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

It is a classic 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.另一方面，如果K具有特征性p> 0，则Artin-Schreier理论断言，p的k的所有环状延伸均通过毗邻等式t^p-t = a的根来获得。 witt载体的理论对特征性p> 0领域的循环扩展进行了优雅的描述。如今，在Galois共同体框架中，证明Kummer，Artin-Schreier和Artin-Schreier-Witt理论是标准的。 For example, the Kummer theory follows from the exact sequence of group schemes called the Kummer sequence and the vanishing theory 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-统一的根源。 Komatsu建立了Kummer理论的一种变体，通过二次扩展扭曲了Kummer理论。在这项研究项目中，Suwa将扭曲的Kummer理论推广到一个环上，阐明了由于Komatsu的扭曲Kummer理论与关于Rikuna引起的环保扩展的通用多项式理论之间的关系。此外，苏瓦（Suwa）建立了一种理论，该理论统一了扭曲的库默理论和artin-schreier理论。在这项工作中，圈式扩展的统一组方案起着重要作用。我们还使用常规表示二次扩展的定期表示，对扭曲的Kummer理论和扭曲的Kummer-Artin-Schreier理论的压缩也有一个很好的描述。[1] T. komatsu-arithmetic rikuna的通用循环多项式和kummer理论的概括。manuscriptaMath 114（2004）265-279 [2] Y. y. rikuna-on rikuna-on Cyclic polynomials.procs.procs.proc。阿米尔。数学。 Soc。 130（2002）2215-2218更少