1994 Fiscal Year Final Research Report Summary
Deformation theory of group schemes and Construction of extensions
| Project/Area Number |
05640063
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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 |
Algebra
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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) |
MOMOSE Fumiyuki Chuo Univ., Dept.of Math., Professor, 理工学部, 教授 (80182187)
ISHII Hitoshi Chuo Univ., Dept.of Math., Professor, 理工学部, 教授 (70102887)
MATSUYAMA Yoshio Chuo Univ., Dept.of Math., Professor, 理工学部, 教授 (70112753)
IWANO Masayoshi Chuo Univ., Dept.of Math., Professor, 理工学部, 教授 (70087013)
SATAKE Ichirou Chuo Univ., Dept.of Math., Professor, 理工学部, 教授 (00133934)
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| Project Period (FY) |
1993 – 1994
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| Keywords | Group scheme / Artin-Schreier / Kummer / algebraic curve / extension |
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| Research Abstract |
The group scheme over a discrete valuation ring which gives a deformation of a Witt group scheme to a torus is completely determined by giving a filtered structure on it. Among those groups schemes, we can specify which group scheme is stangard as a group scheme which gives the unified Kummer-Artin-Shcreier-Witt theory, and morover in lower degree cases we can show the uniqueness of such a normalized stangard group scheme.
The group scheme over a discrete valuation ring which gives a deformation of a Witt group scheme to a torus has a relationship with the unit group scheme of a group-ring scheme. In fact, we can analyze the structure of the unit group schemes of those group-ring schemes, and in a lower dimensional case we can decide the explicit relationship between the stangard deformation group schemes and such unit group schemes.
To construct the group schemes over a discrete valuation ring which gives a deformation of a Witt group scheme to a torus, we need to compute the homomorphism groups and cohomology groups of group schemes over an Artin local ring. When the Artin local ring is F_p-algebra, we could compute completely those groups.
In the future, the works which should be done are to conpactify the group schemes over a discrete valuation ring which gives a deformation of a Witt group scheme to a torus, and to decide the homomorphism groups and cohomology groups of group schemes over an Artin local ring when it is Z/p^n-algebra. We beleave that the fundamental methods of treating those works have been already given.
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