H(F (I), t) = ∑_{n=0}^{∞} [r + \binom{n + d − 2}{d − 1}e_{d−1}(m|I)] t^{n}
= ∑_{n=0}^{∞} [r(1 − e_{d−1}(m|I)) + \binom{n + d − 1}{d − 1}e_{d−1}(m|I)] t^{n}
= \frac{1 − e_{d−1}(m|I)}{(1 − t)^{d−1}} + \frac{e_{d−1}(m|I)}{(1 − t)^{d}}
= \frac{1 + [e_{d−1}(m|I) − 1]t}{(1 − t)^{d}}
= \frac{1 + [μ(I) − d]t}{(1 − t)^{d}}.

Corollary 2.5 Let I be an m-primary ideal in a CM local ring (R, m). Suppose I has minimal mixed multiplicity. Then F (I) is CM if and only if r(I) ≤ 1.

Proof: If r(I) ≤ 1, then F (I) is CM by [Sh]. If F (I) is CM, then r(I) = the degree of the numerator of H(F (I), t) ≤ 1 by the above theorem.

We now consider fiber rings of contracted ideals in two dimensional regular local rings. An ideal I of a two dimensional regular local ring (R, m) is called contracted if there is an x ∈ m \ m^{2} such that IR[m/x] ∩ R = I [ZS, App. 5]. The order of I, o(I) is by definition max{n | I ⊆ m^{n}}. I is contracted if and only if μ(I) = 1 + o(I) [HS, Theorem 2.1]. If I is contracted from S = R[m/x], then for all n, I^{n} is also contracted from S. Any m-primary integrally closed ideal is contracted [ZS, App. 5].

Corollary 2.6 Let (R, m) be a two dimensional regular local ring. Let I be an m-primary contracted ideal of order r. Then

H(F (I), t) = \frac{1 + (r − 1)t}{(1 − t)^{2}}.

Proof: By [V1, Theorem 4.1], e_{1}(m|I) = r = o(I). Since I is an ideal of minimal mixed multiplicity, we can apply Theorem 2.4.

In view of the above theorem 2.5, to produce a non CM fiber cone it is enough to find an ideal having minimal mixed multiplicity whose reduction number exceeds one. We do so in the next example.

Example 2.7 We construct contracted ideals of reduction number greater than one which have non CM fiber cones. Consider the ideals

I = (x^{r}, xy^{r+1}(x, y)^{r−2}, y^{2r+1}).

of k[[x, y]] where k is a field. Since μ(I) = r + 1 = 1 + o(I), I is a contracted ideal and by corollary 2.6,

H(F (I), t) = \frac{1 + (r − 1)t}{(1 − t)^{2}}.

If F (I) were CM, then by corollary 2.5, r(I) ≤ 1. We now show that r(I) > 1 for r ≥ 3. By [V2, Theorem 3.2], r(I) ≤ 1 if and only if e(I) = ℓ(R/I^{2}) − 2ℓ(R/I). Since I is integral over (x^{r}, y^{2r+1}), (x^{r}, y^{2r+1}) is a minimal reduction of I and e(I) = r(2r + 1). Now

I^{2} = (x^{2r}, x^{r+1}y^{r+1}(x, y)^{r−2}, x^{r}y^{2r+1}, x^{2}y^{3r+1}(x, y)^{r−3}, xy^{4r}, y^{4r+2}).

Therefore

ℓ (\frac{R}{I^{2}} ) = ∑_{i=0}^{r−2}(r + 1 + i) + 2r + 1 + ∑_{i=0}^{r−3}(3r + 1 + i) + 4r + 4r + 2
= (r + 1)(r − 1) + \binom{r − 1}{2} + 2r + 1 + (3r + 1)(r − 2) + \binom{r − 2}{2} + 8r + 2
= \binom{r − 1}{2} + \binom{r − 2}{2} + 4r^{2} + 5r

and

ℓ (\frac{R}{I}) = ∑_{i=0}^{r−2}(r + 1 + i) + 2r + 1
= (r + 1)(r − 1) + \binom{r − 1}{2} + 2r + 1
= \binom{r − 1}{2} + r^{2} + 2r.

