PROPERTIES OF THE FIBER CONE OF IDEALS IN LOCAL RINGS

William J. Heinzer Department of Mathematics, Purdue University West Lafayette, Indiana 47907 USA E-mail: heinzer@math.purdue.edu

Mee-Kyoung Kim Department of Mathematics, Sungkyunkwan University Jangangu Suwon 440-746, Korea E-mail: mkkim@math.skku.ac.kr

Abstract. For an ideal I of a Noetherian local ring (R, m) we consider properties of I and its powers as reflected in the fiber cone F (I) of I. In particular, we examine behavior of the fiber cone under homomorphic image R → R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψ_{J} : F_{R}(I) → F_{R′}(IR′). We consider the structure of fiber cones F (I) for which ker ψ_{J} ≠ 0 for each nonzero ideal J of R. If dim F (I) = d > 0, μ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence, we prove that if grade(G_{+}(I)) ≥ d − 1, then F (I) is Cohen-Macaulay and thus a hypersurface.

1. Introduction

For an ideal I in a Noetherian local ring (R, m), the fiber cone of I is the graded ring

F (I) = ⨁_{n≥0} F_{n} = ⨁_{n≥0} I^{n}/ m I^{n} ≅ R[It]/ m R[It],

where R[It] is the Rees ring of I and F_{n} = I^{n}/ m I^{n}. We sometimes write F_{R}(I) to indicate we are considering the fiber cone of the ideal I of the ring R. In terms of the height, ht(I), of I and the dimension, dim R, of R, one always has the inequalities ht(I) ≤ dim F (I) ≤ dim R.

For an arbitrary ideal I ⊆ m of (R, m), the fiber cone F (I) has the attractive property of being a finitely generated graded ring over the residue field k := R/ m

Date: January 9, 2003. 1991 Mathematics Subject Classification. Primary: 13A30, 13C05; Secondary: 13E05, 13H15. Key words and phrases. fiber cone, analytic spread , multiplicity, Cohen-Macaulay ring. We thank Bernd Ulrich for helpful conversations about the topics considered here. The second author is supported by Korea Research Foundation Grant (KRF-2000-015-DP0005).

that is generated in degree one, i.e., F_{n} = F_{1}^{n} for each positive integer n, so F (I) = k[F_{1}].

It is well known in this setting that the Hilbert function H_{F} (n) giving the dimension of I^{n}/ m I^{n} as a vector space over k is defined for n sufficiently large by a polynomial h_{F} (X) ∈ ℚ[X], the Hilbert polynomial of F (I) [Mat, Corollary, page 95], [AM, Corollary 11.2]. A simple application of Nakayama’s lemma, [Mat, Theorem 2.2], shows that the cardinality of a minimal set of generators of I^{n}, μ(I^{n}), is equal to λ(I^{n}/ m I^{n}), the value of the Hilbert function H_{F} (n) of F (I).

An interesting invariant of the ideal I is its analytic spread, denoted ℓ(I), where the analytic spread of I is by definition the dimension of the fiber cone, ℓ(I) = dim F (I) [NR]. The analytic spread measures the asymptotic growth of the minimal number of generators of I^{n} as a function of n. In relation to the degree of the Hilbert polynomial, we have the equality ℓ(I) = 1 + deg h_{F} (X). An ideal J ⊆ I is said to be a reduction of I if there exists a positive integer n such that JI^{n} = I^{n+1}. It then follows that J^{i}I^{n} = I^{n+i} for every postive integer i. If J is a reduction of I, then J requires at least ℓ(I) generators. If the residue field R/ m is infinite, then minimal reductions of I correspond to Noether normalizations of F (I) in the sense that a_{1}, …, a_{r} ∈ I − I^{2} generate a minimal reduction of I if and only if their images \overline{a_{i}} ∈ I/ m I ⊆ F (I) are algebraically independent over R/ m and F (I) is integral over the polynomial ring (R/ m)[\overline{a_{1}}, …, \overline{a_{r}}]. In particular, if R/ m is infinite, then there exist ℓ(I)-generated reductions of I,

For a positive integer s, the fiber cone F (I^{s}) of the ideal I^{s} embeds in the fiber cone F (I) = ⊕_{n=0}^{∞}F_{n} of I by means of F (I^{s}) ≅ ⊕_{n=0}^{∞}F_{ns}. This isomorphism makes F (I) a finitely generated integral extension of F (I^{s}). Thus dim F (I) = dim F (I^{s}) and ℓ(I) = ℓ(I^{s}).

We are particularly interested in conditions that imply the fiber cone F (I) is a hypersurface. Suppose dim F (I) = d > 0 and μ(I) = d + 1. If I has a reduction generated by a regular sequence and if grade(G_{+}(I)) ≥ d − 1, we prove in Theorem 5.6 that F (I) is a hypersurface. We have learned from Bernd Ulrich that this result also follows from results in the paper [CGPU] of Corso-Ghezzi-Polini-Ulrich.

A useful property of the analytic spread ℓ(I) is that it gives an upper bound on the number of elements needed to generate I up to radical. This property of generation up to radical behaves well with respect to analytic spread of a homomorphic image in the following sense:

Lemma 1.1. Suppose I ⊆ m is an ideal of a Noetherian local ring (R, m), where R/ m is infinite. Let a ∈ I and let R′ := R/aR and I′ := IR′. If a′_{1}, …, a′_{s} ∈ R′ are such that rad(a′_{1}, …, a′_{s})R′ = rad I′ and if a_{i} ∈ R is a preimage of a′_{i}, then rad I = rad(a_{1}, …, a_{s}, a)R. In particular, if ℓ(I′) = s, then I can be generated up to radical by s + 1 elements.

Proof. Assume that rad(a′_{1}, …, a′_{s})R′ = rad I′. If x ∈ rad I, then for some positive integer n, we have x^{n} = y ∈ I. Hence the image y′ of y in R′ is in rad(a′_{1}, …, a′_{s})R′. Therefore y and hence also x is in rad(a_{1}, …, a_{s}, a)R.

