2003 Fiscal Year Annual Research Report

Blow-up ringsの環構造の研究

Research Project

Project/Area Number	13640044
Research Institution	Meiji University
Principal Investigator	後藤四郎明治大学, 理工学部, 教授 (50060091)
Co-Investigator(Kenkyū-buntansha)	鴨居祐二明治大学, 商学部, 講師 (80308064) 中村幸男明治大学, 理工学部, 助教授 (00308066) 蔵野和彦明治大学, 理工学部, 教授 (90205188) 居相真一郎北海道教育大学, 教育学部, 助教授 (50333125)
Keywords	Noether局所環 / Buchsbaum局所環 / 整閉イデアル / 重複度 / index of reducibility / 局所コホモロジー / Cohen-Macaulay局所環 / FLC局所環
Research Abstract	この研究は,次の課題とこれに密接に関連する課題について,期間内に成果を挙げることを目標とした。 (1)Rees代数と随伴次数環のGorenstein性を判定する簡便かつ実際的な方法を開発し,その判定法に基づく豊かな具体例を提供する。 (2)随伴次数環のGorenstein性に関連して,Gorenstein局所環内の「優良イデアル」の理論を整備。充実させる。既に,Gorenstein局所環内の「優良イデアル」に関する予備的な研究が完成した。次元dのGorenstein局所環(A,m)内のm-準素イデアルlは,その随伴次数環G(l)がGorenstein環であってa-不変量について等式a(G(l))=1-dが成立つとき「優良である」と呼ぶ。優良イデアルは,Gorenstein局所環内でパラメーター系で生成されたイデアルに次いで特殊であり,優良イデアルの分布状況と構造解析からは,基礎環A自身の構造に関する豊かな情報が期待できるが,体系的な研究は他に類例を見ない。分担者の協力を得て,優良イデアルの理論を深め,関連する問題の解明にも力を注いだ。他に,関連する研究課題として,整閉イデアルとこれに付随する素イデアルの構造に関して,想定外の体系的な深い理論の可能性が見い出された。その応用として,Noether局所環内のパラメターイデアルとsocleに関する端麗な解析が可能となった。 2003年6月にはイタリアとポルトガルで,計2回可換環論の国際会議が開催される。出席して成果の発表を行うと同時に,海外に於けるこの分野の研究の進捗状況を探った。国内研究集会でも計3回の講演を行い,成果の発表を行った。

Research Products

(19 results)

All Other

All Publications (19 results)

[Publications] S.Goto, S.Iai, K.Watanabe: "Good ideals in Gorenstein local rings"Trans.Amer.Math.Soc.. 353. 2309-2346 (2001)
[Publications] S.Goto, K.Nishida: "Finite modules of finite injective dimension over a Noetherian algebra"J.London Math.Soc.. 63. 319-335 (2001)
[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)
[Publications] S.Goto, Y.Nakamura: "Multiplicity and tight closures of parameters"J.Alg.. 244. 302-311 (2001)
[Publications] S.Goto, K.Nishida: "Towards a theory of Bass numbers with application to Gorenstein algebras"Colloquium Mathematicum. 91. 191-253 (2002)
[Publications] S.Goto, M.Kim: "Equimultiple good ideals"J.Math.Kyoto Univ.. 42-1. 21-32 (2002)
[Publications] S.Goto, S.Iai, M.Kim: "Good ideals in Gorenstein local rings obtained by idealization"Proc.Amer.Math.Soc.. 130. 337-344 (2002)
[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)
[Publications] S.Goto, S.Iai: "Gorenstein associated graded rings of analytic deviation two ideals"J.Alg.. 248. 708-723 (2002)
[Publications] S.Goto, F.Hayasaka: "Finite homological dimension and primes associated to integrally closed ideals"Proc.Amer.Math.Soc.. 130. 3159-3164 (2002)
[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)
[Publications] S.Goto, S.Haraikawa: "Good ideals in Artinian Gorenstein local rings obtained by idealization"Tokyo J.Math.. 25-2. 493-499 (2002)
[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)
[Publications] S.Goto, K.Nishida: "Hilbert coefficients and Buchsbaumness of associated graded rings"J.Pure and Appl.Alg.. 181. 61-74 (2003)
[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)
[Publications] S.Goto, Y.Shimoda: "Parametric decomposition of powers of ideals versus regularity of sequences"Proc.Amer.Math.Soc.. (to appear).
[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)
[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.
[Publications] S.Goto, Y.Shimoda: "On the parametric decomposition of powers of parameter ideals in a Noetherian local ring"Tokyo J.Math.. to appear.

2003 Fiscal Year Annual Research Report

Blow-up ringsの環構造の研究

Principal Investigator

後藤 四郎 明治大学, 理工学部, 教授 (50060091)

Research Products

[Publications] S.Goto, S.Iai, K.Watanabe: "Good ideals in Gorenstein local rings"Trans.Amer.Math.Soc.. 353. 2309-2346 (2001)

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

[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)

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

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

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

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

[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)

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

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

[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)

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

[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)

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

[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)

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

[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)

[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.

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

後藤四郎明治大学, 理工学部, 教授 (50060091)