Updated on 2022/02/16

写真a

 
TORII Takeshi
 
Organization
Faculty of Natural Science and Technology Professor
Position
Professor
External link

Degree

  • Doctor (Science) ( Kyoto University )

  • Ph.D. ( Johns Hopkins University )

Research Interests

  • 形式群

  • ボルディズム

  • Stable homotopy

  • 安定ホモトピー

  • Formal Group

  • Bordisum

Research Areas

  • Natural Science / Geometry

Education

  • Kyoto University    

    - 1999

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  • Kyoto University   理学研究科   数学・数理解析

    - 1999

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    Country: Japan

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  • Kyoto University    

    - 1994

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  • Kyoto University   理学部   数学

    - 1994

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    Country: Japan

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Research History

  • Okayama University

    2017

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  • Okayama University

    2007 - 2017

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  • Fukuoka University   Faculty of Science

    2001 - 2007

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Professional Memberships

 

Papers

  • On graded $\mathbb{E}_{\infty}$-rings and projective schemes in spectral algebraic geometry Reviewed

    Mariko Ohara, Takeshi Torii

    to appear in Journal of Homotopy and Related Structures   2021.12

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    We introduce graded $\mathbb{E}_{\infty}$-rings and graded modules over them,
    and study their properties. We construct projective schemes associated to
    connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-rings in spectral
    algebraic geometry. Under some finiteness conditions, we show that the
    $\infty$-category of almost perfect quasi-coherent sheaves over a spectral
    projective scheme $\mathrm{Proj}\,(A)$ associated to a connective
    $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-ring $A$ can be described in terms of
    $\mathbb{Z}$-graded $A$-modules.

    arXiv

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    Other Link: http://arxiv.org/pdf/1803.09389v4

  • On quasi-categories of comodules and Landweber exactness Reviewed

    Takeshi Torii

    Proceedings in Mathematics & Statistics   309   325 - 380   2020

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    In this paper we study quasi-categories of comodules over coalgebras in a
    stable homotopy theory. We show that the quasi-category of comodules over the
    coalgebra associated to a Landweber exact S-algebra depends only on the height
    of the associated formal group. We also show that the quasi-category of
    E(n)-local spectra is equivalent to the quasi-category of comodules over the
    coalgebra A\otimes A for any Landweber exact S_(p)-algebra A of height n at a
    prime p. Furthermore, we show that the category of module objects over a
    discrete model of the Morava E-theory spectrum in the K(n)-local discrete
    symmetric G_n-spectra is a model of the K(n)-local category, where G_n is the
    extended Morava stabilizer group.

    arXiv

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    Other Link: http://arxiv.org/pdf/1612.03265v1

  • Comparison of Morava E-theories Reviewed

    Takeshi Torii

    Mathematische Zeitschrift   266 ( 4 )   933 - 951   2010

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    In this note we show that the n-th Morava E-cohomology group of a finite
    spectrum with action of the n-th Morava stabilizer group can be recovered from
    the (n+1)-st Morava E-cohomology group with action of the (n+1)-st Morava
    stabilizer group.

    arXiv

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    Other Link: http://arxiv.org/pdf/0901.3396v1

  • On degeneration of one-dimensional formal group laws and applications to stable homotopy theory Reviewed

    T Torii

    AMERICAN JOURNAL OF MATHEMATICS   125 ( 5 )   1037 - 1077   2003.10

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:JOHNS HOPKINS UNIV PRESS  

    In this note we study a certain formal group law over a complete discrete valuation ring F[u(n)-1] of characteristic p > 0 which is of height n over the closed point and of height n-I over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law Hn-1 of height n-1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group Sn-1 of Hn-1. We show that the automorphism group S-n of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of Sn-1 to the cohomology Of S-n with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

    Web of Science

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  • Discrete G-spectra and embeddings of module spectra Reviewed

    Takeshi Torii

    JOURNAL OF HOMOTOPY AND RELATED STRUCTURES   12 ( 4 )   853 - 899   2017.12

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:SPRINGER HEIDELBERG  

    In this paper we study the category of discrete G-spectra for a profinite group G. We consider an embedding of module objects in spectra into a category of module objects in discrete G-spectra, and study the relationship between the embedding and the homotopy fixed points functor. We also consider an embedding of module objects in terms of quasi-categories, and show that the two formulations of embeddings are equivalent in some circumstances.

