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.

我们研究的主要目的是制定和发展类似于Kummer和Artin-Schreier理论的理论，这些理论是Galois理论的重要项目以及Etale共同体理论的起点。不用说，加洛瓦理论是现代代数的基础。 Etale同道学的理论为我们提供了算术几何形状的强大工具，该工具在过去半个世纪中已经大大发展了。在过去的半个世纪中，我们为有限的平面组方案获得了几个结果，涉及cleft型Hopf-Galois扩展基础之间的正常基础的关系与有限平面方案的单位组方案之间的关系。这是对Serre的阐述的自然概括，即在田间延伸的正常基础之间的关系和有限基团的组代数组的单位组。

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