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Study of Blowing-up rings

Research Project

Project/Area Number 13640044
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, Professor, 理工学部, 教授 (50060091)

Co-Investigator(Kenkyū-buntansha) KAMOI Yuji  Meiji Univ., School of Commerce, Assistant Prof., 商学部, 講師 (80308064)
NAKAMURA Yukio  Meiji Univ., School of Science and Technology, Associate Prof., 理工学部, 助教授 (00308066)
KURANO Kazuhiko  Meiji Univ., School of Science and Technology, Prof., 理工学部, 教授 (90205188)
IAI Shinichirou  Hokkaido University of Education, School of Education, Associate Prof., 教育学部, 助教授 (50333125)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)
KeywordsBuchsbaum ring / Cohen-Macaulay ring / FLC ring / local cohomology / index of reducibility / multiplicity / system of parameters / Gorenstein ring / Gorenstein環 / Buchsbaum環 / a-invariant / homological dimension / Rees代数 / 射影次元 / 入射次元 / 正則局所環 / 優良イデアル / Noetheian algebra / Bass数
Research Abstract

The purpose of this research is to study the following problems: (1) Find practical criteria for Rees algebras and associated graded rings of ideals to be Gorenstein rings. (2) Develop the theory of good ideals in Gorenstein local rings. As for the second problem we have developed a satisfactory new theory. Let (A, m) be a Gorenstein local ring of dimension d and let I be an m-primary ideal in A. Then we say that I is a good ideal in A, if the associated graded ring G(I) of I is a Gorenstein ring and a(G(I)) = 1-d. Although good ideals are the best ones next to the parameter ideals in A, we lacked deep investigations of the ideals of this kind. The project has performed a basic theory of them. While the project proceeded, unexpected results about integrally closed ideals were discovered, which led us to the other research on the reduction numbers of certain In-primary ideals in Buchsbaurn, or more generally, FLC local rings. The results generalize those on Cohen-Macaulay local rings that were given by Corso-Polini-Huneke-Vasconcelos. We especially summarized these investigations into the following three papers:
[1] S.Goto and H.Sakurai, The equality $1^2 = Q1$ in Buchsbaum rings, Rendiconti del Seminario Matematico dell'Universit di Padova 110(2003),25-56
[2] S.Goto and H.Sakurai, The reduction exponent of socle ideals, associated to parameter ideals in a Buchsbaum local ring of multiplicity two, J. Math. Soc. Japan (to appear)
[3] S.Goto and H.Sakurai, When does the equality $1^2=Q1$ hold true in Buchsbaum rings?, Pieprint 2003

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (81 results)

All Other

All Publications (81 results)

