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Around Kummer-Artin-Screier-Witt theories

围绕 Kummer-Artin-Screier-Witt 理论

基本信息

批准号：
16540040
负责人：
SUWA Noriyuki
金额：
$ 1.22万
依托单位：
Chuo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540040/
关键词：
group scheme Kummer theory Artin-Schreier theory Kummer-Artin-Schreier theory twisted Kummer theory fppf cohomology etale cohomology 代数群 equivariant compactification 生成多項式

项目摘要

It is a classical problem to construct, given a field K and a finite group G, Galois extensions of K with the Galois group G. The most important result is the Kummer theory, which asserts that, if a positive integer n is invertible in K and all the n-th roots of unity are contained in K, all the cyclic extensions of K of degree n is obtained by adjoining a root of an equation t^n=a. On the other hand, if K is of characteristic p>0, the Artin-Schreier theory asserts that all the cyclic extensions of K of degree p is obtained by adjoining a root of an equation t^p-t=a. The theory of Witt vectors gives an elegant description on cyclic extensions of degree p^n of a field of characteristic p>0.Nowadays it is standard to prove the Kummer, Artin-Schreier and Artin-Schreier-Witt theories in the framework of Galois cohomology. For example, the Kummer theory follows from the exact sequence of group schemes called the Kummer sequence and the vanishing theorem of Galois cohomology called Hilbert 9 … More 0. Sekiguchi and Suwa has constructed exact sequences of group schemes, which unify the Kummer sequences and the Artin-Schreier-Witt sequences.Recently another problem interests specialists to remove from the Kummer theory the condition that K contains all the n-th root of unity. Komatsu established a variant of Kummer theory, twisting the Kummer theory by a quadratic extension. In this research project Suwa generalizes the twisted Kummer theory over a ring, clarifying a relation between the twisted Kummer theory due to Komatsu's and the theory on generic polynomials for cyclic extensions due to Rikuna. Moreover Suwa establishes a theory which unifies the twisted Kummer theory and the Artin-Schreier theory.In this work, the unitary group scheme for a quadratic extension of a ring plays an important role. We have gotten also a nice description on compactifications of the twisted Kummer theory and twisted Kummer-Artin-Schreier theory, using the regular representaion of the quadratic extension.[1] T.Komatsu-Arithmetic of Rikuna's generic cyclic polynomial and generalization of Kummer theory. Manuscripta Math 114(2004) 265-279[2] Y.Rikuna-On simple families of cyclic polynomials. Proc. Amer. Math. Soc. 130 (2002) 2215-2218 Less

给定域K和有限群G，构造域K的伽罗瓦扩张和伽罗瓦群G是一个经典问题。最重要的结果是库默理论，它断言，如果一个正整数n在K中可逆，并且所有的n阶单位根都包含在K中，则K的所有n次循环扩张都是通过邻接方程t^n=a的一个根得到的。另一方面，如果K的特征p>0，阿廷-施赖尔理论断言K的所有p次循环扩张都是通过邻接方程t^p-t=a的根得到的。Witt向量理论给出了特征p> 0的域的p^n次循环扩张的优美描述。目前，在Galois上同调的框架下证明库默、Artin-Schreier和Artin-Schreier-Witt理论已成为标准。例如，库默理论是从称为库默序列的群方案的精确序列和称为希尔伯特9的伽罗瓦上同调的消失定理得出的 ...更多信息 0. Sekiguchi和Suwa构造了群概型的正合序列，它统一了库默序列和Artin-Schreier-Witt序列，最近又有一个问题引起了专家们的兴趣，即从库默理论中去掉K包含所有n次单位根的条件。小松建立了一个变种的库默理论，扭曲的库默理论的二次扩展。在这个研究项目中，Suwa概括了扭曲的库默理论在一个环，澄清扭曲的库默理论之间的关系，由于小松的理论和通用多项式的循环扩展由于Rikuna。Suwa还建立了一个统一扭曲的库默理论和Artin-Schreier理论的理论，在这个理论中，环的二次扩张的酉群方案起着重要的作用。利用二次扩张的正则表示，我们还得到了扭曲库默理论和扭曲Kummer-Artin-Schreier理论的紧化的一个很好的刻画. [1]T.Komatsu-Arithmetic of Rikuna's generic cyclic polynomial and generalization of库默theory. Mannapta Math 114（2004）265-279[2] Y. Rikuna-关于循环多项式的简单族。美国数学学会学报130（2002）2215-2218