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.

关于正式群体的卡地亚理论框架，我们对Kummer-Artin-Schreier-Witt理论的描述获得了一些显着的结果。对于z [m] - 代数a，我们还影响了一个额外的w^（m）（m）（a），对witt vectors的经典理论进行了解释。 w^（m）（a）具有W（a）模块的结构，而Frobenius Map f^<（m）>和verschiebung映射在经典案例中定义在w^（m）（a）上。地图（A_0，A_1，A_2，…）→（MA_0，MA_1，MA_2，…）是W（A） - W^（M）（M）（M）（A）的W（A），可与Frobenius Maps和Verschiefungung Maps相关。另一方面，由g^（m）_a表示的小组方案在Kummer-Artin-Schreier-Witt Thoery中起着重要作用，而正式的G^（M）_A的正式完成G^（M）_a _a _a _A的零部分无非是正式的Group Group spf spf a [t]]，与正式的Group Law Law f（x，x，y）= x，y）= x + mx。这是W^（M）（a）对G^（M）_a上P型曲线的Cartier模块的同构的第一个主要结果。第二个主要结果是将w^（m）（a）的自由分辨率描述为d_a-module.furhtermore，如果ε是g^（m）_a的g^（λ）_a的扩展，则Cartier模块C（ε）是w^（λ）（a）的延伸。引入正式功率系列我们给出C（^^__ε）= hom_ <a-gr>（^^ __ W，ε）的明确描述。我们已经将结果写成，作为文章<关于代数和正式群体的扩展的注释V>，它将出现在日记中。