Around Kummer theories, from the view point of group schemes

围绕库默尔理论，从群体方案的角度

• 批准号: 23540027
• 负责人: SUWA Noriyuki
• 金额: $ 1.66万
• 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: 2011
• 资助国家: 日本
• 起止时间: 2011 至 2013
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-23540027/
• 关键词: Kummer理論; Artin-Schreier理論; group scheme; torsor; Hopf代数; Hopf-Galois理論; Ｈopf代数

It is a main purpose of our research to formulate and develop the theories analogous to the Kummer and Artin-Schreier theories, which are important items of the Galois theory and the starting points of the theory of etale cohomology. Needless to say, the Galois theory is a basis of the modern algebra. And the theory of etale cohomology provides us with a powerful tool in arithmetic geometry, which has developed vastly during the past half a century.In particular we obtain several results for finite flat group schemes, concerning to relations between normal bases of a cleft Hopf-Galois extension and the unit group scheme of a finite flat group scheme. This is a natural generalization of Serre's arugument on relations between normal bases of a Galois extension of fields and the unit group of the group algebra of a finite groups.

形成和发展类似于Kummer理论和Artin-Schreier理论的理论是我们研究的一个主要目的。Kummer理论和Artin-Schreier理论是伽罗瓦理论的重要组成部分，也是上同调理论的起点。不用说，伽罗瓦理论是现代代数的基础。在过去的半个世纪里，上同调理论为我们提供了一个强有力的工具，它在算术几何中得到了极大的发展，特别是关于有限平坦群方案的几个结果，关于裂Hopf-Galois扩张的正规基与有限平坦群方案的单位群方案之间的关系。这是Serre关于域的Galois扩张的正规基与有限群的群代数的单位群之间关系的一个自然推广。

项目成果

Kummer theory for algebraic tori and normal basis problem : some examples

代数环面和正规基问题的库默理论：一些例子

• 发表时间: 2013
• 作者: Y.Komori;K.Matsumoto and H. Tsumura;鍬田 政人;Hiroshima Math. J
• 通讯作者: Hiroshima Math. J

Algebraic Number Theory and Related Topics 2011

代数数论及相关主题 2011

• 发表时间: 2013
• 作者: N. Suwa;A. Shiho;K. Sato (eds)
• 通讯作者: K. Sato (eds)

Kummer theory for algebraic tori and normal basis problem

代数环面和正规基问题的 Kummer 理论

• 发表时间: 2013
• 作者: Y.Komori;K.Matsumoto and H. Tsumura;鍬田 政人;Hiroshima Math. J;Masanori Katsurada;M. Katsurada and T. Noda;津村博文;N. Suwa
• 通讯作者: N. Suwa

Algebraic Number Theory and Related Topics 2010

代数数论及相关主题 2010

• 发表时间: 2012
• 作者: M. Kida;N. Suwa;S. Kobayashi (eds)
• 通讯作者: S. Kobayashi (eds)

Artin-Schreier-Witt extensions and normal bases

Artin-Schreier-Witt 扩展和普通碱基

• 发表时间: 2012
• 期刊: Hiroshima Math. J
• 作者: Y. Hamahata;T. Ichikawa;A. Murase;T. Sugano;Masanori Katsurada;K.Matsumoto and H. Tsumura;Masato Kuwata;N. Suwa
• 通讯作者: N. Suwa