Examples given by Huckaba in [Hu, Examples 3.1 and 3.2] establish the surprising fact of the existence of 3-generated height-2 prime ideals I of a 3-dimensional regular local ring R for which dim F (I) = 3 = dim R and for which there exists a principal ideal J = xR ⊆ I such that if R′ := R/xR and I′ := IR′, then dim F_{R′}(I′) = 1 < dim R′ = dim R − 1. This result of Huckaba shows that a statement analogous to Lemma 1.1 for reductions, rather than generators up to radical, is false, that is, it is possible that I′ = I/aR has an s-generated reduction while every reduction of I requires at least s + 2 generators.

These interesting examples are the original motivation for our interest in the behavior of analytic spread in a homomorphic image.

2. Behavior of the fiber cone under homomorphic image.

Setting 2.1. Let J ⊆ m be an ideal of a Noetherian local ring (R, m), let R′ := R/J, and let m′ = m /J . For an ideal I ⊆ m of R let I′ = (I + J)/J = IR′ denote the image of I in R′. There is a canonical surjective ring homomorphism of the fiber cone F_{R}(I) of I onto the fiber cone F_{R′}(I′).

We have R[It] = ⨁_{n≥0} I^{n}t^{n} and R′[I′t] = ⨁_{n≥0}(I′)^{n}t^{n}. Since

(I′)^{n} = (I^{n} + J)/J ≅ I^{n}/(I^{n} ∩ J),

there is a canonical surjective homomorphism of graded rings ϕ_{J} : R[It] → R′[I′t], with ker ϕ_{J} = ⨁_{n≥0}(I^{n} ∩ J)t^{n}.

Since F_{R}(I) = R[It]/ m R[It] and F_{R′}(I′) = R′[I′t]/ m′ R′[I′t], the homomorphism ϕ_{J} : R[It] → R′[I′t] induces a surjective homomorphism ψ_{J} : F_{R}(I) → F_{R′}(I′) which preserves grading. This is displayed in the following commutative diagram for which the rows are exact and the column maps are surjective:

0 ⟶ ⊕_{n≥0}(I^{n} ∩ J)t^{n} ⟶ R[It] ⟶^{ϕ_{J}} R′[I′t] ⟶ 0
↓  ↓  ↓
0 ⟶ ⊕_{n≥0} \frac{(I^{n}∩J)+m I^{n}}{m I^{n}} ⟶ \frac{R[It]}{m R[It]} ⟶^{ψ_{J}} \frac{R′[I′t]}{m′ R′[I′]} ⟶ 0

Since we are interested in the behavior of the fiber cone under homomorphic image, we are especially interested in

ker ψ_{J} = ⊕_{n≥0} \frac{(I^{n} ∩ J) + m I^{n}}{m I^{n}}

Remark 2.2. Let (R, m) be a Noetherian local ring and let I ⊆ m be an ideal of R. Suppose J_{1} ⊆ J_{2} ⊆ m are ideals of R. Let R_{i} := R/J_{i}, i = 1, 2, and let ψ_{i} : F_{R}(I) → F_{R_{i}}(IR_{i}) denote the canonical surjective homomorphisms on fiber cones as in (2.1). Then R_{2} ≅ R_{1}/J′, where J′ = J_{2}/J_{1}, and there exists a canonical surjective homomorphism ψ′ : F_{R_{1}}(IR_{1}) → F_{R_{2}}(IR_{2}) such that ψ_{2} = ψ′ ∘ ψ_{1}.

With notation as in (2.1), if J is a nilpotent ideal of R, then ker ψ_{J} is a nilpotent ideal of F_{R}(I). For suppose x ∈ J is such that x^{s} = 0. If \overline{x} ∈ \frac{(I^{n}∩J)+m I^{n}}{m I^{n}} = F_{n} is the image of x in F (I), then by definition \overline{x}^{s} is the image of x^{s} in F_{sn}, so \overline{x}^{s} = 0. Thus for J a nilpotent ideal of R, we have dim F_{R}(I) = dim F_{R′}(I′) and ℓ(I) = ℓ(I′).

Applying this to the situation considered in (2.2), if s is a positive integer, J_{1} = J^{s} and J_{2} = J, then with ψ′ : F_{R_{1}}(IR_{1}) → F_{R_{2}}(IR_{2}) as in (2.2), it follows that ker ψ′ is a nilpotent ideal and in this situation dim F_{R_{1}}(IR_{1}) = dim F_{R_{2}}(IR_{2}). In particular for the examples of Huckaba [Hu, Examples 3.1 and 3.2] mentioned in the end of Section 1, going modulo a power x^{n}R of the ideal xR also reduces the dimension of the fiber cone F (I) from 3 to 1.

Proposition 2.3. With notation as in Setting 2.1, we have the following implications of Remark 2.2.

(1) If J′ ⊆ J are ideals of R and if ker ψ_{J} = 0, then ker ψ_{J′} = 0.

(2) ker ψ_{J} = 0 if and only if ker ψ_{xR} = 0 for each x ∈ J.

(3) For x ∈ m, we have ker ψ_{xR} = 0 if and only if (I^{n} : x) = (m I^{n} : x) for each n ≥ 0.

Proof. Statements (1) and (2) are clear in view of (2.2) and the description of ker ψ_{J} given in (2.1). For statement (3), we use that I^{n} ∩ xR = x(I^{n} : x). Thus 0 = ker ψ_{xR} = ⊕_{n≥0} \frac{(I^{n}∩xR)+m I^{n}}{m I^{n}} ⟺ (I^{n} ∩ xR) ⊆ m I^{n} for each n ⟺ x(I^{n} :

x) ⊆ m I^{n} for each n ⟺ (I^{n} : x) ⊆ (m I^{n} : x) for each n. This last statement is equivalent to (I^{n} : x) = (m I^{n} : x) for each n.