    DOI: 10.1007/s40062-016-0166-7

    Web of Science

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  • Comparison of power operations in Morava E-theories Reviewed

    Takeshi Torii

    Homology, Homotopy and Applications   19 ( 1 )   59 - 87   2017

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  • Virtual Hodge polynomials of the moduli spaces of representations of degree 2 for free monoids Reviewed

    Kazunori Nakamoto, Takeshi Torii

    Kodai Mathematical Journal   39 ( 1 )   80 - 109   2016

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    In this paper we study the topology of the moduli spaces of representations
    of degree $2$ for free monoids. We calculate the virtual Hodge polynomials of
    the character varieties for several types of $2$-dimensional representations.
    Furthermore, we count the number of isomorphism classes for each type of
    $2$-dimensional representations over any finite field ${\Bbb F}_q$, and show
    that the number coincides with the virtual Hodge polynomial evaluated at $q$.

    arXiv

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    Other Link: http://arxiv.org/pdf/1501.02933v2

  • Every K(n)-local spectrum is the homotopy fixed points of its Morava module Reviewed

    Daniel G. Davis, Takeshi Torii

    Proceedings of the American Mathematical Society   140 ( 3 )   1097 - 1103   2012

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    Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate
    spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava
    K-theory spectrum. Then work of Devinatz and Hopkins and some results due to
    Behrens and the first author of this note, show that if X is a finite spectrum,
    then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point
    spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the
    continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that
    this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for
    all such X, the strongly convergent Adams-type spectral sequence abutting to
    \pi_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts
    to \pi_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).

    arXiv

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    Other Link: http://arxiv.org/pdf/1101.5201v1

  • Rational homotopy type of the moduli of representations with Borel mold Reviewed

    K. Nakamoto, T. Torii

    Forum Mathematicum   24 ( 3 )   507 - 538   2012

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  • Topology of the representation varieties with Borel mold for unstable cases Reviewed

    K. Nakamoto, T. Torii

    Journal of the Australian Mathematical Society   91 ( 1 )   55 - 87   2011

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  • K(n)-localization of the K(n+1)-local En+1-Adams spectral sequences Reviewed

    T. Torii

    Pacific Journal of Mathematics   250 ( 2 )   439 - 471   2011

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  • HKR CHARACTERS, p-DIVISIBLE GROUPS AND THE GENERALIZED CHERN CHARACTER Reviewed

    Takeshi Torii

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY   362 ( 11 )   6159 - 6181   2010.11

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:AMER MATHEMATICAL SOC  

    In this paper we describe the generalized Chern character of classifying spaces of finite groups in terms of Hopkins-Kuhn-Ravenel generalized group characters. For this purpose we study the p-divisible group and its level structures associated with the K(n)-localization of the (n + 1)st Morava E-theory.

    Web of Science

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  • On E-infinity-structure of the generalized Chern character Reviewed

    Takeshi Torii

    BULLETIN OF THE LONDON MATHEMATICAL SOCIETY   42   680 - 690   2010.8

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:OXFORD UNIV PRESS  

    In this note we show that we can lift the generalized Chern character to a morphism of commutative S-algebras. Furthermore, we show that we can take a lifting that is equivariant with respect to the action of a Morava stabilizer group.

    DOI: 10.1112/blms/bdq026

    Web of Science

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  • Milnor operations and the generalized Chern character Reviewed

    Takeshi Torii

    Geometry & Topology Monographs   10   383 - 421   2007

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    We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with
    action of Morava stabilizer group G_n can be recovered from the system of some
    height-(n+1) cohomology groups E^*(Z) with G_{n+1}-action indexed by finite
    subspectra Z. In this note we reformulate and extend the above result. We
    construct a symmetric monoidal functor F from the category of
    E^{vee}_*(E)-precomodules to the category of K_{*}(K)-comodules. Then we show
    that K^*(X) is naturally isomorphic to the inverse limit of F(E^*(Z)) as a
    K_{*}(K)-comodule.

    DOI: 10.2140/gtm.2007.10.383

    arXiv

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    Other Link: http://arxiv.org/pdf/0903.4708v1

  • Algebraic vector bundles on SL(3,C) Reviewed

    K. Nakamoto, T. Torii

    The Rocky Mountain Journal of Mathematics   37 ( 2 )   587 - 596   2007

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  • On relations between 1-lines of Adams-Novikov spectral sequences modulo invariant prime ideals Reviewed

    T. Torii

    Topology and its Applications   150 ( 1-3 )   33 - 57   2005

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  • Topology of the moduli of representations with Borel mold Reviewed

    K Nakamoto, T Torii

    PACIFIC JOURNAL OF MATHEMATICS   213 ( 2 )   365 - 387   2004.2

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:PACIFIC JOURNAL MATHEMATICS  

    We give descriptions of the moduli of representations with Borel mold for free monoids as fibre bundles over the configuration spaces. By using the associated Serre spectral sequences, we study the cohomology rings of the moduli. Also we calculate the virtual Hodge polynomials of them.

