Numerical equivalence on Chow groups of local rings and its applications

局部环Chow群的数值等价及其应用

• 批准号: 15540038
• 负责人: KURANO Kazuhiko
• 金额: $ 2.37万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540038/
• 关键词: Chow group; Grothendieck group; numerical equivalence; standard conjectures; total coordinate ring; Cox ring; Hilbert-Kunz function; Riemann-Roch; デターミナント射; 特異リーマン・ロッホ; グロタンディエク群; チャウ群; 局所環; グロタンデイェク群; 全座標環; リース環; 因子類群; 素元分解環; クルル環

Serre defined the intersection number for two subvarieties of a smooth algebraic variety in an algebraic method. Many researchers tried to extend this definition for two subvarieties of an arbitrary scheme, but in 1985 Dutta, Hochster and MacLaughlin gave an example that implies that it is impossible to do so. For many years, this example has been regarded as a very special bad one. Recently, the development of algebraic K-theory has well explained why such examples exist by the research due to Levin, Roberts and Srinivas. By the project, we proved that the existence of such examples is deeply related to standard conjectures, that is the most important conjecture in the theory of algebraic cycles. Precisely speaking, we defined numerical equivalence on the Chow group of Noetherian local ring, as that on the Chow ring of smooth projective variety. We obtained a lattice if we divided the Chow group by numerical equivalence, and studied its fundamental properties.We proved a formula, that is a natural extension of a relation between the divisor class group of a normal projective variety and the divisor class group of its (normal) homogeneous coordinate ring. Furthermore, we proved that the total coordinate ring of a normal projective variety whose divisor class group is finitely generated free abelian group is factorial.We found a formula that involves the canonical class and the Frobenius direct image for a Noetherian normal local ring of positive characteristic. We proved the vanishing of the second coefficient of the Hilbert-Kunz function of Q-Gorenstein rings.

Serre用代数方法定义了光滑代数簇的两个子簇的交数。许多研究人员试图将这个定义扩展到任意方案的两个子簇，但在1985年Dutta，Hochster和MacLaughlin给出了一个例子，暗示这是不可能的。多年来，这个例子一直被认为是一个非常特别的坏例子。最近，代数K-理论的发展很好地解释了为什么这样的例子存在的研究由于莱文，罗伯茨和Srinivas。通过这个项目，我们证明了这类例子的存在性与标准图有着密切的联系，这是代数圈理论中最重要的猜想。更确切地说，我们定义了Noether局部环的Chow群与光滑投射簇的Chow环的数值等价。通过数值等价划分Chow群得到格，并研究了格的基本性质，证明了正规投射簇的除子群与其（正规）齐次坐标环的除子群之间关系的自然推广.进一步证明了其因子类群为n-生成自由交换群的正规投射簇的全坐标环是阶乘的，并给出了正特征Noether正规局部环的标准类与Frobenius直像的关系式.证明了Q-Gorenstein环的Hilbert-Kunz函数的第二系数为零。

项目成果

Kazuhiko Kurano: "Numerical equivalence defined an Chow groups of Noetherian local rings"Inventiones Mathematicae. 未定.

Kazuhiko Kurano：“数值等价定义了诺特局部环的 Chow 群”数学发明待确定。

The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity

与 Buchsbaum 局部重数环中的参数理想相关的底理想的约简指数

• 发表时间: 2004
• 期刊: J. Math. Soc. Japan 56
• 作者: S.Goto;S.Goto;S.Goto;S.Goto;S.Goto;E.J.Elizondo;K.Kurano;S.Goto;S.Goto
• 通讯作者: S.Goto

Parametric decomposition of power of ideals versus regularity of sequences

理想力量与序列规律性的参数分解

• 发表时间: 2003
• 期刊: Proc. Amer. Math. Soc. 132-4
• 作者: S.Goto;S.Goto;S.Goto;S.Goto;S.Goto;E.J.Elizondo;K.Kurano;S.Goto;S.Goto;E.J.Elizondo;K.Kurano;S.Goto;S.Goto;K.Kurano;S.Goto;S.Goto;S.Goto;S.Goto;S.Goto
• 通讯作者: S.Goto

The a-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

与常规局部环中理想过滤相关的分级环的 a 不变性和 Gorensteinness

• 发表时间: 2003
• 期刊: Proc. Amer. Math. Soc. 131
• 作者: S.Goto;S.Goto;S.Goto;S.Goto;S.Goto;E.J.Elizondo;K.Kurano;S.Goto;S.Goto;E.J.Elizondo;K.Kurano;S.Goto;S.Goto;K.Kurano;S.Goto;S.Goto;S.Goto
• 通讯作者: S.Goto

Asymptotic stability of primes associated to homogeneous components of multigraded modules

与多级模的齐次分量相关的素数的渐近稳定性

• 期刊: J. Algebra to appear
• 作者: S.Goto;S.Goto;S.Goto;S.Goto;S.Goto;E.J.Elizondo;K.Kurano;S.Goto;S.Goto;E.J.Elizondo;K.Kurano;S.Goto;S.Goto;K.Kurano;S.Goto;S.Goto;S.Goto;S.Goto;S.Goto;S.Goto;S.Goto;S.Goto;K.Kurano;S.Goto;E.Hyry;F.Hayasaka
• 通讯作者: F.Hayasaka