Proposition 2.4. Let (R, m) be a Noetherian local ring and let I ⊆ m be an ideal of R. Suppose J_{1} and J_{2} are ideals of R such that rad J_{1} = rad J_{2}. Let R_{i} := R/J_{i}, i = 1, 2, and let ψ_{i} : F_{R}(I) → F_{R_{i}}(IR_{i}) denote the canonical surjective homomorphisms on fiber cones as in (2.1). Then dim F_{R_{1}}(IR_{1}) = dim F_{R_{2}}(IR_{2}) and ℓ(IR_{1}) = ℓ(IR_{2}).

Proof. Since rad(J_{1} + J_{2}) = rad J_{1} = rad J_{2}, it suffices to consider the case where J_{1} ⊆ J_{2}. With notation as in (2.2), ker ψ′ is a nilpotent ideal. Thus dim F_{R_{1}}(IR_{1}) = dim F_{R_{2}}(IR_{2}) and ℓ(IR_{1}) = ℓ(IR_{2}).

As we remarked in Section 1, the dimension of the fiber cone F (I) of an ideal I is the same as the dimension of the fiber cone F (I^{n}) of a power I^{n} of I. Hence, with notation as in (2.1), we have dim F_{R′}(IR′) = dim F_{R′}(I^{n}R′) and ℓ(IR′) = ℓ(I^{n}R′) for each positive integer n.

3. The associated graded ring and the fiber cone.

The associated graded ring of the ideal I plays a role in the behavior of the fiber cone of the image of I modulo a principal ideal as we illustrate in Proposition 3.1 and Example 3.2.

Proposition 3.1. Let I ⊆ m be an ideal of a Noetherian local ring (R, m). For x ∈ m, let x^{∗} denote the image of x in the associated graded ring G(I) = R[It]/IR[It] and let \overline{x} denote the image of x in the fiber cone F (I). If x^{∗} is a regular element of G(I), then F (I)/\overline{x}F (I) ≅ F_{R′}(I′), where R′ = R/xR and I′ = IR′.

Proof. There exists a positive integer s such that x ∈ I^{s} − I^{s+1}. Since x^{∗} is a regular element of G(I) with deg x^{∗} = s, we have (I^{n} ∩ xR) = xI^{n−s} for every n ≥ 0, where I^{n−s} := R if n − s ≤ 0. Hence we have

[ker ψ_{xR}]_{n} = \frac{(I^{n} ∩ xR) + m I^{n}}{m I^{n}} = \frac{xI^{n−s} + m I^{n}}{m I^{n}} = [\overline{x}F (I)]_{n},

for every n ≥ 0. Therefore F (I)/\overline{x}F (I) ≅ F_{R′}(I′).

With notation as in Proposition 3.1, the following example shows that for x ∈ I−m I such that \overline{x} is a regular element of F (I), it may happen that \overline{x}F (I) ⊊ ker ψ_{xR} and F_{R′}(I′) ≅ F̸_{R}(I)/\overline{x}F (I), where R′ = R/xR and I′ = IR′. Proposition 3.1 implies that for such an example x^{∗} ∈ G(I) is necessarily a zero divisor.

Example 3.2. Let k be a field and consider the subring R := k[[t^{3}, t^{4}, t^{5}]] of the formal power series ring k[[t]]. Thus R = k + t^{3}k[[t]] is a complete Cohen-Macaulay one-dimensional local domain. Let I = (t^{3}, t^{4})R. An easy computation implies I^{3} = t^{3}I^{2}. Hence t^{3}R is a principal reduction of I. Since I is 2-generated, it follows from [DGH, Proposition 3.5] that F (I) is Cohen-Macaulay and in fact a complete intersection. Let X, Y be indeterminates over k and define a k-algebra homomorphism ϕ : k[X, Y] → F (I) by setting ϕ(X) = \overline{t^{3}} and ϕ(Y) = \overline{t^{4}}. Then ker ϕ = Y^{3}k[X, Y] and F (I) ≅ k[X, Y]/Y^{3}k[X, Y]. Thus \overline{t^{3}} is a regular element of F (I) and F (I)/\overline{t^{3}}F (I) ≅ k[Y]/Y^{3}k[Y]. Let J = t^{3}R, R′ = R/J and I′ = IR′. Since t^{8} ∈ (I^{2} ∩ J), we have ϕ(Y^{2}) = \overline{t^{8}} ∈ ker ψ_{J} and F_{R′}(I′) ≅ k[Y]/Y^{2}k[Y]. Thus F (I)/\overline{t^{3}}F (I) ≅ F̸_{R′}(I′). In fact, we have ker ψ_{J} = (\overline{t^{3}}, \overline{t^{8}})F (I) and \overline{t^{8}} ∉ \overline{t^{3}}F (I).

We list several observations and questions concerning the dimension of fiber cones and their behavior under homomorphic image.

Discussion 3.3. Let I ⊆ m be an ideal of a Noetherian local ring (R, m). If J ⊆ m is an ideal of R and R′ = R/J, then there exists a surjective ring homomorphism χ_{J} : G_{R}(I) = R[It]/IR[It] → G_{R′}(IR′) of the associated graded ring G_{R}(I) of I onto the associted graded ring G_{R′}(IR′) of IR′ [K, page 150].

We have the following commutative diagram involving the associated graded rings and fiber cones for which the vertical maps α and β are surjective:

G_{R}(I) = R[It]/IR[It] = ⊕_{n≥0}I^{n}/I^{n=1} ⟶^{χ_{J}} R′[I′t]/I′R′[I′t] = ⊕(I′)^{n}/(I′)^{n+1}
α↓  ^{β}↓
F_{R}(I) = R[It]/ m R[It] ⊕_{n≥0} I^{n}/ m I^{n} ⟶^{ψ_{J}} R′[I′t]/ m′ R′[I′] = ⊕_{n≥0}(I′)^{n}/ m(I′)^{n}

If J is nonzero, then ker χ_{J} ≠ 0. It can happen, however, that J is nonzero and yet ker ψ_{J} = 0. This is possible even in the case where I is m-primary. In an example exhibiting this behavior, commutativity of the diagram above implies one must have ker χ_{J} ⊆ ker α.

