Multigraded Rees algebras of 𝔪-primary ideals in local rings of dimension greater than one

Clare D’Cruz

Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India.

clare@math.tifr.res.in

Abstract

Let I_{1}, …, I_{g} be 𝔪-primary ideals in a local ring (R, 𝔪). Set R[It] := R[I_{1}t_{1}, …, I_{g}t_{g}] and M = (𝔪, I_{1}t_{1}, …, I_{g}t_{g}). In this paper we show that if R is a Cohen-Macaulay (CM) local ring of dimension greater than two and if R[It]_{M} is CM with minimal multiplicity, then g = 1.

1 Introduction

The objective of this paper is to investigate the minimal multiplicity property of multigraded Cohen-Macaulay Rees algebras of zero-dimensional ideals in a CM local ring.

Throughout this paper we will assume that (R, 𝔪) is a local ring of positive dimension with infinite residue field. Let I_{1}, …, I_{g} be ideals of positive height in R and let t_{1}, …, t_{g} be indeterminates. The multigraded Rees algebra of R with respect to I_{1}, …, I_{g} is a subring of R[t_{1}, …, t_{g}] denoted by R[I_{1}t_{1}, …, I_{g}t_{g}] and is defined as ⊕_{r_{i}∈IN}(I_{1}t_{1})^{r_{1}} ⋯ (I_{g}t_{g})^{r_{g}}. Here IN denotes the set of non-negative integers. We will use the notation introduced by Herrmann et. al. in [3]. The multigraded Rees algebra will be denoted by R[It]. Let M = (𝔪, I_{1}t_{1}, …, I_{g}t_{g}) denote the maximal homogeneous ideal of R[It].

It is well known that for a CM local ring (R, 𝔪), e(𝔪) ≥ μ(𝔪)−dim R+1 [1], where e(𝔪) denotes the multiplicity of 𝔪 and μ(𝔪) is the minimal number of generators of 𝔪. A CM local ring is said to have minimal multiplicity if equality holds.

Minimal multiplicity of the Rees algebra of one ideal has been extensively studied in the past. In 1988, Huneke and Sally showed that if I is a complete ideal in a twodimensional regular local ring R, then R[It]_{M} is CM with minimal multiplicity [4, Theroem 3.2]. This result was generalized by Verma in 1991. He obtained necessary and sufficient conditions for the Rees algebra of an 𝔪-primary ideal in a CM local ring of dimension two to be CM with minimal multiplicity [14, Theorem 3.2]. This result was further generalized to equimultiple CM ideals of positive height in CM local rings of dimension greater than one [2].

Minimal multiplicity of multigraded Rees algebras of 𝔪-primary ideals in a CM local ring of dimension two was first studied by Verma in [14], [15] and later by Herrmann et. al. in [3]. In [14, Theorem 3.1], sufficient conditions were obtained for the minimal multiplicity of R[It]_{M}. A similar result was obtained by Herrmann et. al.. Infact, they also gave necessary conditions under the extra assumption that R[I_{j}t_{j}] is CM for all j = 1, …, g. In this paper we prove the converse of Theorem 3.1 in [14] (Theorem 3.1). It is not difficult to see that [3, Theorem 5.4, Corollary 5.6] follow from [14, Theorem 3.1] and Theorem 3.1.

In this paper we also show that if R is a CM local ring of dimension d ≥ 3 and if g ≥ 2, then R[It]_{M} is not CM with minimal multiplicity (Theorem 3.2). This result was proved in [3] under some extra assumptions [3, Theorem 5.8, Corollary 5.9, Corollary 5.10, Corollary 5.11, Proposition 5.12].

Our approach here is as follows. We give an estimate on the number of generators of the maximal ideal of R[It]_{M}. To attain this, we need an estimate on the number of generators of a CM ideal in terms of a certain mixed multiplicity (Theorem 2.1).

2 Preliminaries

An ideal J ⊆ I is a reduction of I if there exists a positive integer r such that JI^{r} = I^{r+1} [6]. The ideal J is called a minimal reduction of I if J is minimal with respect to inclusion among all reductions of I. If R/𝔪 is infinite, then any minimal reduction of I is generated by a(I) elements, where a(I) = dim ⨁_{n≥0}(I^{n}/I^{n}𝔪) is called the analytic spread of I [6]. For an ideal I in R, ht I ≤ a(I) ≤ dim R [7]. If J is a reduction of I, then the reduction number of I with respect to J is defined to be

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

The reduction number of I is defined to be

r(I) = min{r_{J}(I) | J is a minimal reduction of I}.