  • [Publications] S.Goto, S.Iai, 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(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, K.Nishida: "Finite modules of finite injective dimension over a Noetherian algebra"J.London Math.Soc.. 63. 319-335 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] D.Aghcheghloo, R.Enshaei, S.Goto, R.Y.Sharp: "Comparison of multigraded and ungraded Cousin complexes"Proc.Edinburgh Math.Soc.. 44. 365-378 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, Y.Nakamura: "Multiplicity and tight closures of parameters"J.Alg.. 244. 302-311 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, K.Nishida: "Towards a theory of Bass numbers with application to Gorenstein algebras"Colloquium Mathematicum. 91. 191-253 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, M.Kim: "Equimultiple good ideals"J.Math.Kyoto Univ.. 42-1. 21-32 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, S.Iai, M.Kim: "Good ideals in Gorenstein local rings obtained by idealization"Proc.Amer.Math.Soc.. 130. 337-344 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, Y.Nakamura: "The bound of the difference between parameter ideals and their tight closures"Tokyo J.Math.. 25-1. 41-48 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, S.Iai: "Gorenstein associated graded rings of analytic deviation two ideals"J.Alg.. 248. 708-723 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals"Proc.Amer.Math.Soc.. 130. 3159-3164 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals II"J.Math.Kyoto Univ.. 42-4. 631-639 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, S.Haraikawa: "Good ideals in Artinian Gorenstein local rings obtained by idealization"Tokyo J.Math.. 25-2. 493-499 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, F.Hayasaka, S.Iai: "The a-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings"Proc.Amer.Math.Soc.. 131. 87-94 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, K.Nishida: "Hilbert coefficients and Buchsbaumness of associated graded rings"J.Pure and Appl.Alg.. 181. 61-74 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, F.Hayasaka, S.Kasuga: "Towards a theory of Gorenstein m-primary integrally closed ideals"J.Herzog and V.Vuletescu (eds.), Commutative Algebra, Singularities and Computer Algebra. 159-177 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, Y.Shimoda: "Parametric decomposition of powers of ideals versus regularity of sequences"Proc.Amer.Math.Soc.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, H.Sakurai: "The equality $I^2 = QI$ in Buchsbaum rings"Rendiconti del Seminario Matematico dell'Universit di Padova. 110. 25-56 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, H.Sakurai: "The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two"J.Math.Soc.Japan. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, Y.Shimoda: "On the parametric decomposition of powers of parameter ideals in a Noetherian local ring"Tokyo J.Math.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, S.Iai, 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
      2003 Final Research Report Summary
  • [Publications] S.Goto, K.Nishida: "Finite modules of finite injective dimension over a Noetherian algebra"J. London Math. Soc. (2). 63. 319-335 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] D.Aghcheghlbo, R.Enshaei, S.Goto, R.Y.Sharp: "Comparison of multigraded and ungraded Cousin complexes"Proc. Edinburgh Math. Soc.. 44. 365-378 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, Y.Nakamura: "Multiplicity and tight closures of parameters"J.Alg.. 244. 302-311 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, K.Nishida: "Towards a theory of Bass numbers with application to Gorenstein algebras"Colloquium Mathematicum. 91. 191-253 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, M.Kim: "Equimultiple good ideals"J. Math. Kyoto Univ.. 42-1. 21-32 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, S.Iai, M.Kim: "Good ideals in Gorenstein local rings obtained by idealization"Proc.Amer.Math.Soc.. 130. 337-344 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, Y.Nakamura: "The bound of the difference between parameter ideals and their tight closures"Tokyo J. Math.. 25-1. 41-48 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, S.Iai: "Gorenstein associated graded rings of analytic deviation two ideals"J.Alg.. 248. 708-723 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals"Proc.Amer.Math.Soc.. 130. 3159-3164 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals II"J. Math. Kyoto Univ.. 42-4. 631-639 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, S.Haraikawa: "Good ideals in Artinian Gorenstein local rings obtained by idealization"Tokyo J. Math.. 25-2. 493-499 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, F.Hayasaka, S.Iai: "The a-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings"Proc.Amer.Math.Soc.. 131. 87-94 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, K.Nishida: "Hilbert coefficients and Buchsbaumness of associated graded rings"J. Pure and Appl. Alg.. 181. 61-74 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, F.Hayasaka, S.Kasuga: "Towards a theory of Gorenstein m-primary integrally closed ideals"Commutative Algebra, Singularities and Computer Algebra (J.Herzog, V.Vuletescu (eds.)) (The Proceedings of NATO Advanced Research Workshop) Sinaia (Romania). September 17-22. 159-177 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, N.Shimoda: "Parametric decomposition of powers of ideals versus regularity of sequences."Proc.Amer.Math.Soc.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, H.Sakurai: "The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two"J.Math.Soc.Japan. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, Y.Shimoda: "On the parametric decomposition of powers of parameter ideals in a Noetherian local ring"Tokyo J. Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] S.Goto, S.Iai, K.Watanabe: "Good ideals in Gorenstein local rings"Trans.Amer.Math.Soc.. 353. 2309-2346 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, K.Nishida: "Finite modules of finite injective dimension over a Noetherian algebra"J.London Math.Soc.. 63. 319-335 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] D.Aghcheghloo, R.Enshaei, S.Goto, R.Y.Sharp: "Comparison of multigraded and ungraded Cousin complexes"Proc.Edinburgh Math.Soc.. 44. 365-378 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, Y.Nakamura: "Multiplicity and tight closures of parameters"J.Alg.. 244. 302-311 (2001)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, K.Nishida: "Towards a theory of Bass numbers with application to Gorenstein algebras"Colloquium Mathematicum. 91. 191-253 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, M.Kim: "Equimultiple good ideals"J.Math.Kyoto Univ.. 42-1. 21-32 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, S.Iai, M.Kim: "Good ideals in Gorenstein local rings obtained by idealization"Proc.Amer.Math.Soc.. 130. 337-344 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, Y.Nakamura: "The bound of the difference between parameter ideals and their tight closures"Tokyo J.Math.. 25-1. 41-48 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, S.Iai: "Gorenstein associated graded rings of analytic deviation two ideals"J.Alg.. 248. 708-723 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals"Proc.Amer.Math.Soc.. 130. 3159-3164 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals II"J.Math.Kyoto Univ.. 42-4. 631-639 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, S.Haraikawa: "Good ideals in Artinian Gorenstein local rings obtained by idealization"Tokyo J.Math.. 25-2. 493-499 (2002)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, F.Hayasaka, S.Iai: "The a-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings"Proc.Amer.Math.Soc.. 131. 87-94 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, K.Nishida: "Hilbert coefficients and Buchsbaumness of associated graded rings"J.Pure and Appl.Alg.. 181. 61-74 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, F.Hayasaka, S.Kasuga: "Towards a theory of Gorenstein m-primary integrally closed ideals"J.Herzog and V.Vuletescu (eds.), Commutative Algebra, Singularities and Computer Algebra. 159-177 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, Y.Shimoda: "Parametric decomposition of powers of ideals versus regularity of sequences"Proc.Amer.Math.Soc.. (to appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, H.Sakurai: "The equality $I^2=QI$ in Buchsbaum rings"Rendiconti del Seminario Matematico dell'Universit di Padova. 110. 25-56 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, H.Ssakurai: "The reduction exponent of socle ideals associated to parameter ideals in a Buchsbaum local ring of multiplicity two."J.Math.Soc.Japan. to appear.