Example 3.4. Let k be a field and let R = k[x, y]_{(x,y)}, where x^{2} = xy = 0. Let I = yR and let J = xR. Then ker ψ_{J} = ⨁_{n≥0} \frac{(xR∩y^{n}R)+m y^{n}R}{m y^{n}R} = 0, but J = xR ≠ 0.

A reason for the existence of examples such as Example 3.4 is given in Proposition 3.5.

Proposition 3.5. Suppose (R, m) is a Noetherian local ring and I is an m-primary ideal. If the fiber cone F (I) is an integral domain, then ker ψ_{J} = 0 for every ideal J of R such that dim(R/J) = dim R. In particular, if I is m-primary and F (I) is an integral domain, then there exists a prime ideal J of R such that ker ψ_{J} = 0.

Proof. Let R′ := R/J. Since I is m-primary, dim F (IR′) = dim R′. Thus dim R′ = dim R implies dim F (IR′) = dim F (I). Since F (I) is an integral domain, it follows that ker ψ_{J} = 0. The last statement follows becaues there exists a prime ideal J of R such that dim R = dim(R/J).

Propositon 3.5 and Example 3.4 show that with notation as in (3.1), it can happen that x^{∗} ∈ G(I) is not a regular element and yet ker ψ_{xR} = \overline{x}F (I).

In Section 4 we consider fiber cones F (I) such that ker ψ_{J} ≠ 0 for each nonzero ideal J.

4. Maximal fiber cones with respect to homomorphic image.

Suppose (R, m) is a Noetherian local ring and I ⊆ m is an ideal of R. If J is a nonzero ideal of R such that ker ψ_{J} = ⨁_{n≥0} \frac{(J∩I^{n})+m I^{n}}{m I^{n}} is the zero ideal of F (I), then we have F_{R}(I) = F_{R′}(IR′), where R′ := R/J; so the fiber cone F (I) is realized as a fiber cone of a proper homomorphic image R′ of R. If there fails to exist such an ideal J, i.e., if ker ψ_{J} ≠ 0 for each nonzero ideal J, then we say that F (I) is a maximal fiber cone of R.

We record in Remark 4.1 some immediate consequences of the inequality dim F_{R′}(IR′) ≤ dim R′.

Remark 4.1. With notation as in (2.1), we have:

(1) If J is such that dim R′ < dim R and if dim F_{R}(I) = dim R, then ker ψ_{J} ≠ 0.

(2) If I is m-primary and J is not contained in a minimal prime of R, then ker ψ_{J} ≠ 0.

(3) If R is an integral domain and dim F (I) = dim R, then F (I) is a maximal fiber cone.

(4) If R is an integral domain, then F (I) is a maximal fiber cone for every m-primary ideal I of R.

We are interested in describing all the maximal fiber cones of R. Thus we are interested in conditions on I and R in order that there exist a nonzero ideal J of R such that ker ψ_{J} = 0. In considering this question, by Proposition 2.3, one may assume that J = xR is a nonzero principal ideal. Thus the question can also be phrased:

Question 4.2. Under what conditions on I and R does it follow for each nonzero element x ∈ m that ker ψ_{xR} ≠ 0?

Discussion 4.3. Information about Question 4.2 is provided by the work of Rees in [R]. In particular, [R, Theorem 2.1] implies that if x ∈ m is such that (I^{n} : x) = I^{n} for each positive integer n, then F_{R}(I) = F_{R′}(I′), where R′ := R/xR and I′ := IR′. Thus for x ∈ m a sufficient condition for ker ψ_{xR} = 0 is that (I^{n} : x) = I^{n} for each positive integer n. It is readily seen that this colon condition on x is equivalent to x ∉ I and the image of x in the associated graded ring G(I) = R[It]/IR[It] is a regular element. More generally, if x ∈ I^{s} − I^{s+1} and if the image x^{∗} of x in G(I) is a regular element, then by Proposition 3.1 ker ψ_{xR} = \overline{x}F (I). Thus if we also have x ∈ m I^{s}, then ker ψ_{xR} = 0. Example 3.4 shows that this sufficient condition for ker ψ_{xR} = 0 is not a necessary condition.

Proposition 2.3 gives a necessary and sufficient condition on a principal ideal J = xR in order that ker ψ_{xR} = 0, namely that (I^{n} : x) = (m I^{n} : x) for each integer n ≥ 0. By Proposition 2.3, if ker ψ_{xR} = 0, then also ker ψ_{yxR} = 0 for every y ∈ R.

If I = yR is a non-nilpotent principal ideal of R, we give in Corollary 4.5 necessary and sufficient conditions for F (I) to be a maximal fiber cone.

Proposition 4.4. Suppose (R, m) is a Noetherian local ring and I = yR ⊆ m is a non-nilpotent principal ideal of R. For x ∈ m, we have ker ψ_{xR} = 0 ⟺ y^{n} ∉ xR for each positive integer n.

Proof. We have ker ψ_{xR} = 0 ⟺ (y^{n}R ∩ xR) ⊆ m y^{n}R for each positive integer n, and y^{n} ∉ xR ⟺ (y^{n}R ∩ xR) ⊊ y^{n}R ⟺ (y^{n}R ∩ xR) ⊆ m y^{n}R.

Corollary 4.5. Let (R, m) be a Noetherian local ring and I = yR ⊆ m be a nonnilpotent principal ideal of R. Then F (I) is a maximal fiber cone if and only if R is a one-dimensional integral domain.

