A Note on the Coefficients of the Abstract Hilbert Function
A Note on the Coefficients of the Abstract Hilbert Function
复制标题
关于抽象希尔伯特函数系数的注记
DOI：
10.1112/jlms/s1-35.2.209
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发表时间：
1960
影响因子：
1.2
通讯作者：
D. Northcott
中科院分区：
文献类型：
--
作者：
D. Northcott
设Q是具有极大理想m的^-维（d> 0）局部环，q是m-准素理想.此外，当If是一个^-模时，设LQ（M）表示它的长度，这是通过子模的合成序列以通常的方式$来定义的。然后，一旦n超过某个值，LQ（Q/qn）就等于n的多项式。这个多项式通常被称为q的希尔伯特特征函数。希尔伯特函数的次数等于d，并且首项具有形式eQndld I，其中e0是称为q的重数的正整数。大量的注意力已经支付给这个数字，有可用的||关于它的大量结果。其他系数的真实的意义，然而，还不知道。当d= 1时，特征函数采用以下形式。eon-k和D.柯比和作者^ f已经导致了一个更好的理解所发挥的作用，由k出现在这里作为常数项，但也可以被认为是签署的系数nd~ x。本文将一维半正则局部环的某些结果推广到任意维数的半正则局部环。我们还记得d维（d> 1）局部环Q称为半正则环，如果存在d个非单位元av a2，.，如：（2）（3）（4）（5）（6）（7）（8）（9）（10）（a1）：at-（a2>...，a^）对于i= 1，2，.，D.这样的环有许多重要的性质，在[8]中可以找到关于更基本性质的简单说明。注意，在一维情况下，说Q是半正则的等价于断言不是每个非单位都是零因子。现在假设Q是d维（d^ l）半正则局部环并且q是由系统u1} u2，.生成的理想，的参数。则 ** q的重数等于LQ（Q/（I）;因此，在术语
Let Q be a^-dimensional (d> 0) local ring with maximal ideal m and let q be an m-primary ideal. Furthermore, when If is a^-module, let LQ (M) denote its length this being defined by means of composition series of submodules in the usual way $. Then once n has exceeded a certain value, LQ (Q/qn) becomes equal to a polynomial in n. This polynomial is usually referred to as the Hilbert characteristic function of q. The degree of the Hilbert function is equal to d and the leading term has the form eQndld\, where e0 is a positive integer called the multiplicity of q. A good deal of attention has been paid to this number and there is available|| a substantial body of results concerning it. The real significance of the other coefficients is, however, not yet known. When d= 1 the characteristic function takes the form. eon—k, and recent papers by D. Kirby and the author^ f have led to a much better understanding of the role played by k which appears here as the constant term but which may also be thought of as the signed coefficient of nd~ x. In the present paper, some of the results obtained in the one-dimensional case are extended to semi-regular local rings of arbitrary dimension. We recall that a d-dimensional (d> 1) local ring Q is said to be semiregular if there exist d non-units av a2,..., ad (say) such that (alf a2,..., a^): at—(av a2>..., a^) for i= 1, 2,..., d. Such rings have a great many important properties and a simple account of the more elementary ones will be found in [8]. Observe that, in the one-dimensional case, to say that Qis semi-regular is equivalent to asserting that not every non-unit is a zero-divisor. Suppose now that Q is a d-dimensional (d^ l) semi-regular local ring and that q is an ideal generated by a system u1} u2,..., ud of parameters. Then** the multiplicity of q is equal LQ (Q/(\); hence, in the terminology of