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Study of blow-up rings

气胀环的研究

基本信息

批准号：
11640049
负责人：
GOTO Shiro
金额：
$ 2.3万
依托单位：
MEIJI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640049/
关键词：
Buchsbaum ring Cohen-Macaulay ring Gorenstein ring canonical module Rees algebra associated graded ring Buchsbaun環 Cohen-Macaulay環 Gorenstein環 Rees代数 Buchsbaum環 S^1-manifold ヒルベルト類体

项目摘要

Let I be an m-primary ideal in a Gorenstein local ring (A, m) with dim A = d and assume that I contains a parameter ideal Q in A as a reduction. Then we say that I is good ideal in A if G = 【symmetry】_n≧_0I^n/I^<n+1> is a Gorenstein ring with a(G) = 1 - d. The associated graded ring G of I is a Gorenstein ring with a(G) = -d if and only if I = Q.Therefore, good ideals in our sense are good ones next to the parameter ideals Q in A.A basic theory of good ideals is developed by this project. We have that I is a good ideal in A if and only if I^2= QI and I = Q : I.Firstly a criterion for finite-dimensional Gorenstein graded algebras A over fields k to have the nonempty sets X_A of good ideals shall be given. Secondly in the case where d=1 we will give a correspondence theorem between the set X_A and the set Y_A of certain overrings of A.A characterization of good ideals of the case where d = 2 will be given in terms of the goodness in their powers. Thanks to Kato's Rieman-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show the structure of the set X_A of good ideals in A heavily depends on d = dim A.The set X_A may be empty if d ≦ 2, while X_A is necessarily infinite if d ≧ 3. To analyze this phenomenon we shall lastly explore monomial good ideals in the polynomial ring k[X_1, X_2, X_3] in three variables over a field k. Let I be an ideal in a Gorenstein local ring A.Then I is said to be an equimultiple good ideal if I contains a reduction Q = (a_1, a_2, …, a_s) generated by s elements in A and if the associated graded ring G(I)=【symmetry】_n≧_0I^n/I^<n+1> of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = ht_AI.The structure of the sets X_<A,s> (s ≧ 0) of equimultiple good ideals I with ht_AI = s. Some of the results in the case where s = dim A are successfully generalized to those of equimultiple case with improvements.

设I是Gorenstein局部环(A，m)中的m-素数理想，其中dim A=d，并假设I包含A中的一个参数理想Q作为约简.如果G=[对称性]_n≧_0I^n/I^&lt；n+1&gt；是具有a(G)=1-d的Gorenstein环，则称I是A中的好理想.相应的分次环G是具有a(G)=-d的Gorenstein环当且仅当I=Q.因此，我们意义上的好理想是A中仅次于参数理想Q的好理想.证明了I是A中的好理想当且仅当I^2=QI且I=Q：I。首先给出域k上有限维Gorenstein分次代数A有好理想的非空集X_A的一个判据。其次，在d=1的情况下，我们将给出A的某些上环的集合X_A与集合Y_A之间的对应定理，并用它们的幂的良好性来刻画d=2的好理想。借助于加藤的Rieman-Roch定理，我们能够对二维Gorenstein有理局部环中的好理想进行分类。作为结论，我们将证明A中好理想的集合X_A的结构在很大程度上取决于d=dim A。当d≦2时，X_A可能为空，而当d≧3时，X_A必然是无穷的。为了分析这一现象，我们将在域k上的三元多项式环k[X_1，X_2，X_3]中探索单项好理想。设I是Gorenstein局部环A中的理想，则称I是等重好理想，如果I包含约化Q=(a_1，a_2，…，a_S)，且若I的相应分次环G(I)=[对称性]_n≧_0i^n/i^&lt；n+1&gt；是具有a(G(I))=1-S的Gorenstein环，其中S=ht_Ai，则X_&lt；A，S&gt；(S≧_0)具有等重好理想的集合的结构，其中ht_AI=S.当S=dim A时，某些结果被成功地推广到等重情形.