Proof. By Proposition 4.4, for x ∈ m we have y ∈ rad xR ⟺ ker ψ_{xR} ≠ 0. Suppose F (I) is a maximal fiber cone. Then by definition, ker ψ_{xR} ≠ 0 for each nonzero x ∈ m. Since y is not nilpotent, there exists a minimal prime P of R such that y ∉ P. It follows that P = 0, for if not, then there exists a nonzero x ∈ P and y ∈ rad xR ⊆ P implies y ∈ P. Thus R is an integral domain. Moreover, this same argument implies y is in every nonzero prime of R. Since R is Noetherian, it follows that dim R = 1. For yR has only finitely many minimal primes and every minimal prime of yR has height one by the Altitude Theorem of Krull [N, page 26] or [Mat, page 100]. If there exists P ∈ Spec R with ht P > 1, then the Altitude Theorem of Krull implies P is the union of the height-one primes contained in P. This implies there exist infinitely many height-one primes contained in P. Since y is contained in only finitely many height-one primes, this is impossible. Thus dim R = 1. Since R is local, m is the only nonzero prime of R.

Conversely, if R is a one-dimensional Noetherian local integral domain, then (4.1) implies that F (I) is a maximal fiber cone for every non-nilpotent principal ideal I = yR ⊆ m.

Question 4.6. If F (I) is a maximal fiber cone of R, does it follow that dim F (I) = dim R?

Proposition 4.7. Suppose (R, m) is a Noetherian local ring and I ⊆ m is an ideal of R. If dim F (I) := n = ht(I) < dim R and if F (I) is an integral domain, then F (I) is not a maximal fiber cone. In particular, if I is of the principal class, i.e., I = (a_{1}, …, a_{n})R, where ht(I) = n, and if ht(I) < dim R, then F (I) is not a maximal fiber cone of R.

Proof. Choose x ∈ m such that x is not in any minimal prime of I. Then L := (I, x)R has height n + 1. Let \overline{x} denote the image of x in the fiber cone F_{R}(L). Then F_{R}(L) is a homomorphic image of a polynomial ring in one variable F_{R}(I)[z] over F_{R}(I) by means of a homomorphism mapping z → \overline{x}. Since dim F (I) = n and F (I) is an integral domain, it follows that F (I)[z] ≅ F (L) by means of an isomorphism taking z → \overline{x}. Let J = xR and R′ := R/J. Then ht(IR′) = ht(L/xR) = n, so dim F_{R′}(IR′) ≥ n. Since ψ_{J} : F_{R}(I) → F_{R′}(IR′) is surjective and F_{R}(I) is an n-dimensional integral domain, it follows that ψ_{J} : F_{R}(I) → F_{R′}(IR′) is an isomorphism. In particular, if I is of the principal class, then F (I) is a polynomial ring in n variables over the field R/ m, so F (I) is an integral domain with dim F (I) = ht(I).

If I is generated by a regular sequence, then I is of the principal class. Thus if F (I) is a maximal fiber cone and I is generated by a regular sequence, then by Proposition 4.7, dim F (I) = dim R.

We observe in Proposition 4.8 a situation where the integral domain hypothesis of Proposition 4.7 applies.

Proposition 4.8. Let A = k[X_{1}, X_{2}, ⋯, X_{d}] = ⨁_{n=0}^{∞} A_{n} be a polynomial ring in d variables over a field k and let m = (X_{1}, X_{2}, ⋯, X_{d})A denote its homogeneous maximal ideal. Suppose I = (f_{1}, f_{2}, ⋯, f_{n})A, where f_{1}, f_{2}, ⋯, f_{n} are homogeneous polynomials all of the same degree t. Let R = A_{m}. Then F (IR) is an integral domain. Thus if F (IR) is a maximal fiber cone, then dim F (IR) = d.

Proof. We have

k[f_{1}, f_{2}, ⋯, f_{n}] = k ⊕ I_{1} ⊕ I_{2} ⊕ ⋯,

where I_{i} = I^{i} ∩A_{it} for i > 0. Since I^{i}/mI^{i} ≅ I^{i} ∩A_{it} for i ≥ 0, we have the following isomorphisms:

k[f_{1}, f_{2}, ⋯, f_{n}] ≅ ⊕_{i=0}^{∞}(I^{i}/ m I^{i}) ≅ ⊕_{i=0}^{∞}(I^{i}R/ m I^{i}R) = F (IR).

Therefore F (IR) is an integral domain. The result now follows from Proposition 4.7.

Corollary 4.9. With notation as in Proposition 4.8, if dim F (I) = ht I and F (IR) is a maximal fiber cone, then I is m-primary.

Proof. We have dim F (IR) = d by Proposition 4.8. Since I is homogeneous ideal and ht I = d, m is the unique homogeneous minimal prime of I, Therefore I is m-primary.

Question 4.10. Let (R, m) be a Noetherian local ring and let I ⊆ m be an ideal of R. If dim F (I) = ht I and F (I) is a maximal fiber cone, does it follow that I is m-primary?

Remark 4.11. Without the assumption in Question 4.10 that dim F (I) = ht I, it is easy to give examples where F (I) is a maximal fiber cone and yet I is not m-primary. For example, with notation as in Proposition 4.8, if d > 1 and I = (X_{1}^{2}, X_{1}X_{2}, …, X_{1}X_{d})A, then ht(IR) = 1, but dim F (IR) = d and F (IR) is a maximal fiber cone.

5. When is the fiber cone a hypersurface?

Setting 5.1. Let I ⊆ m be an ideal of a Noetherian local ring (R, m). In this section we consider the structure of the fiber cone F (I) = ⊕_{n≥0}F_{n} in the case where dim F (I) = d > 0 and μ(I) = d + 1. If a_{1}, …, a_{d+1} is a basis for F_{1} = I/ m I as a vector space over the field k := R/ m, then there exists a presentation ϕ : k[X_{1}, …, X_{d+1}] → F (I) of F (I) as a graded k-algebra homomorphic image of a polynomial ring in d + 1 variables over k defined by setting ϕ(X_{i}) = a_{i}, for i = 1, …, d + 1. Moreover, F (I) is a hypersurface if and only if ker ϕ is a principal ideal [K, Examples 1.2].

