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The Kummer-Artin-Schreier-Witt theory and the deformations of the Art theory

Research Project

Project/Area Number 08640059
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionCHUO UNIVERSITY

Principal Investigator

SEKIGUCHI Tsutomu  Chuo Univ., Fac.of Sci.& Engi., Prof., 理工学部, 教授 (70055234)

Co-Investigator(Kenkyū-buntansha) TOKIZAWA Masamichi  Chuo Univ., Fac.of Sci.& Engi., As.Prof., 理工学部, 教授 (50055117)
YAMAMOTO Makoto  Chuo Univ., Fac.of Sci.& Engi., Prof., 理工学部, 教授 (10158305)
MOMOSE Fumiyuki  Chuo Univ., Fac.of Sci.& Engi., Prof., 理工学部, 教授 (80182187)
MATSUYAMA Yoshio  Chuo Univ., Fac.of Sci.& Engi., Prof., 理工学部, 教授 (70112753)
佐武 一郎  中央大学, 理工学部, 教授 (00133934)
諏訪 紀幸  東京電機大学, 工学部, 教授 (10196925)
三松 佳彦  中央大学, 理工学部, 助教授 (70190725)
Project Period (FY) 1996 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 1998: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1997: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1996: ¥1,100,000 (Direct Cost: ¥1,100,000)
KeywordsWitt vectors / Artin-Hasse / Kummer theory / extension group / group scheme / Artin-Schreier-Witt theory / Witt群 / Artin-Schrecer-Wittl理論 / Artin-Schreier-Witt理論
Research Abstract

The deformation groups W_n of the group scheme W_n of Witt vectors of length ri to the torus G^n_ over valuation ring A are given inductively as the elements of Ext^1(W_<n-1>, g^<(lambda)>) starting from W_1=g^<(lambda)>. The extension groups Ext^1(W_<n-1>, G^n_) are essentially isomorphic to Hom(W_<n-1>, G_<m, A>) and hence the deformation groups W_n are determined by calculating the groups Hom(W_<n-1>, G_<m, A>) and Ext^1(W_<n-1>, G_<m, A>).
In the previous research, we showed the following.
Let p be a prime integer, and A be a Z(p)-algebra. Then we can construct the deformations of the Arti Hasse exponential series, and using these deformed Artin-Hasse exponential series, we can show that t groups Hom(g^<(lambda)>, G_<m, A>) and Ext^1 (g^<(lambda)>, G_<m, A>) are isomorphic to the kernel and the cokernel of the twist Frobenius endomorphism F^<(lambda)> = F-[lambda^<P-1>] : W*W, respectively.
In this research, we tried to generalize these isomorphy to higher dimensional cases. In fact, for a givgroup scheme W_n deforming W_nto G^n_, we can construct an
endomorphism psi : W_n * W_n consisting some endomorphisms of W containing deformed Frobenius endomorphisms of type F^<(lambda)>, and we can sha that the groups Hom(W_n, G_<m, A>) nad Ext^1 (W_n, G_<m, A>) are isomorphic to the kernel and the cokernel of respectivly.

Report

(4 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report
  • 1996 Annual Research Report
  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] 関口 力: "A note on eptensions of algedrsie and formal groups, III" Tohoku Math, J.49. 241-257 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 関口 力: "代数群と形式代数群の変形の例について" 数理解析研究所講究録(ホップ代数と量子群). 997. 44-57 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 関口 力: "Wn, _AのGm, _Aによる拡大について" 数理解析研究所講究録(リジッド幾何学と群作用). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 関口 力: "A note on ehtcnsions of algebraie and formal groups, IV" Chuo Math.Preprint Series. 48. 1-28 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Sekiguchi & N.Suwa: "A note on extensions of alge-braic and formal groups III" Tohoku J.of Math.49. 241-257 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Sekiguchi & N.Suwa: "A note on extensions of alge-braic and formal groups IV" Preprint Series CHUO MATH. NO.48. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Sekiguchi & N.Suwa: "On some examples of defor-mations of algebraic groups or formal groups, in Japanese" RIMS Kokyuroku 997, "Hopf alge-bras and Quantum groups". 44-57 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Sekiguchi & N.Suwa: "On the extensions of W_<n, A> by G_<m, A>" to appear in RIMS Kokyuroku "Rigid geometry and group action". (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 関口 力: "A note on extensions of algelerair and foumal groups III" Tohoku Math.J.49. 241-257 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 関口 力: "代数群と形式代数群の変形の例について" 数理解析研究所講究録(ホップ代数と量子群). 997. 44-57 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 関口 力: "Wn,AのGm,Aによる拡大について" 数理解析研究所講究録(リジッド幾何学と群作用). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 関口 力: "A not on ertensions of algobraic and journal groups,IV" Chuo Math:Prepient Series. 48. 1-28 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] 関口 力: "Anote on extensions of algebraic and formal groups III" Tohoku Math.J.49. 241-257 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 百瀬文之: "Modular conjecture for Q-curves and QM-curves" Waseda Univ.Technical Refort. 97・5. 1-26 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 関口力: "A note on extensions of algelraic and fornal growpsIII" Thohoku J.(1998年掲載予定).

    • Related Report
      1996 Annual Research Report
  • [Publications] 関口力: "代数群と形式代数群の変形の例について" 数理解析研究所講究録(ホップ代表と量子群). (掲載予定).

    • Related Report
      1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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