Reduction numbers and initial ideals
Reduction numbers and initial ideals
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约简数和初始理想
DOI：
10.1090/s0002-9939-02-06607-8
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
A. Conca
中科院分区：
文献类型：
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作者：
A. Conca
标准分级代数 A 的约简数 r(A) 是最小整数 k，使得存在 A 的齐次最大理想 m 的最小约简 J，使得 Jmk = mk+1。 Vasconcelos 推测 r(R/I) ≤ r(R/in(I))，其中 in(I) 是多项式环 R 中关于项阶的理想 I 的初始理想。这篇笔记的目的是为了证明这个猜想。 1. 归约数和初始理想 设 K 为无限域，A = ⊕ i∈NAi 为齐次 K 代数，即 R/I 形式的代数，其中 R = K[x1, . 。 。 , xn] 是多项式环，I 是齐次理想。 A 的约简数 r(A) 是最小整数 k，使得存在 A 的齐次最大理想 m 的最小约简 J，使得 J m = m。不难看出，r(A)是使得A/J在k处的希尔伯特函数不消失的最大整数k；这里 J 是由 d = dimA 通用线性形式生成的 A 的理想值。 Vasconcelos 推测 [10，猜想 7.2] r(R/I) ≤ r(R/ inτ (I)) 其中 inτ (I) 是 I 关于项阶 τ 的初始理想。 Bresinsky 和 Hoa [2] 已针对一般初始理想 Ginτ (I)，或更一般地，当 inτ (I) 是 Borel 固定时证明了该猜想。 Trung [8] 表明 r(R/I) = r(R/GinRL(I))，其中 GinRL(I) 是 I 相对于逆词典顺序 RL（简称 revlex）的一般初始理想。这篇笔记的目的是从总体上证明这个猜想。这篇论文写完后，我们得知Trung[9]已经用一种完全不同的方法独立解决了一般猜想。我们证明的是 Vasconcelos 猜想的以下推广：定理 1.1。令 p 为整数，0 ≤ p ≤ n，令 inτ (I) 为 I 关于项阶 τ 的初始理想。令 J 为由 p 个通用线性形式生成的理想。那么 R/I + J 的希尔伯特函数≤ R/ inτ (I) + J 的函数，即，对于所有 j ∈ N，dimK [R/I + J ]j ≤ dimK [R/ inτ (I) + J ]j。编辑收到，2001 年 9 月 24 日，修订版，2001 年 10 月 29 日。2000 年数学学科分类。初级 13P10、13A30；中学 13F20。
The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that Jmk = mk+1. Vasconcelos conjectured that r(R/I) ≤ r(R/in(I)) where in(I) is the initial ideal of an ideal I in a polynomial ring R with respect to a term order. The goal of this note is to prove the conjecture. 1. Reduction numbers and initial ideals Let K be an infinite field and let A = ⊕ i∈NAi be a homogeneous K-algebra, that is, an algebra of the form R/I where R = K[x1, . . . , xn] is a polynomial ring and I is a homogeneous ideal. The reduction number r(A) of A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that J m = m. It is not difficult to see that r(A) is the largest integer k such that the Hilbert function of A/J at k does not vanish; here J is the ideal of A generated by d = dimA generic linear forms. Vasconcelos conjectured [10, Conjecture 7.2] that r(R/I) ≤ r(R/ inτ (I)) where inτ (I) is the initial ideal of I with respect to a term order τ . The conjecture has been proved by Bresinsky and Hoa [2] for the generic initial ideal Ginτ (I), or, more generally, when inτ (I) is Borel-fixed. Trung [8] showed that r(R/I) = r(R/GinRL(I)) where GinRL(I) is the generic initial ideal of I with respect to the degree reverse lexicographic order RL (revlex for short). The goal of this note is to prove the conjecture in general. After this paper was written we were informed that Trung [9] has independently solved the conjecture in general by a completely different method. What we prove is the following generalization of Vasconcelos’ conjecture: Theorem 1.1. Let p be an integer, 0 ≤ p ≤ n, and let inτ (I) be the initial ideal of I with respect to a term order τ . Let J be an ideal generated by p generic linear forms. Then the Hilbert function of R/I + J is ≤ that of R/ inτ (I) + J , that is, dimK [R/I + J ]j ≤ dimK [R/ inτ (I) + J ]j for all j ∈ N. Received by the editors September 24, 2001 and, in revised form, October 29, 2001. 2000 Mathematics Subject Classification. Primary 13P10, 13A30; Secondary 13F20.