Coefficient Ideals in and Blowups of a Commutative Noetherian Domain
Coefficient Ideals in and Blowups of a Commutative Noetherian Domain
复制标题
交换诺特域中的系数理想和放大
DOI：
10.1006/jabr.1993.1261
复制
发表时间：
1993
影响因子：
0.9
通讯作者：
Kishor Shah
中科院分区：
文献类型：
--
作者：
W. Heinzer;Bernard L. Johnston;David Lantz;Kishor Shah
在具有单位的交换noether域R中，与非零理想I相关联的Ratliff-Rush理想为Ĩ =∈∞n=1 (in +1:RIn = {is∩R:S∈B (I)}，其中B (I) = {R[I/a]P:a∈I - 0, P∈Spec(R[I/a])}是I的放大。我们观察到在具有相同关联Ratliff-Rush理想的理想类中，某些理想是极小的甚至是唯一的。如果（R， M）是局部的、拟非混合的、解析非分支的，并且如果I是M-原初的，那么我们证明了I的系数理想I{k}，即包含I的最大理想I，其Hilbert多项式在最高k项上与I的Hilbert多项式一致，也是由B (I)通过类似于“s2 -化”的过程得到的一个放大B (I)(k)缩并而来。这使我们可以推广系数的概念。我们在二维正则局部环的特定背景下研究了这些概念，观察了这些概念与完全理想的Zariski理论的相互作用。
Abstract The Ratliff-Rush ideal associated to a nonzero ideal I in a commutative Noetherian domain R with unity is Ĩ = ⋃∞n=1 (In+1:RIn = ⋂ {IS∩R:S∈ B (I)}, where B (I) = {R[I/a]P:a∈I−0, P∈Spec(R[I/a])} is the blowup of I. We observe that certain ideals are minimal or even unique in the class of ideals having the same associated Ratliff-Rush ideal. If (R, M) is local, quasi-unmixed, and analytically unramified, and if I is M-primary, then we show that the coefficient ideal I{k} of I, i.e., the largest ideal containing I whose Hilbert polynomial agrees with that of I in the highest k terms, is also contracted from a blowup B (I)(k), which is obtained from B (I) by a process similar to "S2-ification." This allows us to generalise the notion of coefficient ideas. We investigate these ideas in the specific context of a two-dimentional regular local ring, observing the interaction of these notions with the Zariski theory of complete ideals.