Lemma 5.2. Let (R, m) be a Noetherian local ring having infinite residue field R/ m := k, and let I ⊆ m be an ideal of R such that dim F (I) = d > 0 and μ(I) = d + 1. Let r = r(I) denote the reduction number of I and let ϕ : k[X_{1}, …, X_{d+1}] → F (I) be a presentation of the fiber cone F (I) as in Setting 5.1. Then the minimal degree of a nonzero form f ∈ ker ϕ is r + 1.

Proof. The map ϕ from the graded ring A = k[X_{1}, …, X_{d+1}] = ⊕_{n≥0}A_{n} onto the graded ring F (I) = ⊕_{n≥0}F_{n} = ⊕_{n≥0}(I^{n}/ m I^{n}) is a surjective graded k-algebra homomorphism of degree 0. Let K := ker ϕ = ⊕_{n≥0}K_{n}. For each positive integer n we have a short exact sequence

0 → K_{n} → A_{n} → F_{n} → 0

of finite-dimensional vector spaces over k. Since I has reduction number r, it follows from [ES, Theorem, page 440] that dim_{k} F_{i} = μ(I^{i}) = \binom{i+d}{d} for i = 0, 1, …, r and dim_{k} F_{r+1} = μ(I^{r+1}) < \binom{r+d+1}{d}. Since dim A_{i} = \binom{i+d}{d} for all i, it follows that K_{i} = 0 for i = 0, …, r and K_{r+1} ≠ 0. Hence the minimal degree of a nonzero form f ∈ ker ϕ is r + 1.

Remark 5.3. Let A = k[X_{1}, …, X_{n}] be a polynomial ring in n variables X_{1}, …, X_{n} over a field k. For an ideal K of A, it is well known that ht(K) = 1 if and only if dim(A/K) = n − 1 [K, Corollary 3.6, page 53]. Moreover, K is principal if and only if ht(P) = 1 for each associated prime P of K. If K = (g_{1}, …, g_{m})A and g is a greatest common divisor of g_{1}, …, g_{m}, then K = gJ, where ht(J) > 1. Thus K is principal if and only if J = A. If K is homogeneous, then g_{1}, …, g_{m} may be taken to be homogeneous; it then follows that g is homogeneous and K = gJ, where J is homogeneous with ht(J) > 1. If K = rad K, then each associated prime of K is a minimal prime and K is principal if and only if ht(P) = 1 for each minimal prime P of K.

Proposition 5.4. Let (R, m) be a Noetherian local ring with infinite residue field k = R/ m and let I ⊆ m be an ideal of R such that dim F (I) = d > 0 and μ(I) = d + 1. Let ϕ : A = k[X_{1}, …, X_{d+1}] → F (I) be a presentation of F (I) as a graded homomorphic image of a polynomial ring as in Setting 5.1. Let f ∈ K := ker ϕ be a nonzero homogeneous form of minimal degree. Then the following are equivalent.

(1) ker ϕ = fA, i.e., F (I) is a hypersurface.

(2) ht P = 1 for each P ∈ Ass K.

(3) F (I) is a Cohen-Macaulay ring.

(4) deg f = e(F (I)), the multiplicity of F (I).

Proof. That (1) is equivalent to (2) is observed in Remark 5.2. It is clear that (1) implies (3) and it follows from [BH, (2.2.15) and (2.1.14)] that (3) implies (2). To see the equivalence of (3) and (4), we use [DRV, Theorem 2.1]. By Lemma 5.2, deg f = r + 1, where r is the reduction number of I.

Since dim F (I) = d, there exists a minimal reduction J = (x_{1}, …, x_{d})R of I and y ∈ I such that I = J + yR. By [DRV, Theorem 2.1], F (I) is Cohen-Macaulay if and only if

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

Since for 0 ≤ n ≤ r, λ(\frac{I^{n}}{JI^{n−1}+m I^{n}}) = 1, the sum on the right hand side of the displayed equation is r + 1 = deg f. This proves the equivalence of (3) and (4).

Remark 5.5. With notation as in Proposition 5.4, we have the following inequality e(F (I)) ≤ deg f, where e(F (I)) is the multiplicity of F (I). Hence by Proposition 5.4, F (I) is not Cohen-Macaulay ⟺ e(F (I)) < deg f.

Proof. Let J = (x_{1}, …, x_{d})R be a minimal reduction of I. Then JF (I) is generated by a homogeneous system of parameters for F (I) and

λ(\frac{F (I)}{JF (I)}) = ∑_{n=0}^{r} λ(\frac{I^{n}}{JI^{n−1} + m I^{n}}).

Let M denote the maximal homogeneous ideal of F (I). Then

e(F (I)) = e(F (I)_{M}) ≤ λ(\frac{F (I)_{M}}{JF (I)_{M}}) = λ(\frac{F (I)}{JF (I)}).

Thus e(F (I)) ≤ deg f = r + 1. Hence by Proposition 5.4, F (I) is not Cohen-Macaulay if and only if e(F (I)) < deg f.

Theorem 5.6. Let (R, m) be a Noetherian local ring with infinite residue field k = R/ m and let I ⊆ m be an ideal of R such that dim F (I) = d > 0 and μ(I) = d+1. Suppose there exists a minimal reduction J of I generated by a regular sequence. Assume that grade(G_{+}(I)) ≥ d − 1. Then F (I) is Cohen-Macaulay and thus a hypersurface.

Proof. For x ∈ R, let x^{∗} denote the image of x in G(I) and let \overline{x} denote the image of x in F (I). There exists a minimal reduction J = (x_{1}, …, x_{d}) ⊆ I and x_{d+1} ∈ I such that

(I) {x_{1}, …, x_{d}} is a regular sequence in R.

(II) {x_{1}, …, x_{d}, x_{d+1}} is a minimal set of generators of I.

(III) {x_{1}^{∗}, …, x_{d−1}^{∗}} is a regular sequence in G(I).