Therefore if r ≥ 3, then

ℓ (\frac{R}{I^{2}} ) − 2ℓ (\frac{R}{I}) = 4r^{2} + 5r + \binom{r − 2}{2} − 2 [r^{2} + 2r + \binom{r − 1}{2}]
= 2r^{2} + r − (r − 2)
< 2r^{2} + 2r
= e(I)

Example 2.8 [cf. [HJLS, (6.3)],[HM, Corollary (3.5) and Example (3.9)] There exist examples of m-primary ideals in R = k[[x, y]] of arbitrary reduction number and whose fiber ring is CM. Let I = (x^{r}, x^{r−1}y, y^{r}), r ≥ 1. Then F (I) is CM for all r ≥ 1. Since I is generated by homogeneous polynomials of equal degree in R, F (I) ≅ k[x^{r}, x^{r−1}y, y^{r}]. It is easy to see that k[x^{r}, x^{r−1}y, y^{r}] ≅ k[t_{1}, t_{2}, t_{3}]/(t_{2}^{r} − t_{1}^{r−1}t_{3}). Hence F (I) is CM and its Hilbert series is

H(F (I), t) = \frac{1 + t + ⋯ + t^{r−1}}{(1 − t)^{2}} .

It follows from theorem 1.2 that the reduction number of I is independent of the reduction chosen and is equal to r − 1.

Example 2.9 Let R = k[[x, y, z]], I = (x^{3}, y^{3}, z^{3}, xy, xz, yz). This ideal was studied by Huneke and Lipman [L]. We claim that I has minimal mixed multiplicity. It is easy to see that

1. J = (x^{3} + yz, y^{3} + z^{3} + xz, xz + xy) is a minimal reduction of I and JI = I^{2}.

2. (yz, y + z, x) is a joint reduction of the set of ideals (I, m, m) and Im^{2} = yzm^{2} + (y + z)Im + xIm.

3. (yz, xy + xz, x + y + z) is a joint reduction of the set of ideals (I, I, m) and I^{2}m = yzIm + (xy + xz)Im + (x + y + z)I^{2}.

Hence

e(I) = e(J) = 11
e_{1}(m|I) = e(yz, y + z, x) = 2
e_{2}(m|I) = e(yz, xy + xz, x + y + z) = 4.

Since μ(I) = 6 = e_{2}(m|I) + 2, I has minimal mixed multiplicity. Hence by corollary 2.5, F (I) is CM.

3 Cohen-Macaulayness of fiber cones of ideals generated by quadratic sequences.

Let R be a standard graded ring over a field, let M denote the unique graded maximal ideal of R, and let I be an ideal generated by a homogeneous quadratic sequence. Under some technical assumptions on the quadratic sequence, we show that a certain deformation of the fiber cone F (I) := R[It]/MR[It] has a nice form. From this we can deduce the Cohen-Macaulayness of F (I) and also a formula for its Hilbert series. (The Hilbert series can also be calculated from the formula in [RV] for the bigraded Hilbert series of R[It]—just set X = 1 in that formula.) These results apply to the examples treated in [RV], namely, straightening-closed ideals in graded algebras with straightening law, Huckaba-Huneke ideals of analytic deviation 1 and 2, and Moralès-Simis ideals defining the homogeneous co-ordinate rings of certain projective space curves.

The main result of this section is stated in terms of stable linearizations of quadratic sequences. We recall the relevant definitions from [RS]. A subset Λ of a finite poset (Ω, ≤) is an ideal if

λ ∈ Λ, ω ∈ Ω, and ω ≤ λ  ⟹  ω ∈ Λ.

If Λ is an ideal of Ω and ω ∈ Ω\Λ is such that λ ∈ Λ for every λ ⪇ ω, then (Λ, ω) is a pair of Ω. Given a set {x_{ω} | ω ∈ Ω} of elements of a ring R and Λ ⊆ Ω, denote by X_{Λ} the ideal (x_{λ} | λ ∈ Λ) of R (X_{Λ} = 0 if Λ is empty) and by I the ideal X_{Ω} = (x_{ω} | ω ∈ Ω).

Definition 3.1 A set {x_{ω} | ω ∈ Ω} ⊆ R is a quadratic sequence if for every pair (Λ, ω) of Ω there exists an ideal Θ of Ω such that

1. (X_{Λ} : x_{ω}) ∩ I = X_{Θ}.

2. x_{ω}X_{Θ} ⊆ X_{Λ}I.

Such an ideal Θ is said to be associated to the pair (Λ, ω). This association need not be unique—the set {x_{ω} | ω ∈ Ω} of generators of I may not be unshortenable—but X_{Θ} is unique by 1.

Definition 3.2 A linearization of a poset Ω of cardinality n is a bijective map # : Ω ⟶ [1, n] := {1, …, n} such that ω ≤ ω′ ⟹ #(ω) ≤ #(ω′).

