On the deformations of cyclic Galois coverings of algebraic curves

本研究的最终目的是将特征为p（0）的域上的完全非奇异曲线C及其p^n阶自同构σ的对（C，sigma）提升为特征为零的域上的对。为此，我们必须建立一个理论的变形维特群环面。n维Witt群W_n是W_<n-1>n通过G_a的扩张，它包含Z/p^n作为Z/p^<n-1>通过Z/p的扩张，我们所要求的变形应保持Witt群的这样的扩张。在本研究中，我们发现，要解决这类变形，需要Artin局部环上群概型的扩张群的消失定理。利用这个消失定理，我们证明了我们可以控制W_n作为W_<n-1>by G_a的扩张的变形，特殊化映射的满射性，W_n到环面的变形的存在性，保持滤子和常子群方案Z/p^n。利用这些定理，我们可以精确地构造Artin-SchreierWitt正合列到库默型正合列的变形.这些变形的正合列正好给出了Artin-Schreier-Witt理论和库默理论的统一理论。事实上，我们可以通过计算这些变形的Witt群的第一上同调群来检验统一理论。我们的理论是足够一般的，但不幸的是，我们不能从这个方向决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，更精确地考察群概型的同构变形，我们可以看到我们的变形定义在环Z &lt;（p）&gt;[mu_p^n]上。此外，这些变形应该从群环的单位群给出。从这个观点出发，我们也可以给出一些部分结果.

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.