On the deformations of cyclic Galois coverings of algebraic curves

关于代数曲线循环伽罗瓦覆盖的变形

基本信息

批准号：
02640075
负责人：
SEKIGUCHI Tsutomu
金额：
$ 0.45万
依托单位：
Chuo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1990
资助国家：
日本
起止时间：
1990 至 1991
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-02640075/
关键词：
Witt group Artin-Schreier Kummer algebraic curve extension Group scheme クンマ-拡大拡大群群変形

项目摘要

Our final aim of this research is to lift a pair (C, sigma), of a complete nonsingular curve C and its automorphism sigma of order p^n over a field of characteristic p(0), to a pair over a field of characteristic zero. For this purpose, we must construct a theory of deformations of Witt groups to tori. The n-dimensional Witt group W_n is an extension of W_<n-1> by G_a, and it contains Z/p^n as the extension of Z/p^<n-1> by Z/p. The deformations we require should preserves thus ffitrations of Witt groups. In this research, we found out that to hnadle this kind of deformations we needed a kind of vanishing theorem df extension groups of group schemes over an Artin local rings. More-over, using this vanishing theorem, we showed that we could control the deformations of W_n as an extension of W_<n-1> by G_a, the surjectivity of a specialization map, and the existence of deformations of W_n to a torus keeping the filtrations and the constant subgroup scheme Z/p^n. Using tyhese theorems, we could precisely construct the deformations of Artin-SchreierWitt exact sequences to exact sequences of kummer type. These deformed exact sequences give exactly the unified theory of the Artin-Schreier-Witt theory and the Kummer theory. In fact, we can check the unified theory by computing the first cohomology group for these deformed Witt groups. Our theory is sufficiently general, but unfortunately we could not decide the defining rings of these deformations from this direction, and for this purpose we must develop another kind of method. In fact, looking the deformations of isogenies of group schemes more precisely, we can see that our deformation of W_n is defined over the ring Z_<(p)>[mu_p^n]. Moreover, these deformations should be given from the unit groups of group rings. From this view point, we could also give some partial results.

我们的这项研究的最终目的是将一对（C，Sigma）提升为完全非曲线C及其在特征p（0）领域上的阶p^n的自动形态sigma，以在特征零的领域上方。为此，我们必须构建Witt组变形的理论。 n维witt组w_n是g_a的w_ <n-1>的扩展，它包含z/p^n作为z/p^<n-1>的扩展，z/p^<n-1>由z/p。我们需要的变形应保留，从而对WITT组的fitriation。在这项研究中，我们发现，对于这种变形，我们需要一种消失的定理DF扩展小组组成方案，这是在Artin本地环上。更重要的是，使用该消失的定理，我们表明我们可以控制W_N的变形作为g_a的W_ <n-1>的扩展，G_A，专业图的过度映射，以及W_N的变形为圆环的变形存在，以保持过滤和恒定的子组方案z/p^n。使用TYHESE定理，我们可以精确地构建Artin-SchreierWitt精确序列的变形，以与Kummer类型的精确序列构建。这些变形的精确序列完全赋予了Artin-Schreier-Witt理论和Kummer理论的统一理论。实际上，我们可以通过计算这些变形的WITT组的第一个共同体学组来检查统一理论。我们的理论足够一般，但是不幸的是，我们无法从这个方向决定这些变形的定义环，为此，我们必须开发另一种方法。实际上，更精确地查看小组方案的异基因变形，我们可以看到我们的W_N的变形是通过环z _ <（p）> [mu_p^n]定义的。此外，这些变形应来自组环的单位组。从这个角度来看，我们还可以给出一些部分结果。