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Study of blow-up rings

Research Project

Project/Area Number 11640049
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

GOTO Shiro  Meiji University, School of Science and Technology, Department of Mathematics, Professor, 理工学部, 教授 (50060091)

Co-Investigator(Kenkyū-buntansha) IAI Shin-ichiro  Meiji University, School of Science and Technology, Assistant, 理工学部, 助手
NAKAMURA Yukio  Meiji University, School of Science and Technology, Department of Mathematics, Lecturer, 理工学部, 講師 (00308066)
桂田 祐史  明治大学, 理工学部, 助教授 (80224484)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
KeywordsBuchsbaum ring / Cohen-Macaulay ring / Gorenstein ring / canonical module / Rees algebra / associated graded ring / Buchsbaun環 / Cohen-Macaulay環 / Gorenstein環 / Rees代数 / Buchsbaum環 / S^1-manifold / ヒルベルト類体
Research Abstract

Let I be an m-primary ideal in a Gorenstein local ring (A, m) with dim A = d and assume that I contains a parameter ideal Q in A as a reduction. Then we say that I is good ideal in A if G = 【symmetry】_n≧_0I^n/I^<n+1> is a Gorenstein ring with a(G) = 1 - d. The associated graded ring G of I is a Gorenstein ring with a(G) = -d if and only if I = Q.Therefore, good ideals in our sense are good ones next to the parameter ideals Q in A.A basic theory of good ideals is developed by this project. We have that I is a good ideal in A if and only if I^2= QI and I = Q : I.Firstly a criterion for finite-dimensional Gorenstein graded algebras A over fields k to have the nonempty sets X_A of good ideals shall be given. Secondly in the case where d=1 we will give a correspondence theorem between the set X_A and the set Y_A of certain overrings of A.A characterization of good ideals of the case where d = 2 will be given in terms of the goodness in their powers. Thanks to Kato's Rieman-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show the structure of the set X_A of good ideals in A heavily depends on d = dim A.The set X_A may be empty if d ≦ 2, while X_A is necessarily infinite if d ≧ 3. To analyze this phenomenon we shall lastly explore monomial good ideals in the polynomial ring k[X_1, X_2, X_3] in three variables over a field k. Let I be an ideal in a Gorenstein local ring A.Then I is said to be an equimultiple good ideal if I contains a reduction Q = (a_1, a_2, …, a_s) generated by s elements in A and if the associated graded ring G(I)=【symmetry】_n≧_0I^n/I^<n+1> of I is a Gorenstein ring with a(G(I)) = 1 - s, where s = ht_AI.The structure of the sets X_<A,s> (s ≧ 0) of equimultiple good ideals I with ht_AI = s. Some of the results in the case where s = dim A are successfully generalized to those of equimultiple case with improvements.

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • Research Products

    (43 results)

All Other

All Publications (43 results)

