Study of excellent rings and its neighborhood

• Project/Area Number: 16540032
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: KAWASAKI Takesi Tokyo Metropolitan University, Department of Mathematics and Information Science, assistant professor, 大学院理工学研究科, 助手 (40301410)
• Project Period (FY): 2004 – 2006
• Project Status: Completed (Fiscal Year 2006)
• Budget Amount: ¥3,600,000 (Direct Cost: ¥3,600,000); Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000); Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000); Fiscal Year 2004: ¥1,400,000 (Direct Cost: ¥1,400,000)
• Keywords: excellent rings / Cohen-Macaulay rings / Cousin complex / canonical module / annihilator theorem / Briancon-Skodaの定理 / 局所コホモロジー / コーエン・マコーレー環

Research Abstract

(1) In [K], I gave a necessarily and sufficient condition for a Noetherian ring to have an arithmetic Cohen-Macaulayfication.
I refine its construction by introducing a new notion, called p-standard sequences.
(2) In [K], computed the depth of the canonical module. I compute local cohomology modules of the canonical module.
(3) In 1978, Faltings prove the annihilator theorem. In 1982, he conjectured that the annihilator theorem holds under weaker assumption. I give an affirmative answer to his conjecture.
(4) In 1991, Hukene [H] proved the uniform Artin-Rees theorem and the uniform Briancon-Skoda theorem. He also conjectured that these theorem hold under weaker assumption. I give an affirmative answer to [H, Conjecture 2.13] concerning these theorems.
[F1] G.Faltings, Ueber die Annulatoren lokaler Kohomologiegruppen, Arch.Math. (Basel) 30 (1978), 473--476.
[F2] G.Faltings, Der Endlichkeitssatz in der lokalen Kohomologie, Math.Ann.255 (1981), 45--56.
[H] C.Huneke, Uniform bounds in noetherian rings, Invent.Math. 107 (1992), 203--223.
[K] T.Kawasaki, Finiteness of Cousin cohomologies.