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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.

本研究的最终目的是将特征为p(0)域上的完全非奇异曲线C及其p^n阶自同构的一对（C, sigma）提升到特征为0域上的一对（C, sigma）。为此，我们必须建立维特群对环面变形的理论。n维Witt群W_n是W_<n-1>经G_a的扩展，它包含Z/p^n作为Z/p^<n-1>经Z/p的扩展。我们所要求的变形应该保留维特群的这种特性。在本研究中，我们发现要处理这类变形，我们需要一种关于群方案在局部环上的可拓群的消失定理。此外，利用该消失定理，我们证明了W_n的变形可以控制为W_<n-1>的扩展G_a，专化映射的满射性，以及W_n的变形存在于保持滤波和恒定子群格式Z/p^n的环面。利用这些定理，我们可以精确地构造Artin-SchreierWitt精确序列到kummer型精确序列的变形。这些变形的精确序列给出了Artin-Schreier-Witt理论和Kummer理论的统一理论。事实上，我们可以通过计算这些变形Witt群的第一个上同群来检验统一理论。我们的理论是足够普遍的，但不幸的是，我们不能从这个方向确定这些变形的定义环，为此我们必须发展另一种方法。事实上，更精确地观察群格式的等同性变形，我们可以看到W_n的变形是在环Z_<(p)>[mu_p^n]上定义的。而且，这些变形必须由群环的单位群给出。从这个角度来看，我们也可以给出一些部分的结果。