1989 Fiscal Year Final Research Report Summary
On the deformations of cyclic Galoi coverings of algebraic curves
| Project/Area Number |
62540066
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Chuo University |
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Principal Investigator |
SEKIGUCHI Tsutomu Chuo Univ., Dept. of Math., Professor, 理工学部, 教授 (70055234)
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| Co-Investigator(Kenkyū-buntansha) |
MATSUYAMA Yoshio Chuo Univ., Dept. of Math., Professor, 理工学部, 教授 (70112753)
ISHII Hitoshi Chuo Univ., Dept. of Math., Professor, 理工学部, 教授 (70102887)
IWANO Masahiro Chuo Univ., Dept. of Math., Professor, 理工学部, 教授 (70087013)
KURIBAYASHI Akikazu Chuo Univ., Dept. of Math., Professor, 理工学部, 教授 (40055033)
SEKINO Kaoru Chuo Univ., Dept. of Math., Professor, 理工学部, 教授 (40054994)
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| Project Period (FY) |
1987 – 1989
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| Keywords | Witt group / Artin-Schreier / Kummer / Algebraic curve / Extension / Group scheme |
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| Research Abstract |
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.
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