Let I_{1}, …, I_{g} be 𝔪-primary ideals in local ring R of dimension d. Then for r_{1}, …, r_{g} large, ℓ_{R}(I_{1}^{r_{1}}I_{2}^{r_{2}} ⋯ I_{g}^{r_{g}}/I_{1}^{r_{1}+1}I_{2}^{r_{2}} ⋯ I_{g}^{r_{g}}) is a polynomial of degree d − 1 and can be written in the form

∑_{q_{1}+···+q_{g}=d−1} e(I_{1}^{[q_{1}+1]}|I_{2}^{[q_{2}]}| ⋯ |I_{g}^{[q_{g}]})\binom{r_{1} + q_{1}}{q_{1}} ⋯ \binom{r_{g} + q_{g}}{q_{g}} + lower degree terms,

where e(I_{1}^{[q_{1}+1]}| ⋯ |I_{g}^{[q_{g}]}) are positive integers and they are called the mixed multiplicities of the set of ideals (I_{1}, …, I_{g}) [11]. For g = 2, we will use the notation

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

Rees introduced joint reductions in order to understand mixed multiplicities [8]. Let I_{1}, …, I_{d} be 𝔪-primary ideals. A set of elements x_{1}, …, x_{d} is a 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 showed that if R/𝔪 is infinite then joint reductions exist [8].

In the next theorem we give an estimate on the number of generators of 𝔪-primary ideals in a CM local ring. This will be used in section 3 to give an estimate on the number of generators of the maximal ideal of R[It]_{M}.

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

μ(I) ≤ e_{d−1}(𝔪|I) + d − 1.

Proof: We induct on d. The case d = 1 has been proved by J. Sally [10, pg 49]. If d > 1, then there exists a nonzerodivisor y ∈ I and a positive integer s_{0} so that for s ≥ s_{0} and for all r ≥ 0,

yR ∩ 𝔪^{r}I^{s} = y𝔪^{r}I^{s−1}

[8, Lemma 1.2]. Let “-” denote the image in \overline{R} = R/yR. Then e_{d−2}(\overline{𝔪}|\overline{I}) = e_{d−1}(𝔪|I) [5, Lemma 2.5]. By induction hypothesis

μ(I) = μ(\overline{I}) + 1 ≤ e_{d−2}(\overline{𝔪} | \overline{I}) + d − 1 = e_{d−1}(𝔪|I) + d − 1.□

It is possible to give a lower bound on mixed multiplicities in terms of multiplicity of a certain ideal in R. The next lemma will be used in the proof of the main theorem.

Lemma 2.2 Let (R, 𝔪) be a local ring and let I_{1}, …, I_{g} be 𝔪-primary ideals of R. Then for all nonnegative integers q_{1}, …, q_{g}; q_{1} + ⋯ + q_{g} = d − 1,

e(I_{1}^{[q_{1}+1]}|I_{2}^{[q_{2}]}| ⋯ |I_{g}^{[q_{g}]}) ≥ e(I_{1} + ⋯ + I_{g}) ≥ 1.

Proof: Since I_{1}, …, I_{g} are 𝔪-primary, by Rees interpretation of mixed multiplicities, there exists a joint reduction x_{1}, …, x_{d} of q_{1} + 1 copies of I_{1}, q_{2} copies of I_{2}, …, q_{g} copies of I_{g}, such that e(x_{1}, …, x_{d}) = e(I_{1}^{[q_{1}+1]}|I_{2}^{[q_{2}]}| … |I_{g}^{[q_{g}]}). Since (x_{1}, …, x_{d}) ⊆ I_{1} + ⋯ + I_{g}, e(x_{1}, …, x_{d}) ≥ e(I_{1} + ⋯ + I_{g}). □

3 Rees algebras with minimal multiplicity

In this section we prove our main results. We first state a few facts on Rees algebras.