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, Y.Shimoda: "On the parametric decomposition of powers of parameter ideals in a Noetherian local ring"Tokyo J.Math.. to appear.

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Goto, M.-K.Kim: "Equimultiple good ideals"J.Math.Kyoto Univ.. 42-1. 21-32 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, S.Iai, M.-K.Kim: "Good Ideals in Gorenstein local rings obtained by idealization"Proc.Amer.Math.Soc.. 130. 337-344 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, S.Haraikawa: "Good Ideals in Artinian Gorenstein local rings obtained by idealization"Tokyo J.Math.. 25. 493-499 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, S.Iai: "Gorenstein associated graded rings of analytic deviation two ideals"J.Alg.. 248. 708-723 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals"Proc.Amer.Math.Soc.. 130. 3159-3164 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals, II"J.Math.Kyoto Univ.. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, F.Hayasaka, S.Iai: "The $\roman{a}$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings"Proc.Amer.Math.Soc.. 131. 87-94 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, K.Nishida: "Hilbert coefficients and Buchsbaumness of associated graded rings"J.Pure and Appi.Alg.. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, Y.Shimoda: "Parametric decomposition of powers of ideals versus regularity of sequences"Proc.Amer.Math.Soc.. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, F.Hayasaka, S.Kasuga: "Towards a theory of Gorenstein $\frak{m}$-primary integrally closed ideals"The proceedings of NATO Advanced Workshop, Sinaia(Romania), September 17-21, 2002. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Goto, Y.Nakamura: "The bound of the difference between parameter ideals and their tight closures"Tokyo J.Math.. 25. 41-48 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 後藤四郎, 櫻井秀人: "いつ等式I^2=QIが成り立つか?"明治大学理工学部研究報告. 26. 19-22 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 後藤四郎, 櫻井秀人: "重複度2のBuchsbaum環と等式$I^2=QI$"明治大学理工学部研究報告. 27. 13-17 (2002)

    • Related Report
      2002 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
      2001 Annual Research Report
  • [Publications] 後藤四郎, 西田憲司: "Finite modules of finite injective dimension over a Noetherian algebra"J. London Math. Soc.. 63. 319-335 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] D.Aghcheghloo, R.Enshaei, S.Goto, R.Y.Sharp: "Comparison of multigraded and ungraded Cousin complexes"Proc. Edinburgh Math. Soc.. 44. 365-378 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 後藤四郎, 早坂太: "射影次元有限の整閉イデアルに付随する素イデアルの講造"明治大学理工学部研究報告. 25. 25-27 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 後藤四郎, 中村幸男: "Multiplicity and tight closures of parameters"J. Alg.. 244. 302-311 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 後藤四郎, 居相真一郎: "Towards a theory of Bass numbers with application to Gorenstein algebras"Colloquium Mathematicum. 91. 191-253 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Goto, M.-K.Kim: "Equimultiple good ideals"J. Math. Kyoto Univ.. 42-1. 21-32 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Goto, S.Iai, M.Kim: "Good ideals in Gorenstein local rings obtained by idealization"Proc. Amer. Math. Soc.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] 後藤四郎, 中村幸男: "The bound of the difference between parameter ideals and their tight closures"Tokyo J. Math.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] 後藤四郎, 早坂太: "Finite homological dimension and primes associated to integrally closed ideals"Proc. Amer. Math. Soc.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] 後藤四郎, 早坂太, 居相真一郎: "The $\roman{a}$-invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings"Proc. Amer. Math. Soc.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] 後藤四郎, 櫻井秀人: "いつ等式1^2=QIが成り立つか?"明治大学理工学部研究報告. (to appear).

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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