Study on homological conjectures and their applications

• Project/Area Number: 22840042
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Meiji University
• Principal Investigator: SHIMOMOTO Kazuma 明治大学, 研究・知財戦略機構, ポスト・ドクター (70588780)
• Project Period (FY): 2010 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥1,846,000 (Direct Cost: ¥1,420,000、Indirect Cost: ¥426,000); Fiscal Year 2011: ¥1,001,000 (Direct Cost: ¥770,000、Indirect Cost: ¥231,000); Fiscal Year 2010: ¥845,000 (Direct Cost: ¥650,000、Indirect Cost: ¥195,000)
• Keywords: 数論 / 代数幾何への応用 / フロベニウス写像 / 密着閉包 / F-特異点 / ホモロジカル予想 / 岩澤理論 / モジュラー形式 / p-進ヘッケ環 / ベルティニ型定理 / 可換環論 / 特異点 / ゼータ関数 / フロベウス写像

Research Abstract

I made an extensive study on homological conjectures in commutative algebra and developed some algebraic techniques. These techniques were applied to consider applications to singularities arising from algebraic geometry and the main conjecture of Iwasawa theory on modular forms.

Research Products

• [Journal Article] F-coherent rings with applications to tight closure theory (2011)
  • Author(s): 下元数馬
  • Journal Title: Journal of Algebra Volume: 338巻 Issue: 1 Pages: 24-34
  • DOI: 10.1016/j.jalgebra.2011.05.006
  • ISSN: 0021-8693
• [Presentation] Galois cohomology and Selmer group over normal domains (2011)
  • Author(s): 下元数馬
  • Organizer: The 7-th Japan-Vietnam Seminar
  • Place of Presentation: ベトナム・クイニョン大学
  • Year and Date: 2011-12-13
• [Presentation] Local Bertini theorem and the Iwasawa main conjecture (2011)
  • Place of Presentation: ユタ大学
  • Year and Date: 2011-02-26
• [Presentation] Workshop on almost purity at Ann Arbor (2011)
  • Place of Presentation: ミシガン大学
• [Presentation] (2011)
  • Organizer: The 7-th Japan-Vietnam Seminar
  • Place of Presentation: クイニョン大学(ベトナム)
• [Presentation] 局所ベルティニ定理と岩澤予想への応用 (2010)
  • Author(s): 下元数馬
  • Organizer: 第32回可換環論シンポジウム
  • Place of Presentation: 神奈川県三浦郡葉山町湘南国際村
  • Year and Date: 2010-12-12