Let I_{1}, …, I_{g} be m-primary ideals in a local ring (R, m) of dimension d. By a result of Valla dim R[It]_{M} = d + g [12]. Notice that e(MR[It]_{M}) = e(M) and μ(MR[It]_{M}) = μ(M). Hence, if R[It]_{M} is CM with minimal multiplicity, then e(M) = μ(M) − (d + g) + 1.

Since M = (𝔪, I_{1}t_{1}, …, I_{g}t_{g}) and M^{2} = (𝔪^{2}, 𝔪I_{j}t_{j}(1 ≤ j ≤ g), I_{i}I_{j}t_{i}t_{j}(1 ≤ i ≤ j ≤ g)),

μ(M) = μ(𝔪) + ∑_{j=1}^{g} μ(I_{j}).  (1)

We state the formula for e(M) [15, Theorem 1.4]:

e(M) = ∑_{q_{0}+q_{1}+···+q_{g}=d−1} e(𝔪^{[q_{0}+1]}|I_{1}^{[q_{1}]}| ⋯ |I_{g}^{[q_{g}]}).  (2)

In 1991, Verma gave sufficient conditions for the multigraded Rees algebra to be CM with minimal multiplicity [14, Theorem 3.1]. In this paper we prove that these conditions are necessary also.

Theorem 3.1 Let (R, 𝔪) be a CM local ring of dimension two. Let I_{0} = 𝔪, I_{1}, …, I_{g} be 𝔪-primary ideals in R. Suppose R[It]_{M} is CM with minimal multiplicity. Then (i) R has minimal multiplicity, (ii) for all j = 1, … g, r(I_{j}) ≤ 1, and μ(I_{j}) = e_{1}(𝔪|I_{j})+1.

Proof: Suppose R[It]_{M} is CM with minimal multiplicity. Since the residue fields of R and R[It]_{M} are infinite and R[It]_{M} is CM with minimal multiplicity, JM = M^{2} [9, Theorem 1]. Let K be a minimal reduction of 𝔪. Then J = (K, I_{1}t_{1}, …, I_{g}t_{g}) is a reduction of M. Comparing the graded pieces of JM and M^{2} we get K𝔪 = 𝔪^{2}. Hence R has minimal multiplicity [9, Theorem 1]. Repeating the above argument one can show that r(I_{j}) ≤ 1 (1 ≤ j ≤ g). Since e(M) = μ(M) − (2 + g) + 1, by (2) and (1) we get

e(𝔪) + ∑_{j=1}^{g} e_{1}(𝔪|I_{j}) = μ(𝔪) + ∑_{j=1}^{g} μ(I_{j}) − (2 + g) + 1
≤ e(𝔪) + ∑_{j=1}^{g} e_{1}(𝔪|I_{j}) [by Theorem 2.1].  (3)

Thus equality holds in (3). This gives μ(I_{j}) = e_{1}(𝔪|I_{j}) + 1 for all j = 1, …, g. □

We now prove the main result of this paper.

Theorem 3.2 Let I_{1}, …, I_{g} be 𝔪-primary ideals in a CM local ring (R, 𝔪) of dimension greater than two. If R[It]_{M} is CM with minimal multiplicity, then g = 1.

Proof: Let d = dim R. By (2) we have

e(M) = e(𝔪) + ∑_{j=1}^{g} e_{d−1}(𝔪|I_{j}) + ∑_{q_{i}≠d−1;0≤i≤g^{q_{0}+q_{1}+···+qg=d−1}} e(𝔪^{[q_{0}+1]}|I_{1}^{[q_{1}]}| ⋯ |I_{g}^{[q_{g}]})

≥ e(𝔪) + ∑_{j=1}^{g} e_{d−1}(𝔪|I_{j}) + [\binom{g + d − 1}{d − 1} − (g + 1)] [by Lemma 2.2].(4)

By (1) and Theorem 2.1 we have

μ(M) − (d + g) + 1 = μ(𝔪) + ∑_{j=1}^{g} μ(I_{j}) − (d + g) + 1
≤ e(𝔪) + ∑_{j=1}^{g} e_{d−1}(𝔪|I_{j}) + g(d − 2).  (5)

Since R[It]_{M} is CM with minimal multiplicity, e(M) = μ(M)−(d+g)+1 and hence by (4) and (5)

\binom{g + d − 1}{d − 1} − (g + 1) ≤ g(d − 2) i.e., \binom{g + d − 1}{d − 1} − g(d − 1) − 1 ≤ 0.

