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Study on subrings of polynomial rings

Research Project

Project/Area Number 16540018
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionUniversity of Fukui

Principal Investigator

ONODA Nobuharu  University of Fukui, Engineering, Applied Physics, Professor, 工学部, 教授 (40169347)

Project Period (FY) 2004 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
KeywordsInternational collaboration / India / polynomial ring / finite generation / fiber ring / valuation ring
Research Abstract

The purpose of the study is to consider the following problem on subrings A of a polynomial ring in n-variables over a commutative ring R :
(1) Find conditions for A to be finitely generated over R.
(2) Find conditions for A to be a polynomial ring or an A^<[r]->fibration over R.
For the first problem, in collaboration with Dr.Amartya K.Dutta, I investigated the case where R is a discrete valuation ring and n=1, and gave a condition for the closed fiber of A over R to be finitely generated. In connection with this result, I studied Noetherian subrings A of a polynomial ring in one variable over a unique factorization domain R, and gave a condition for A to be finitely generated over R. Furthermore I proved that, under this condition, A is a polynomial ring.
For the problem (2), I investigated a faithfully flat integral domain A over a unique factorization domain R such that generic and codimension one fibers of A over R are polynomial rings in one variable. I proved that such A is a direct limit of certain algebras, and using this result I gave a condition for A to be a polynomial ring.
Concerning the problem (2), I studied the following problem with Professor T.Asanuma (Toyama University) :
Let R be a valuation ring with quotient field K and let V be a valuation ring of an algebraic function field K(x,y) in one variable over K such that V dominates R. Find out the algebraic structure of the residue field of V.
For this problem, we investigated the case where K(x,y) is a hyperelliptic function field defined by y^2=x^n+ax+b, and proved that among the valuation rings V of K(x,y) dominating R, there exists at most one V such that the residue field of V is not a rational function field in one variable over the residue field of R. Furthermore, for such V, we determined the defining equation of the residue field of V.

Report

(3 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • Research Products

    (12 results)

All 2006 2005 2004

All Journal Article (12 results)

  • [Journal Article] Valuation rings of algebraic function fields in one variable2006

    • Author(s)
      T.Asanuma, N.Onoda
    • Journal Title

      Affine Algebraic Geometry

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Valuation rings of algebraic function fields in one variable2006

    • Author(s)
      T.Asanuma, N.Onoda
    • Journal Title

      Affine Algebraic Geometry (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Commutative group algebras generated by idempotents2006

    • Author(s)
      H.Kawai, N.Onoda
    • Journal Title

      Mathematics Journal of Toyama University

    • NAID

      110004681245

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Generic fibrations by A^1 and A^* over discrete valuation rings2005

    • Author(s)
      T.Asanuma, N.Onoda
    • Journal Title

      Contemporary Mathematics 369

      Pages: 47-62

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Generic fibrations by affine curves over discrete valuation rings2005

    • Author(s)
      T.Asanuma, N.Onoda
    • Journal Title

      第26回可換環論シンポジウム報告集

      Pages: 36-44

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Commutative group algebras generated by idempotents2005

    • Author(s)
      H.Kawai, N.Onoda
    • Journal Title

      Toyama Mathematical Journal 28

      Pages: 41-54

    • NAID

      110004681245

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generic fibrations by A^1 and A^* over discrete valuation rings2005

    • Author(s)
      T.Asanuma, N.Onoda
    • Journal Title

      Contemporary Mathematics Vol.369

      Pages: 47-62

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generic fibrations by affine curves over discrete valuation rings2005

    • Author(s)
      T.Asanuma, N.Onoda
    • Journal Title

      26th Symposium on Commutative Algebra

      Pages: 36-44

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Commutative group algebras generated by idempotents2005

    • Author(s)
      H.Kawai, N.Onoda
    • Journal Title

      Toyama Mathematical Journal Vol.28

      Pages: 41-54

    • NAID

      110004681245

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Generic fibrations by affine curves over discrete valuation rings2005

    • Author(s)
      T.Asanuma, M.Onoda
    • Journal Title

      第26回可換環論シンポジウム報告集

      Pages: 36-44

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Idealizers, Complete Integral Closures and Almost Pseudo-valuation Domains2004

    • Author(s)
      N.Onoda, T.Sugatani, et al.
    • Journal Title

      Kyungpook Mathematical Journal 44

      Pages: 557-563

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Idealizers, Complete Integral Closures and Almost Pseudo-valuation Domains2004

    • Author(s)
      N.Onoda, T.Sugatani, et al.
    • Journal Title

      Kyungpook Mathematical Journal Vol.44

      Pages: 557-563

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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