Chromatic redshift and homotopical algebraic geometry

• Project/Area Number: 22540087
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Okayama University
• Principal Investigator: TORII Takeshi 岡山大学, 大学院・自然科学研究科, 准教授 (30341407)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 安定ホモトピー圏 / クロマティック赤方偏移 / ホモトピー論的代数幾何 / ホモトピー固定点スペクトラム / Adamsスペクトル系列 / HKR指標 / Morava E理論 / Chern指標 / 可換S代数 / ホモトピカル代数幾何 / クロマティックレベル / 形式群 / ホモトピー固定点

Research Abstract

I studied the relationship among the layers of the chromatic filtration in the stable homotopy category bymeans of homotopical algebraic geometry. I obtained the relationship between the Hecke operators on the Morava E-theories with different heights. I also showed that any spectrum which is local with respect to the Morava K-theory can be obtained as the homotopy fixed point spectrum for its Morava module in collaboration with D.G.Davis. Furthermore, I obtained results on the embeddings of the module categories for the Galois extensions of commutative S-algebras.

Research Products

• [Journal Article] Rational homotopy type of the moduli of representations with Borel mold (2012)
  • Author(s): Nakamoto Kazunori and Torii Takeshi
  • Journal Title: Forum Math Volume: 24 no. 3 Pages: 507-538
  • Peer Reviewed
• [Journal Article] Every K(n)-local spectrum is the homotopy fixed points of its Morava module (2012)
  • Author(s): Daniel G Davis and Torii Takeshi
  • Journal Title: Proc. Amer. Math. Soc Volume: 140 no. 3 Pages: 1097-1103
  • Peer Reviewed
• [Journal Article] Rational homotopy type of the moduli of representations with Borel mold (2012)
  • Author(s): Kazunori Nakamoto and Takeshi Torii
  • Journal Title: Forum Mathematicum Volume: 24 Issue: 3 Pages: 507-538
  • DOI: 10.1515/form.2011.071
  • Peer Reviewed
• [Journal Article] Every K(n)-local spectrum is the homotopy fixed points of its Morava module (2012)
  • Author(s): Daniel G Davis and Takeshi Torii
  • Journal Title: Proceedings of the American Mathematical Society Volume: 140 Issue: 3 Pages: 1097-1103
  • DOI: 10.1090/s0002-9939-2011-11189-4
  • Peer Reviewed
• [Journal Article] Topology of the representation varieties with Borel mold for unstable cases (2011)
  • Author(s): Nakamoto Kazunori and Torii Takeshi
  • Journal Title: J. Aust. Math. Soc Volume: 91 no. 1 Pages: 55-87
  • Peer Reviewed
• [Journal Article] K(n)-localization of the K(n+1)-local E_{n+1}-Adams spectral sequences (2011)
  • Author(s): Torii Takeshi
  • Journal Title: Pacific J. Math Volume: 250 no. 2 Pages: 439-471
  • Peer Reviewed
• [Journal Article] K(n)-localization of the K(n+1)-local E_{n+1}-Adams spectral sequences (2011)
  • Author(s): Torii Takeshi
  • Journal Title: Pacific Journal of Mathematics Volume: 250 Issue: 2 Pages: 439-471
  • DOI: 10.2140/pjm.2011.250.439
  • Peer Reviewed
• [Journal Article] Comparison of Morava E-theories (2010)
  • Author(s): Torii Takeshi
  • Journal Title: Math. Z Volume: 266 no. 4 Pages: 933-951
  • Peer Reviewed
• [Journal Article] On E_∞-structure of the generalized Chern character (2010)
  • Author(s): Torii Takeshi
  • Journal Title: Bull. Lond. Math. Soc Volume: 42 no. 4 Pages: 680-690
  • Peer Reviewed
• [Journal Article] HKR characters, p-divisible groups and the generalized Chern character (2010)
  • Author(s): Tori i Takeshi
  • Journal Title: Trans. Amer. Math. Soc Volume: 362 no. 11 Pages: 6159-6181
  • Peer Reviewed
• [Journal Article] Comparison of Morava E-theories (2010)
  • Author(s): Torii Takeshi
  • Journal Title: Mathematische Zeitschrift Volume: 266 Pages: 933-951
  • Peer Reviewed
• [Journal Article] On E_∞-structure of the generalized Chern character (2010)
  • Author(s): Torii Takeshi
  • Journal Title: Bulletin of the London Mathematical Society Volume: 42 Pages: 680-690
  • Peer Reviewed
• [Journal Article] HKR characters, p-divisible groups and the generalized Chern character (2010)
  • Author(s): Torii Takeshi
  • Journal Title: Transactions of the American Mathematical Society Volume: 362 Pages: 6159-6181
  • Peer Reviewed
• [Presentation] On K(1)-local topological modular forms (2012)
  • Author(s): Tori i Takeshi
  • Organizer: SGAD2012 Geometrical perspective of topological modular forms
  • Place of Presentation: 東京大学
  • Year and Date: 2012-11-15
• [Presentation] 可換S代数のGalois拡大と加群の圏の埋め込みについて (2012)
  • Author(s): 鳥居 猛
  • Organizer: ホモトピー論シンポジウム
  • Place of Presentation: 山口大学
  • Year and Date: 2012-11-03
• [Presentation] escent for structured modules (2012)
  • Author(s): 鳥居 猛
  • Organizer: (非)可換代数とトポロジー
  • Place of Presentation: 信州大学
• [Presentation] Power operations and the generalized Chern character (2011)
  • Author(s): Torii Takeshi
  • Organizer: 第4回東アジア代数的トポロジー国際会議(EACAT4)
  • Place of Presentation: 東京大学
  • Year and Date: 2011-12-09
• [Presentation] Power operations and the generalized Chern character (2011)
  • Author(s): Torii Takeshi
  • Organizer: Structured Ring Spectra, Universitat Hamburg
  • Place of Presentation: Germany
  • Year and Date: 2011-08-02
• [Presentation] On the K(n)-local category (2011)
  • Author(s): Tori i Takeshi
  • Organizer: 低次元トポロジーと数論III
  • Place of Presentation: 九州大学西新プラザ
  • Year and Date: 2011-03-17
• [Presentation] 可換S代数のGalois拡大と加群の圏の埋め込みについて
  • Author(s): 鳥居 猛
  • Organizer: ホモトピー論シンポジウム
  • Place of Presentation: 山口大学
• [Presentation] On K(1)-local topological modular forms
  • Author(s): 鳥居 猛
  • Organizer: SGAD2012 Geometrical perspective of topological modular forms
  • Place of Presentation: 東京大学