Let R′ = R/(x_{1}, …, x_{d−1})R, let m′ = m /(x_{1}, …, x_{d−1})R and let I′ = IR′. By Condition II, I′ is a 2-generated ideal having a principal reduction generated by the image x′_{d} of x_{d}. Condition I implies that x′_{d} is a regular element of R′. Hence by [DGH, Proposition 3.5], F_{R′}(I′) is Cohen-Macaulay.

As observed in (2.1), the kernel of the canonical map ψ : F_{R}(I) → F_{R′}(I′) is

⊕_{n≥0} \frac{(I^{n} ∩ (x_{1}, …, x_{d−1})) + m I^{n}}{m I^{n}} .

Condition III and Proposition 3.1 imply

ker ψ = ⊕_{n≥0} \frac{(x_{1}, …, x_{d−1})I^{n−1} + m I^{n}}{m I^{n}} = (\overline{x_{1}}, …, \overline{x_{d−1}})F (I).

Hence

\frac{F (I)}{(\overline{x_{1}}, …, \overline{x_{d−1}})} ≅ F_{R′}(I′)

and to show F (I) is Cohen-Macaulay, it suffices to show \{\overline{x_{1}}, …, \overline{x_{d−1}}\} is a regular sequence in F (I). By the generalized Vallabrega-Valla criterion of Cortadellas and Zarzuela [CZ, Theorem 2.8], to show \{\overline{x_{1}}, …, \overline{x_{d−1}}\} is a regular sequence in F (I), it suffices to show

(x_{1}, …, x_{d−1}) ∩ m I^{n+1} = (x_{1}, …, x_{d−1}) m I^{n}, for all n ≥ 0.

“ ⊇′′ is clear. We prove “ ⊆′′ by induction on n. (Case i ) n = 0 : Let u ∈ (x_{1}, …, x_{d−1}) ∩ m I. Thus u = ∑_{i=1}^{d−1} r_{i}x_{i} = ∑_{j=1}^{d+1} α_{j}x_{j}, where r_{i} ∈ R and α_{j} ∈ m. Therefore

(r_{1} − α_{1})x_{1} + ⋯ + (r_{d−1} − α_{d−1})x_{d−1} − α_{d}x_{d} − α_{d+1}x_{d+1} = 0.

Since {x_{1}, …, x_{d+1}} is a minimal generating set for I, each r_{i} − α_{i} ∈ m. Since α_{i} ∈ m, r_{i} ∈ m. Hence u = ∑_{i=1}^{d−1} r_{i}x_{i} ∈ m(x_{1}, …, x_{d−1}). (Case ii ) 1 ≤ n < r, where r = r_{J}(I) is the reduction number of I with respect to J: We have (x_{1}, …, x_{d−1}) ∩ m I^{n+1} = (x_{1}, …, x_{d−1}) ∩ (I^{n+1} ∩ m I^{n+1}) = ((x_{1}, …, x_{d−1}) ∩ I^{n+1}) ∩ m I^{n+1} = ((x_{1}, …, x_{d−1})I^{n} ∩ m I^{n+1}, the last equality by Condition III.

Hence u ∈ (x_{1}, …, x_{d−1}) ∩ m I^{n+1} implies u ∈ ((x_{1}, …, x_{d−1})I^{n} ∩ m I^{n+1}. Thus u = ∑_{i=1}^{d−1} x_{i}g_{i}, where g_{i} ∈ I^{n} and u = H(x_{1}, …, x_{d+1}), where H(X_{1}, …, X_{d+1}) ∈ R[X_{1}, …, X_{d+1}] is a homogeneous polynomial with coefficients in m of degree n+1. Let G_{i}(X_{1}, …, X_{d+1}) ∈ R[X_{1}, …, X_{d+1}] be a homogeneous polynomial of degree n such that G_{i}(x_{1}, …, x_{d+1}) = g_{i}.

Let τ : R[X_{1}, …, X_{d+1}] → R[It], where τ(X_{i}) = x_{i}t be a presentation of the Rees algebra R[It]. Consider the following commutative diagram.

0 ⟶ ker(τ) ⟶ R[X_{1}, …, X_{d+1}] ⟶^{τ} R[It] ⟶ 0
π_{1}↓  ^{π_{2}}↓  ^{π_{3}}↓
0 ⟶ ker(ϕ) ⟶ (R/ m)[X_{1}, …, X_{d+1}] ⟶^{ϕ} F (I) ⟶ 0

Since ∑_{i=1}^{d−1} x_{i}g_{i} − H(x_{1}, …, x_{d+1}) = 0, the homogeneous polynomial

∑_{i=1}^{d−1} X_{i}G_{i}(X_{1}, …, X_{d+1}) − H(X_{1}, …, X_{d+1}) ∈ ker τ.

Since H(X_{1}, …, X_{d+1}) has coefficients in m, we have

0 = π_{3}τ(∑_{i=1}^{d−1} X_{i}G_{i} − H) = ϕπ_{2}(∑_{i=1}^{d−1} X_{i}G_{i} − H) = ϕπ_{2}(∑_{i=1}^{d−1} X_{i}G_{i}).

Hence π_{2}(∑_{i=1}^{d−1} X_{i}G_{i}) ∈ ker ϕ. Since ∑_{i=1}^{d−1} X_{i}G_{i} is of degree n + 1 ≤ r, Lemma 5.2 implies π_{2}(∑_{i=1}^{d−1} X_{i}G_{i}) = 0. Therefore the coefficients of ∑_{i=1}^{d−1} X_{i}G_{i} are in m. Evaluating this polynomial by mapping X_{i} ↦ x_{i} gives u = ∑_{i=1}^{d−1} x_{i}g_{i} ∈ (x_{1}, …, x_{d_{1}}) m I^{n}. ( Case iii ) n ≥ r : Since n ≥ r, we have I^{n+1} = JI^{n} = (x_{1}, …, x_{d})I^{n}. Let u ∈ (x_{1}, …, x_{d−1}) ∩ m I^{n+1} = (x_{1}, …, x_{d−1}) ∩ m(x_{1}, …, x_{d})I^{n}. Thus u = ∑_{i=1}^{d−1} r_{i}x_{i} = ∑_{j=1}^{d} α_{j}x_{j}, where each r_{i} ∈ R and each α_{j} ∈ m I^{n}. Hence α_{d}x_{d} = ∑_{i=1}^{d−1}(r_{i} − α_{i})x_{i} and this implies α_{d} ∈ ((x_{1}, …, x_{d−1}) : x_{d}) = (x_{1}, …, x_{d−1}), the last equality because of Condition I. Hence