We will prove by induction on g and d that the above inequality does not hold true if g ≥ 1. If d = 3, then

\binom{g + 2}{2} − 2g − 1 = \binom{g}{2} > 0 since g ≥ 2.

If g = 2, then

\binom{d + 1}{2} − 2(d − 1) − 1 = \frac{d(d − 3) + 2}{2} > 0 since d ≥ 3

If d ≥ 4 and g ≥ 3, then by induction hypothesis

\binom{g + d − 1}{d − 1} = \binom{g + d − 2}{d − 1} + \binom{g + d − 2}{d − 2}
> (g − 1)(d − 1) + 1 + g(d − 2) + 1
= g(d − 1) + (g − 1)(d − 2) + 1
> g(d − 1) + 1.
□

The next corollary has been proved by Verma. Here we give a short proof. The author is grateful to the referee for his suggestion.

Corollary 3.3 [13, Theorem 3.4] Let (R, 𝔪) be a CM local ring of dimension greater than two and let r be any natural number. Then R[𝔪^{r}t]_{M} is CM with minimal multiplicity if and only if R is a regular local ring and r = 1.

Proof: Let d = dim R. Suppose R[𝔪^{r}t]_{M} is CM with minimal multiplicity. Then e(M) = μ(M) − d. Hence

e(𝔪) + e_{d−1}(𝔪|𝔪^{r}) + d − 2 ≤ e(𝔪) + e_{1}(𝔪|𝔪^{r}) + … + e_{d−1}(𝔪|𝔪^{r})
= e(M)  [Put g = 1 in (4)]
= μ(M) − d
≤ e(𝔪) + e_{d−1}(𝔪|𝔪^{r}) + d − 2 [Put g = 1 in (5)].

It follows that 1 = e_{1}(𝔪|𝔪^{r}) = r e_{1}(𝔪|𝔪) = r e(𝔪). This is possible only if r = 1 and e(𝔪) = 1. Conversely, since R is a regular local ring, R[𝔪t] is CM [12]. It is easy to see that e(M) = μ(M) − d. □

Acknowledgments: The author is very grateful to the referee for his comments. The author also wishes to thank J. K. Verma for his valuable suggestions.

References

[1] Abhyankar, S. S.: Local rings of high embedding dimension, Amer. J. Math. 89 (1967), 1073-1077.

[2] D’Cruz, C.: Regular local rings and Cohen-Macaulay Rees algebras with minimal multiplicity, preprint.

[3] Herrmann, M., Hyry, E., Ribbe, J. and Tang, Z.: Reduction numbers and multiplicities of multi-graded structures, J. Algebra 197 (1997) 311-341.

[4] Huneke, C., and Sally, J.: Birational extensions in dimension two and integrally closed ideals, J. Algebra 115 (1988), 481-500.

[5] Katz, D. and Verma, J. K.: Extended Rees algebras and mixed multiplicities, Math. Z. 202 (1989) 111-128.

[6] Northcott, D. G. and Rees, D: Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158.

[7] Rees, D.: Rings associated with ideals and analytic spread, Math. Proc. Cambridge Philos. Soc. 89 (1981), 423-432.

[8] Rees, D.: Generalizations of reductions and mixed multiplicities, J. London Math. Soc. 29 (1984), 397-414.

[9] Sally, J. D.: On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ. 17 (1977), 19-21.

[10] Sally, J. D.: “Number of generators of ideals in local rings”, Lecture Notes in Pure and Applied Mathematics, Vol. 35, Dekker, New York, 1978.

[11] Teissier, B.: Cycles èvanescents, section planes, et conditions de Whitney, Singularitiés à Cargése, 1972. Astérisque 7-8 (1973) 285-362.

[12] Valla, G.: Certain graded algebras are always Cohen-Macaulay, J. Algebra 42 (1976) 537-548.

[13] Verma, J. K.: Rees algebras with minimal multiplicity, Comm. Algebra 17 (1989), 2999-3024.

[14] Verma, J. K.: Joint reductions and Rees algebras, Math. Proc. Cambridge Philos. Soc. 109 (1991), 335-342.

[15] Verma, J. K.: Multigraded Rees algebras and mixed multiplicities, J. Pure Appl. Algebra 77 (1992), 219-228.