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1994 Fiscal Year Final Research Report Summary

Deformation theory of group schemes and Construction of extensions

Research Project

Project/Area Number 05640063
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionCHUO UNIVERSITY

Principal Investigator

SEKIGUCHI Tsutomu  Chuo Univ., Dept.of Math., Professor, 理工学部, 教授 (70055234)

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)
Project Period (FY) 1993 – 1994
KeywordsGroup scheme / Artin-Schreier / Kummer / algebraic curve / extension
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.

  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] 関口 力: "Anote on eptensions of algebraic and formal groups II" Mathematische Zeitschrifh. 217. 447-457 (1994)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 関口 力: "On the unified Kummer-Artin-Scherier-Witt theory" Chuo Math.Preprint Series. 41. 1-43 (1994)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 関口 力: "Theorie de Kummer-Artin-Schreier of applications" Journal de Theorie des Nombres de Bordeaux. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 関口 力: "On the structure of the group schome Z〔Z/p^n〕^x" Conpositio Mathematica. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Sekiguchi: ""A note on extensions of algebraic and formal groups II"" Mathematische Zeitschrift. Vol.217. 447-457 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Sekiguchi: ""On the unified Kummer-Artin-Schreier-Witt theory"" Chuo Math. Preprint Series. Vol.41. 1-43 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Sekiguchi: ""Theorie de Kummer-Artin-Schreier et applications"" Preprint. (to appear in Journal de Theorie des Nombres de Brodeaux).(1994)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Sekiguchi: ""On the structure of the group scheme Z[Z/p^n]^x"" Preprint. (to appear in Compositio Mathematica).

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1996-04-15  

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