  • [Publications] 後藤四郎: "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity"J.Alg.. 213. 604-661 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎・西田憲司: "Catenarity in module finite algebras"Proc. Amer. Math. Soc.. 127. 3495-3502 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎・西田憲司: "Minimal jnjective resolutions of Cohen-Macaulay isolated singularities"Arch. Math.. 73. 249-255 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎・原井川聡: "イデアル化によって得られたArtin Gor enstein局所環内のgood idealsの構造と分布について"明治大学科学技術研究所紀要. 38. 9-24 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎: "Cohen-Macaulayness versus negativity of a-invariants in Rees algebr as associated to ideals of minimal multiplicity"J.Pure and Applied Alg.. 152. 93-107 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎・居相真一郎: "Embeddings of certain graded rings into their canonical modules"J.Alg.. 228. 377-396 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎・原井川聡・居相真一郎: "Complete intersection in overrings of a certain one-dimensional Gor enstein graded local ring"J.Alg.. 233. 772-790 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎・居相真一郎・渡辺敬一: "Good ideals in Gorenstein local rings"Trans. Amer. Math. Soc.. 353. 2309-2346 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎・M.Kim: "Equimultiple good ideals"J.Math. Kyoto Univ.. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] D.Aghcheghloo. R.Enshaei, S.Goto, and R.Y.Sharp: "Comparison of multigraded and ungraded Cousin complexes"Proc. Edinburgh Math. Soc.. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎・居相真一郎・M.Kim: "Good ideals in Gorenstein local rings obtained by idealization"Proc. Amer. Math. Soc.. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 春日里枝・早坂太・後藤四郎: "1次元Noether局所環内の整閉なGorenstein m-準素イデアルの構造と分布について"明治大学理工学部研究報告. to appear.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto: "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity"J.Alg.. 213. 604-661 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and K.Nishida: "Catenarity in module finite algebras"Proc.Amer.Math.Soc.. 127. 3495-3502 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and K.Nishida: "Minimal injective resolutions of Cohen-Macaulay isolated singularities"Arch.Math.. 73. 249-255 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and S.Iai: "Embeddings of certain graded rings into their canonical modules"J.Alg.. 228. 377-396 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto: "Cohen-Macaulayness versus negativity of a-invariants in Rees algebras associated to ideals of minimal multiplicity"J.Pure and Applied Alg.. 152. 93-107 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto, S.Iai, and K.Watanabe: "Good ideals in Gorenstein local rings"Trans.Amer.Math.Soc.. 353. 2309-2346 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto, S.Haraikawa, and S.Iai: "Complete intersection in overrings of a certain one-dimensional Gorenstein graded local ring"J.Alg.. 233. 772-790 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] D.Aghcheghloo, R.Enshaei, S.Goto, and R.Y.Sharp: "Comparison of multigraded and ungraded Cousin complexes"Proc.Edinburgh Math.Soc.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and M.Kim: "Equimultiple good ideals"J.Math.Kyoto Univ.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and K.Nishida: "Finite modules of finite injective dimension over a Noetherian algebra"J.London Math.Soc.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto, S.Iai, and M.Kim: "Good ideals in Gorenstein local rings obtained by idealization"Proc.Amer.Math.Soc.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and K.Nishida: "Towards a theory of Bass numbers with application to Gorenstein algebras"(Preprint). (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals"(Preprint). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and Y.Nakamura: "On the multiplicity and the tight closure of parameter ideals"(Preprint). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and S.Iai: "Gorenstein associated graded rings of analytic deviation two ideals"(Preprint). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and S.Haraikawa: "Good ideals in Artinian Gorenstein local rings obtained by idealization"(Preprint). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Goto and Y.Nakamura: "Finiteness of sup _ql_R(q*/q) and sup_q {e_q(R) - l_R(R/q*)}"(Preprint). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 後藤四郎: "Cohen-Macaulayness versus negativity of a-invariants in Rees algebr as associated to ideals of minimal multiplicity"J.Pure and Applied Alg.. 152. 93-107 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 後藤四郎,居相真一郎: "Embeddings of certain graded rings into their canonical modules"J.Alg.. 228. 377-396 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 後藤四郎,原井川聡,居相真一郎: "Complete intersection in overrings of a certain one-dimensional Gorenstein graded local ring"J.Alg.. 233. 772-790 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 後藤四郎,居相真一郎,渡辺敬一: "Good ideals in Gorenstein local rings"Trans.Amer.Math.Soc.. 353. 2309-2346 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 後藤四郎,M.Kim: "Equimultiple good ideals"J.Math.Kyoto Univ.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] D.Aghcheghloo,R.Enshael,S.Goto,and R.Y.Sharp.: "Comparison of multigraded and ungraded Cousin complexes"Proc.Edinburgh Math.Soc.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] 後藤四郎,居相真一郎,M.Kim: "Good ideals in Gorenstein local rings obtained by idealization"Proc.Amer.Math.Soc.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] 後藤四郎,西田憲司: "Catenarity in module-finite algebras"Proc.Amer.Math.Soc. 127. 3495-3502 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 後藤四郎,西田憲司: "Minimal injective resolutions of Cohen-Macaulay isolated singularities"Archiv der Mathematik. 73. 249-255 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 後藤四郎: "Buchsbaumness in Rees algebras associated to ideals of minimal multiplicity"J.Alg.. 213. 604-661 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] 後藤四郎: "Cohen-Macaulayness versus negativity of a-invariants in Rees algebras associated to ideals of minimal multiplicity"Journal of Pure and Applied Algebra. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] 原井川聡,後藤四郎: "イデアル化によって得られたArtin Gorenstein局所環内のgood idealsの構造と分布について"明治大学科学研究所紀要. 38. 9-24 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] D.Aghcheghloo,R.Enshaei,S.Goto,and R.Y.Sharp,: "Comparison of multigraded and ungraded Cousin complexes"Proc.Edinburgh Math,Soc.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] 後藤四郎,居相真一郎: "Embeddings of certain graded rings into their canonical modules"Journal of Algebra. (to appear).

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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