characteristic cycles and ramification of etale sheaves

• Project/Area Number: 19H01780
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: The University of Tokyo
• Principal Investigator: Saito Takeshi 東京大学, 大学院数理科学研究科, 教授 (70201506)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥16,900,000 (Direct Cost: ¥13,000,000、Indirect Cost: ¥3,900,000); Fiscal Year 2023: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2022: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2021: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2020: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2019: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
• Keywords: 局所体 / 分岐群 / Frobenius--Witt微分形式 / 特異台 / Hasse--Arfの定理 / エタール層 / 分岐 / 特性サイクル / Deligne--Kato公式 / 隣接輪体 / Frobenius--Witt微分 / マイクロ台 / 余接束 / F横断性 / 正則性判定法 / エタール・コホモロジー

Outline of Research at the Start

正標数の代数多様体のエタール・コホモロジーは，数論幾何の基本的な研究対象である。エタール層の特性サイクルは，余接束上に定義され，層のEuler数などの不変量が記述されている.
本研究の目的は，この特性サイクルの理論を発展させるとともにその適用範囲を広げ，数論幾何へ応用することである．スキームの分岐について理解を深めることが，研究の主な方法である．特性サイクルの記述，数論的な不変量への拡張，混標数のスキームへの拡張などを目標とする.

Outline of Final Research Achievements

First, on the ramification groups of local fields, I proved that the graded quotients are abelian and annihilated by p. I also constructed injections from the character groups of the graded quotients to the group of certain differential forms.
Further, globalizing the construction of the latter groups, I constructed the Frobenius-Witt cotangent bundle on the characteristic p fiber of a regular scheme of mixed characteristic. On this vector bundle, I defined the singular support of an etale sheaf. I also gave a regularity criterion of schemes.

Academic Significance and Societal Importance of the Research Achievements

局所体の分岐の古典理論においては、剰余体が完全という仮定が必要であったが、高次元の多様体やスキームの分岐を調べるには、この仮定を取り除くことが必要である。分岐群の次数商の構造を解明することで、局所体の分岐理論の一般化の基礎的な部分が完成した。
この研究において、Frobenius--Witt微分形式の定義を発見した。これによって、正標数の多様体上のエタール層の特異台や特性サイクルの理論を、より整数論的な対象である混標数のスキーム上に拡張する道が開けた。

Research Products

• [Int'l Joint Research] IHES(フランス)
• [Int'l Joint Research] シカゴ大学(米国)
• [Journal Article] Graded quotients of ramification groups of local fields with imperfect residue fields (2023)
  • Author(s): Saito Takeshi
  • Journal Title: American Journal of Mathematics Volume: 145 Issue: 5 Pages: 1389-1464
  • DOI: 10.1353/ajm.2023.a907702
  • Peer Reviewed
• [Journal Article] A characterization of ramification groups of local fields with imperfect residue fields (2023)
  • Author(s): Takeshi Saito
  • Journal Title: proceedings of International conference on arithmetic geometry 2020, TIFR. Volume: N/A Pages: 421-433
  • Peer Reviewed
• [Journal Article] Cotangent bundles and micro-supports in mixed characteristic case (2022)
  • Author(s): Takeshi Saito
  • Journal Title: Algebra & Number Theory Volume: 16-2 Issue: 2 Pages: 335-368
  • DOI: 10.2140/ant.2022.16.335
  • Peer Reviewed
• [Journal Article] Frobenius-Witt differentials and regularity (2022)
  • Author(s): Takeshi Saito
  • Journal Title: Algebra & Number Theory Volume: 16-2 Issue: 2 Pages: 369-391
  • DOI: 10.2140/ant.2022.16.369
  • Peer Reviewed
• [Journal Article] REFINED SWAN CONDUCTORS OF ONE-DIMENSIONAL GALOIS REPRESENTATIONS (2019)
  • Author(s): KATO KAZUYA、LEAL ISABEL、SAITO TAKESHI
  • Journal Title: Nagoya Mathematical Journal Volume: 236 Pages: 1-49
  • DOI: 10.1017/nmj.2019.13
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Coincidence of two Swan conductors of abelian characters (2019)
  • Author(s): KATO KAZUYA、SAITO TAKESHI
  • Journal Title: Epijournal de Geometrie Algebrique Volume: 3 Pages: 1-16
  • Peer Reviewed / Int'l Joint Research
• [Presentation] On the Hasse-Arf theorem (2023)
  • Author(s): Takeshi Saito
  • Organizer: Arithmetic and Cohomology of Algebraic Varieties
  • Int'l Joint Research
• [Presentation] On the Hasse-Arf theorem (2023)
  • Author(s): 斎藤 毅
  • Organizer: 第19回北陸数論研究集会 「超幾何関数の数論とその周辺」
• [Presentation] Upper ramification groups of local fields with imperfect residue fields (2022)
  • Author(s): Takeshi Saito
  • Organizer: Franco-Asian Summer School on Arithmetic Geometry
  • Int'l Joint Research / Invited
• [Presentation] Micro support of a constructible sheaf in mixed characteristic (2021)
  • Author(s): Takeshi Saito
  • Organizer: Conference en honneur de Luc Illusie
  • Int'l Joint Research / Invited
• [Presentation] Wild ramification and the cotangent bundle in mixed characteristic (2020)
  • Author(s): Takeshi Saito
  • Organizer: Eighth Pacific Rim Conference
  • Int'l Joint Research / Invited
• [Presentation] Graded Quotients of Ramification Groups of a Local Field with Imperfect Residue Field, (2020)
  • Author(s): Takeshi Saito
  • Organizer: International conference on arithmetic geometry 2020
  • Int'l Joint Research / Invited
• [Presentation] Characteristic cycle of a constructible sheaf (2019)
  • Author(s): Takeshi Saito
  • Organizer: Arithmetic Geometry in Carthage
  • Int'l Joint Research / Invited
• [Presentation] CC (2019)
  • Author(s): Takeshi Saito
  • Organizer: Wild Ramification and Irregular Singularities
  • Int'l Joint Research / Invited
• [Funded Workshop] Arithmetic Geometry in Carthage (2019)