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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

在给定域K和有限群G的情况下，用G构造K的伽罗瓦扩展是一个经典问题。最重要的结果是Kummer理论，它断言，如果正整数n在K中可逆，并且K中包含所有单位的n次根，则K的所有n次循环扩展都可以通过相邻方程t^n=a的根得到。另一方面，如果K的特征为p>0，则Artin-Schreier理论断言，K的所有p次循环扩展都是通过邻接方程t^p-t=a的根得到的。Witt向量理论给出了特征为p>的域的p^n次循环扩展的一个优美的描述。目前，在伽罗瓦上同调的框架下证明Kummer、Artin-Schreier和Artin-Schreier- witt理论已成为标准。例如，Kummer理论是从群方案的精确序列（称为Kummer序列）和伽罗瓦上同调的消失定理（称为Hilbert）推导出来的。Sekiguchi和Suwa构造了群格式的精确序列，它统一了Kummer序列和Artin-Schreier-Witt序列。最近另一个引起专家兴趣的问题是从Kummer理论中去掉K包含所有单位的n次方根的条件。小松建立了Kummer理论的变体，通过二次推广扭转Kummer理论。在本研究项目中，Suwa推广了环上的扭曲Kummer理论，阐明了Komatsu的扭曲Kummer理论与Rikuna的环扩展的一般多项式理论之间的关系。此外，Suwa还建立了一个将扭曲的Kummer理论与Artin-Schreier理论相结合的理论。在这项工作中，环的二次扩展的酉群格式起着重要的作用。我们也很好地描述了扭曲Kummer理论和扭曲Kummer- artin - schreier理论的紧化，使用了二次扩展的正则表示Rikuna一般循环多项式的算法及Kummer理论的推广。数学学报，114(2004):265-279[j] . rikuna -关于循环多项式的简单族。Proc,阿米尔。数学。Soc. 130 (2002) 2215-2218