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2013 年度 実績報告書

離散付置環上のモチビックコホモロジー

研究課題

研究課題/領域番号 23340004
研究機関名古屋大学

研究代表者

ガイサ トーマス  名古屋大学, 多元数理科学研究科, 教授 (30571963)

研究期間 (年度) 2011-04-01 – 2016-03-31
キーワードススリン・ホモロジー / 多様体の類体論 / Rojtmanの定理
研究概要

In the previous year I was able to complete research on three papers. In the first one I proved that Suslin homology satisfies descent for proper hyperenvelopes, and for l-hyperenvelopes if l is invertible in the coefficients. I gave an axiomatic proof, which can also be applied to other theories in the future. This research is posted on the ArXiv, and submitted for publication.
In the second project I applied this method to prove Rojtman's theorem for normal schemes over algebraically closed fields. The theorem states that the torsion in the 0th Suslin homology is isormorphic to the torsion of the Albanese semi-abelian variety. The theorem is an improvement from previous results, which either used properness, smoothness, or characteristic 0. This result is also posted on the ArXiv and submitted for publication. A survey of this results is accepted for publication at the RIMS journal.
Finally, I concluded research on a joint project with Alexander Schmidt in Heidelberg, where we proved a version of the Hurewicz theorem for varieties (singular or not) over algebraically closed fields. The theorem states that the abelianized tame fundamental group mod m is isomorphic to Suslin homology with mod m coefficients. This paper is also posted on ArXiv, and submitted for publication.

現在までの達成度 (区分)
現在までの達成度 (区分)

2: おおむね順調に進展している

理由

I was able to prove several theorems which are related to the project, and required some time to write the research papers. But overall the project advances as expected.

今後の研究の推進方策

At first I want to finish another project with Alexander Schmidt. We show that over a finite field, a modified version of Suslin homology (called Weil-Suslin homology) surjects onto the tame abelianized enlarged fundamental group, for any variety over a finite field. We also show that this fundamental group is a finitely generated abelian group, that the kernel of the surjection is the divisible subgroup of Weil-Suslin homology (under resolution of singularities), and that the map is an isormophism under Parshin's conjecture. The mathematics is in place, but it will require some time to write the paper.
After this I am planing to revisit a result of Grothendieck, where he shows that the finiteness of the Brauer group of a surface fibered over a curve over a finite field is equivalent to the finiteness of the Tate-Shafarevich group of the Jacobian of the special fiber of the curve. To prove this equivalence, one needs exactly to consider motivic cohomology over a stricly henselian valuation ring. I hope to obtain some results for different weights using the conjecture of Saito and Sato.

  • 研究成果

    (9件)

すべて その他

すべて 学会発表 (6件) (うち招待講演 6件) 備考 (3件)

  • [学会発表] Duality and class field theory

    • 著者名/発表者名
      Thomas Geisser
    • 学会等名
      Conference on Homotopical Methods in Algebraic Geometry
    • 発表場所
      USC Los Angeles
    • 招待講演
  • [学会発表] Higher class field theory for schemes over finite fields

    • 著者名/発表者名
      Thomas Geisser
    • 学会等名
      Conference on Global Fields
    • 発表場所
      Moscow
    • 招待講演
  • [学会発表] Tame class field theory of singular schemes

    • 著者名/発表者名
      Thomas Geisser
    • 学会等名
      Workshop on Reciprocity Sheaves
    • 発表場所
      Yatsugatake
    • 招待講演
  • [学会発表] Class field theory over algebraically closed fields

    • 著者名/発表者名
      Thomas Geisser
    • 学会等名
      九州代数的数論
    • 発表場所
      九州大学
    • 招待講演
  • [学会発表] Class field theory of singular schemes

    • 著者名/発表者名
      Thomas Geisser
    • 学会等名
      Conference in honor of Uwe Jannsen's 60th birthday
    • 発表場所
      Universitaet Regensburg
    • 招待講演
  • [学会発表] Rojtman's theorem for normal schemes"

    • 著者名/発表者名
      Thomas Geisser
    • 学会等名
      Conference on motivic and etale homotopy theory
    • 発表場所
      Universitaet Heidelberg
    • 招待講演
  • [備考] Tame Class Field Theory for Sing. Varieties

    • URL

      http://arxiv.org/abs/1309.4068

  • [備考] Homological Descent for Motivic Homology Theories

    • URL

      http://arxiv.org/abs/1401.7775

  • [備考] Rojtman's theorem for normal schemes

    • URL

      http://arxiv.org/abs/1402.1831

URL: 

公開日: 2015-05-28  

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