On the deformations of cyclic Galoi coverings of algebraic curves

本研究的目的是构造特征域p(&gt；O)上完备非奇异曲线C的一对(C，sigma)的提升，以及它的p^n(n&lt；大于或等于&gt；1)阶的自同构sigma。对于n=1的情形，我们用Lang的类场理论得到了肯定的结果。本文的重要思想之一是构造Artin-Schreier理论到Kummer理论的变形，即构造加法群到乘法群的变形。因此，当我们试图解决n&lt；大于或等于&gt；2的问题时，首先必须构造Witt群W_n到环面(G_M)^n的变形，我们在1988年和1989年完全决定了当n=2时的这种变形。1990年，我们发现了Artin局部环上加群的环面扩张的一个消失性定理，并利用它研究了Artin局部环上群方案的扩张。此外，利用消失定理，我们成功地构造了Artin-Schreier-Witt正合列到Kummer型正合列的形变。在未来，我们必须构建Artin-Schreier-Witt理论和Kummer理论的统一理论，并将其应用于我们原来的问题。

Our aim of this research is to construct a lifting of a couple (C,sigma) of a complete non-singular curve C over a field of characteristic p (>O), and its automorphism sigma of order p^n (n<greater than or equal>1). For the case of n = 1, we have obtained an affirmative result by using Lang's class field theory. In the argument, one of the important ideas if to construct a deformation of the Artin-Schreier theory to the Kummer theory, i.e., to construct a deformation of an additive group to a multiplicative group. Therefore when we try to solve the problem for n<greater than or equal>2, first of all, we must construct the deformations of the Witt group W_n to a torus (G_m)^n. We decided Completely such deformations for n=2 in 1988 and 1989. In 1990, we discovered a vanishing theorem for extensions of additive groups by a torus over an Artin local ring, and usinit we treated the extensions of group schemes over a Artin local ring. Moreover, by using the vanishing theorem, we succeeded in constructing the deformations of the Artin-Schreier-Witt exact sequence to an exact sequence of Kummer type. In future, we must construct a unified theory of the Artin-Schreier-Witt and the Kummer theories, and apply it to our original problem.