A filtration of the sally module and the associated graded ring of an ideal
A filtration of the sally module and the associated graded ring of an ideal
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理想的 sally 模块和相关分级环的过滤
DOI：
10.1080/00927870008826896
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
C. Polini
中科院分区：
文献类型：
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作者：
C. Polini
设（R;m）是一个诺瑟局部环设我是一个R-理想环。I的伴生梯度环G = grI(R)在奇异点分解的研究中起着重要作用。它的相关性在于这样一个事实，即它在代数上代表了一个品种沿着一个亚品种膨胀的特殊纤维。一个常被解决的问题是找到暗示G深度下界的数值条件。例如，在[7,8]和[3]中，这个深度是用I的希尔伯特系数来测量的。为了更好地解释这些结果，让我们引入一些符号：对于某个整数r，如果Ir+1 = JIr，则理想JI称为I的约简数。这样的最小r称为I相对于J的约简数，并记为rJ(I)。如果R是具有无限剩余域的Cohen - Macaulay， I是一个m-初等理想，那么I的任何最小约简（相对于包含）都是由正则序列生成的。I的Hilbert-Samuel函数是数值函数HI(n) = λ(R=In)（其中λ（）表示长度），它测量所有n 1中R=In的长度的增长。如果d表示R的维数，众所周知，对于n0， HI(n)是n中的d次多项式
Let (R;m) be a Noetherian local ring and let I be an R-ideal. The associated graded ring of I, G = grI(R), plays a significant role in the study of resolution of singularities. Its relevance lies upon the fact that it represents algebraically the exceptional fiber of the blowup of a variety along a subvariety. A commonly addressed issue is to find numerical conditions that imply lower bounds on the depth of G . In [7, 8] and [3], for instance, this depth has been measured by using the Hilbert coefficients of I. To better explain these results, let us introduce some notation: An ideal J I is called a reduction of I if Ir+1 = JIr for some integer r. The least such r is called the reduction number of I with respect to J, and denoted rJ(I). If R is Cohen– Macaulay with infinite residue field and I is an m-primary ideal, then any minimal (with respect to inclusion) reduction of I is generated by a regular sequence. The Hilbert–Samuel function of I is the numerical function HI(n) = λ(R=In) (where λ( ) denotes length) that measures the growth of the length of R=In for all n 1. If d denotes the dimension of R, it is well-known that for n 0, HI(n) is a polynomial in n of degree d