Let {x_{ω} | ω ∈ Ω} be a quadratic sequence and # : Ω ⟶ [1, n] be a fixed linearization. Identify Ω with [1, n] via #. Then ([1, j − 1], j) is a pair of Ω for every j ∈ [1, n]. Let

Θ_{j} = an ideal of Ω associated to ([1, j − 1], j)
I_{j} = ((x_{1}, …, x_{j−1}) : x_{j})
Δ = {(j, k) | 1 ≤ j ≤ k ≤ n, x_{j} · x_{k} ∈ (x_{1}, …, x_{j−1})}.

For k ∈ [1, n], set

Ψ_{k} = [1, k − 1] ⋃_{{j≤k \atop (j,k)∉Δ}} Θ_{j} and 𝔄_{k} = X_{Ψ_{k}}.

Note that Ψ_{k} is an ideal of Ω and that 𝔄_{k} is independent of the choices of Θ_{j}.

Definition 3.3 A linearization # : Ω ⟶ [1, n] of the indexing poset Ω of a quadratic sequence is stable if I_{k} = (𝔄_{k} : x_{k}) for every k, 1 ≤ k ≤ n.

We can now state the main result of this section:

Proposition 3.4 Let R be a standard graded algebra over a field k, that is, R = ⊕_{j≥0}R_{j} = R_{0}[R_{1}] with R_{0} = k. Let {x_{ω}|ω ∈ Ω} ⊆ R be a quadratic sequence consisting of homogeneous elements of R, let # : Ω → {1, …, n} be a stable linearization, and suppose that

deg(x_{1}) ≤ ⋯ ≤ deg(x_{n}).

Further assume that x_{1}, …, x_{n} form an unshortenable set of generators of the ideal I := (x_{1}, …, x_{n}), that is, I ≠ (x_{1}, …, x̂_{j}, …, x_{n}). Let M := ⊕_{j>0}R_{j} denote the irrelevant maximal ideal of R. Then

k[Δ] := k[T_{1}, …, T_{n}]/(T_{j}T_{k}|(j, k) ∈ Δ)

is a deformation of the fiber cone F (I) := R[It]/MR[It] of I. In particular (1) F (I) is Cohen-Macaulay if k[Δ] is so, and (2) F (I) has the same Hilbert series as k[Δ].

Proof. We borrow the notation and set-up of §1 of [RS] (see page 541). Let J denote the kernel of the presentation map from the polynomial ring A := R[T_{1}, …, T_{n}] onto the Rees ring R[It]. This map is defined by T_{i} ↦ x_{i}t. The presentation ideal of the fiber cone F (I) as a quotient of A is then (J, M). Let F be the filtration on A specified on page 541 of [RS]. This filtration F is a refinement of the (M, T_{1}, …, T_{n})-adic filtration on A. Theorem 1.4 of [RS] asserts that

(∗) gr_{F}(J) = (I_{1}T_{1}, …, I_{n}T_{n}, T_{j}T_{k}|(j, k) ∈ Δ),

where I_{j} := ((x_{1}, …, x_{j−1}) : x_{j}). So it is natural to expect the following:

(†) gr_{F}(J, M) = (M, T_{j}T_{k}|(j, k) ∈ Δ).

The theorem clearly follows from (†) and we prove (†) below.

Since M is homogeneous with respect to F and T_{j}T_{k} ∈ gr_{F}(J) from (*), it follows that gr_{F}(J, M) ⊇ (M, T_{j}T_{k}|(j, k) ∈ Δ). To prove the other inclusion, it suffices to prove the statement (‡) below. Fix notation as in 3.2 of [RS]. Let F be a form of degree d in J. Let Q be the subset of P (n, d) such that F = ∑_{p∈Q} a_{p}T^{p} where 0 ≠ a_{p} ∈ R for p ∈ Q. Assume further that the a_{p} are homogeneous. Let q be the such element of Q that the initial form F_{∗} of F with respect to F equals a_{q}T^{q}, and let s be the such element of Q that a_{s} ∉ M and the term F_{s} = a_{s}T^{s} has the least degree in IN_{0}^{n+1} among those with a_{p} ∉ M. We claim that

(‡) T_{j}T_{k} divides T^{s} for some (j, k) ∈ Δ.

We prove (‡) by “induction on s − q”. If s = q, then from (*) we know that either a_{s} ∉ I_{j} for some j or (‡) holds. But I_{j} ⊆ M by unshortenability, so a_{s} ∉ I_{j} by the definition of s, so (‡) holds. Now suppose that deg(s) > deg(q). By (*), either a_{q} ∈ I_{j} for some j, or T_{j}T_{k} divides T^{q} for some (j, k) ∈ Δ. Accordingly there are two cases.