α_{d} ∈ (x_{1}, …, x_{d−1}) ∩ m I^{n} = (x_{1}, …, x_{d−1}) m I^{n−1},

the last equality because of our inductive hypothesis. Thus u = ∑_{j=1}^{d} α_{j}x_{j} = ∑_{j=1}^{d−1} α_{j}x_{j}+α_{d}x_{d} ∈ (x_{1}, …, x_{d−1}) m I^{n}+(x_{1}, …, x_{d−1}) m I^{n−1}I = (x_{1}, …, x_{d−1}) m I^{n}. This completes the proof that \{\overline{x_{1}}, …, \overline{x_{d−1}}\} is a regular sequence in F (I), and thus the proof of Theorem 5.6

6. The Cohen-Macaulay property of one-dimensional fiber cones

We record in this short section several consequences of a result of D’Cruz, Raghavan and Verma [DRV, Theorem 2.1] for the Cohen-Macaulay property of the fiber cone of a regular ideal having a principal reduction.

Proposition 6.1. Let (R, m) be a Noetherian local ring and let I ⊆ m be a regular ideal having a principal reduction aR. Let r = r_{aR}(I) be the reduction number of I with respect to aR. Then the following are equivalent.

(1) F (I) is a Cohen-Macaulay ring.

(2) λ(\frac{aI^{n}+m I^{n+1}}{m I^{n+1}} ) = λ(\frac{I^{n}}{m I^{n}} ) for 1 ≤ n ≤ r − 1.

Proof. (1) ⇒ (2). Suppose F (I) is a Cohen-Macaulay ring. Then \overline{a}(= a+m I) is a regular element of F (I) with deg \overline{a} = 1. Let F (I) = ⊕_{n≥0}F_{n}, where F_{n} = I^{n}/ m I^{n}, and consider the graded k-algebra homomorphism ϕ\overline{a} : F_{n} → F_{n+1} given by ϕ\overline{a}(\overline{x}) = \overline{x} · \overline{a}, for every \overline{x} ∈ F_{n}. Since \overline{a} is a regular element of F (I), dim_{k} F_{n} = dim_{k}(\overline{a}F_{n}). For 1 ≤ n ≤ r − 1, we have

λ(\frac{aI^{n} + m I^{n+1}}{m I^{n+1}} ) = λ(\overline{a}(\frac{I^{n}}{m I^{n}} )) = dim_{k}(\overline{a}F_{n}) = dim_{k}(F_{n}) = λ(\frac{I^{n}}{m I^{n}} ).

(2) ⇒ (1). Suppose that λ(\frac{aI^{n}+m I^{n+1}}{m I^{n+1}} ) = λ(\frac{I^{n}}{m I^{n}} ), for 1 ≤ n ≤ r − 1. Since a is a non-zero-divisor R, I^{n+r}/ m I^{n+r} ≅ I^{r}/ m I^{r}, for every n ≥ 1. Hence e(F (I)) = λ(I^{r}/ m I^{r}). To see the Cohen-Macaulay property of F (I), we use [DRV, Theorem 2.1]. We have the following:

∑_{n=0}^{r} λ(\frac{I^{n}}{aI^{n−1} + m I^{n}} ) = λ(\frac{R}{m}) + ∑_{n=1}^{r} λ(\frac{I^{n}}{aI^{n−1} + m I^{n}} )

= λ(\frac{R}{m}) + ∑_{n=1}^{r} [λ(\frac{I^{n}}{m I^{n}} ) − λ(\frac{aI^{n−1} + m I^{n}}{m I^{n}} )]

= λ(\frac{R}{m}) + ∑_{n=1}^{r} [λ(\frac{I^{n}}{m I^{n}} ) − λ(\frac{I^{n−1}}{m I^{n−1}} )]

= λ(\frac{I^{r}}{m I^{r}} )

= e(F (I)).

Hence by [DRV, Theorem 2.1], F (I) is a Cohen-Macaulay ring.

As an immediate consequence of Proposition 6.1 we have

Corollary 6.2. Let (R, m) be a Noetherian local ring and I be a regular ideal having a principal reduction aR with r_{aR}(I) = 2. If μ(I) = n, then

F (I) is Cohen-Macaulay ⟺ λ(\frac{aI + m I^{2}}{m I^{2}} ) = n.

Example 6.3 shows that Proposition 6.1 and Corollary 6.2 may fail to be true without the assumption on the length of \frac{aI^{n}+m I^{n+1}}{m I^{n+1}} .

Example 6.3. Let k be a field and consider the subring R = k[[t^{3}, t^{7}, t^{11}]] of the formal power series ring k[[t]]. Let I = (t^{6}, t^{7}, t^{11})R. An easy computation implies t^{6}I ≠ I^{2} and t^{6}I^{2} = I^{3}. Hence r_{t^{6}R}(I) = 2. Note that \overline{t^{6}}F (I) is a homogeneous system of parameter of F (I). But \overline{t^{6}}\overline{t^{11}} = (t^{6} + m I)(t^{11} + m I) = t^{17} + m I^{2} = 0, and hence F (I) is not a Cohen-Macaulay ring. And λ(\frac{t^{6}I+m I^{2}}{m I^{2}} ) = λ(\frac{I^{2}}{m I^{2}} ) − λ(\frac{I^{2}}{t^{6}I+m I^{2}} ) = 2 < 3.

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