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Applications of the Kummer-Artin-Schreier-Witt theory to Number Theory and to Algebraic Geometry

Kummer-Artin-Schreier-Witt 理论在数论和代数几何中的应用

基本信息

批准号：
12640041
负责人：
SUWA Noriyuki
金额：
$ 1.15万
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：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640041/
关键词：
Kummer Theory Witt vector Artin-Hasse exponential series Artin-Schreier-Witt theory algebraic group formal group Cartier theory Artin-Hasse exponential series Artin-Hasse exponention series

项目摘要

We have gotten some remarkable results concerning to a description of the Kummer-Artin-Schreier-Witt theory in the framework of the Cartier theory for formal groups. We intorduce an additive group W^(M) (A) for a Z[M]-algebra A, paraphrasing the classical theory of Witt vectors. W^(M)(A) has a structure of W(A)-module, and the Frobenius map F^<(M)> and the Verschiebung map are defined on W^(M)(A) as in the classical case. The map(a_0, a_1,a_2, … ) →(Ma_0,Ma_1,Ma_2, … ) is a W(A)-homomorphism of W^(M)(A) to W(A), com-patible with the Frobenius maps and the Verschiebung maps. On the other hand, the group scheme denoted by G^(M)_A plays an important role in the Kummer-Artin-Schreier-Witt thoery, and the formal completion G^(M)_A along the zero section of G^(M)_A is nothing but the formal group Spf A[[T]] with a formal group law f(X,Y) = X + Y + MXY. It is the first main result that W^(M)(A) is isomorphic to the Cartier module of p-typical curves on G^(M)_A. The second main result is a description of free resolution of W^(M)(A) as a D_A-module.Furhtermore, if ε is an extension of G^(Λ)_A by G^(M)_A, the Cartier module C(ε) is an extension of W^(Λ)(A) by W^(M)(A). Introducing a formal power serieswe give an explicit description of C(^^^__ε) = Hom_<A-gr>(^^^__W,ε). We have written up our results as the article <A note on extensions of algebraic and formal groups, V>, which will appear in a journal.

我们在卡地亚形式群理论框架内描述 Kummer-Artin-Schreier-Witt 理论方面取得了一些显着的成果。我们为 Z[M]-代数 A 引入加法群 W^(M) (A)，解释维特向量的经典理论。 W^(M)(A) 具有 W(A) 模的结构，并且与经典情况一样，Frobenius 映射 F^<(M)> 和 Verschiebung 映射在 W^(M)(A) 上定义。映射(a_0, a_1,a_2, … ) →(Ma_0,Ma_1,Ma_2, … ) 是 W^(M)(A) 到 W(A) 的 W(A) 同态，与 Frobenius 映射和 Verschiebung 映射兼容。另一方面，G^(M)_A 表示的群方案在 Kummer-Artin-Schreier-Witt 理论中起着重要作用，沿着 G^(M)_A 的零部分的形式完成 G^(M)_A 只不过是具有形式群律 f(X,Y) = X + Y + MXY 的形式群 Spf A[[T]]。第一个主要结果是W^(M)(A)同构于G^(M)_A上的p-典型曲线的Cartier模。第二个主要结果是将W^(M)(A)的自由解析描述为D_A模块。此外，如果ε是G^(Λ)_A通过G^(M)_A的扩展，则卡地亚模块C(ε)是W^(Λ)(A)通过W^(M)(A)的扩展。引入形式幂级数，我们给出 C(^^^__ε) = Hom_<A-gr>(^^^__W,ε) 的显式描述。我们已经将我们的结果写成文章<关于代数和形式群的扩展的注释，V>，它将出现在期刊上。