First assume that a_{q} ∈ I_{j} for some j. Then there exists a linear form F′ = a_{q}T_{j} − a_{1}T_{1} −⋯−a_{j−1}T_{j−1} in J with deg(a_{q})+deg(x_{j}) = deg(a_{1})+deg(x_{1}) = ⋯ = deg(a_{j−1})+ deg(x_{j−1}). By our assumption that deg(x_{1}) ≤ … ≤ deg(x_{n}), we have deg(a_{1}) ≥ … ≥ deg(a_{j−1}) ≥ deg(a_{q}). Since a_{q} ∈ M by assumption, it follows that a_{1}, …, a_{j−1} are also in M. Consider G = F − F′T^{q}/T_{j}. Since F′_{∗} = a_{q}T_{j}, it follows that F_{∗} = (F′T^{q}/T_{j})_{∗}, so that deg(G_{∗}) < deg(F_{∗}). Since F′T^{q}/T_{j} belongs to M[T_{1}, …, T_{n}], it follows that G_{s} is the term of least degree in G with coefficient not in M. By induction, we have T_{j}T_{k} divides T^{s} for some (j, k) in Δ. We are done.

Now assume that T_{j}T_{k} divides T^{q} for some (j, k) in Δ. By the definition of Δ and axiom 2 in the definition of quadratic sequence, there exists a 2-form F′ in J with F′_{∗} = T_{j}T_{k}. Now consider G = F − a_{q}F′T^{q}/T_{j}T_{k}, and argue just as in the first case.

Example 3.5 (Defining ideals of monomial projective space curves lying on the quadric xw − yz = 0 as in [MS]) This example is treated in §2 of [RS] (see pages 558, 559) and in §3.3 of [RV]. We have

k[Δ] = k[T_{1}, …, T_{b−c+2}]/(T_{3}, …, T_{b−c+1})^{2}

so that k[Δ] is Cohen-Macaulay and its Hilbert series is given by

H(k[Δ]; t) = \frac{1 + (b − c − 1)t}{(1 − t)^{3}}.

Example 3.6 (Huckaba-Huneke ideals of analytic deviation 1 and 2.) This example is treated in [RV] (see §3.4 of that paper). We have

k[Δ] = k[T_{1}, …, T_{n}]/(T_{m+1}, …, T_{n})^{2}

so that k[Δ] is Cohen-Macaulay and its Hilbert series is given by

H(k[Δ]; t) = \frac{1 + (n − m)t}{(1 − t)^{m}}

Here m is the analytic spread of the ideal and n is the minimal number of generators of the ideal.

Example 3.7 (Straightening-closed ideals in graded algebras with straightening law.) These examples are treated in §2 of [RS] and in §3.2 of [RV]. Let R be a graded algebra with straightening law on a finite poset Π over a field k and let M denote the graded maximal ideal of R. If Ω is a straightening-closed ideal of Π which admits a linearization # : Ω → {1, …, n} satisfying

deg(x_{1}) ≤ ⋯ ≤ deg(x_{n}),

then k[Δ] is the face ring of the poset Ω. Thus F (I) has the same Hilbert series as the face ring of Ω. And F (I) is Cohen-Macaulay if Ω is a Cohen-Macaulay poset. While these conclusions are well-known for most particular examples of straightening-closed ideals, what appears to be new is the conclusion from Remark 1.3 that the reduction number is independent of the minimal reduction and can be read off the Hilbert series of F (I). We illustrate this last conclusion by means of an example: Let (Z_{ij}|1 ≤ i ≤ 2, 1 ≤ j ≤ n) be a generic 2 × n matrix (n ≥ 3), R = k[Z_{ij}] the polynomial ring over a field k in these 2n indeterminates, Ω the poset of 2 × 2 minors of the matrix Z, and I the ideal of R generated by the elements of Ω. The Hilbert series of F (I) is (see [CHV])

H(F (I), t) = \frac{h_{0} + h_{1}t + ⋯ + h_{n−3}t^{n−3}}{(1 − t)^{2n−3}}

where

h_{i} = \binom{n − 2}{i}^{2} − \binom{n − 3}{i − 1}\binom{n − 1}{i + 1}.

Thus the reduction number of I is n − 3.

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Clare D’Cruz School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India. email: clare@math.tifr.res.in K. N. Raghavan, SPIC Mathematical Institute, 92, G. N.. Chetty Road, T. Nagar, Chennai 600 017 India. email: knr@ssf.ernet.in J. K.Verma, Department of Mathematics, Indian Institute of Technology, Bombay, Mumbai 400076, India. email: jkv@math.iitb.ernet.in