    Web of Science

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  • The geometric fixed point spectrum of (Z/p)^k Borel cohomology for E_n and its completion Reviewed

    T. Torii

    Contemporary Mathematics   293, 343-369   2002

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  • Topological realization of level structures of the formal group law over E(n) Reviewed

    T Torii

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   39 ( 3 )   577 - 587   1999.10

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:KINOKUNIYA CO LTD  

    Web of Science

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  • Level structure over E n and stable splitting by Steinberg idempotent Reviewed

    T Torii

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   39 ( 3 )   589 - 596   1999.10

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:KINOKUNIYA CO LTD  

    Web of Science

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  • Topological realization of the integer ring of local field Reviewed

    T Torii

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   38 ( 4 )   781 - 788   1998.12

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    Language:English   Publishing type:Research paper (scientific journal)   Publisher:KINOKUNIYA CO LTD  

    Web of Science

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MISC

  • On higher monoidal $\infty$-categories

    Takeshi Torii

    2021.10

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    In this paper we introduce a notion of $\mathbf{O}$-monoidal
    $\infty$-categories for a finite sequence $\mathbf{O}^{\otimes}$ of
    $\infty$-operads, which is a generalization of the notion of higher monoidal
    categories in the setting of $\infty$-categories. We show that the
    $\infty$-category of coCartesian $\mathbf{O}$-monoidal $\infty$-categories and
    right adjoint lax $\mathbf{O}$-monoidal functors is equivalent to the opposite
    of the $\infty$-category of Cartesian $\mathbf{O}_{\rm rev}$-monoidal
    $\infty$-categories and left adjoint oplax $\mathbf{O}_{\rm rev}$-monoidal
    functors, where $\mathbf{O}^{\otimes}_{\rm rev}$ is a sequence obtained by
    reversing the order of $\mathbf{O}^{\otimes}$.

    arXiv

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    Other Link: http://arxiv.org/pdf/2111.00158v1

  • On duoidal $\infty$-categories

    Takeshi Torii

    2021.6

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    A duoidal category is a category equipped with two monoidal structures in
    which one is (op)lax monoidal with respect to the other. In this paper we
    introduce duoidal $\infty$-categories which are counterparts of duoidal
    categories in the setting of $\infty$-categories. There are three kinds of
    functors between duoidal $\infty$-categories, which are called bilax, double
    lax, and double oplax monoidal functors. We make three formulations of
    $\infty$-categories of duoidal $\infty$-categories according to which functors
    we take. Furthermore, corresponding to the three kinds of functors, we define
    bimonoids, double monoids, and double comonoids in duoidal $\infty$-categories.

    arXiv

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    Other Link: http://arxiv.org/pdf/2106.14121v1

  • An application of Hochschild cohomology to the moduli of subalgebras of the full matrix ring II

    K. Nakamoto, T. Torii

    Ring theory 2019   176 - 187   2021

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  • An application of Hochschild cohomology to the moduli of subalgebras of the full matrix ring

    K, Nakamoto, T. Torii

    Proceedings of the 51st Symposium on Ring Theory and Representation Theory   110 - 118   2019

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  • The moduli of subalgebras of the full matrix ring of degree 3

    K. Nakamoto, T. Torii

    Proceedings of the 50th Symposium on Ring Theory and Representation Theory   137 - 149   2018

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  • Equivariance of generalized Chern characters

    Takeshi Torii

    2009.4

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    In this note some generalization of the Chern character is discussed from the
    chromatic point of view. We construct a multiplicative G_{n+1}-equivariant
    natural transformation \Theta from some height (n+1) cohomology theory E^*(-)
    to the height n cohomology theory K^*(-)\hat{\otimes}_F L, where K^*(-) is
    essentially the n-th Morava K-theory. As a corollary, it is shown that the
    G_n-module K^*(X) can be recovered from the G_{n+1}-module E^*(X). We also
    construct a lift of \Theta to a natural transformation between characteristic
    zero cohomology theories.

    arXiv

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    Other Link: http://arxiv.org/pdf/0904.1647v1

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Research Projects

  • Moduli of representations and related topics (4)

    Grant number:20K03509  2020.04 - 2024.03

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)

    中本 和典, 鳥居 猛, 面田 康裕, 奥山 真吾

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    Grant amount:\4420000 ( Direct expense: \3400000 、 Indirect expense:\1020000 )

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  • 一般の安定ホモトピー論における余加群の研究

    Grant number:17K05253  2017.04 - 2022.03

    日本学術振興会  科学研究費助成事業 基盤研究(C)  基盤研究(C)

    鳥居 猛

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    Grant amount:\4680000 ( Direct expense: \3600000 、 Indirect expense:\1080000 )

    一般の安定ホモトピー論におけるガロア群と導来淡中双対性および導来群スキームの表現のモジュライについて研究するために、デュオイダル圏およびデュオイダル圏におけるホップ亜代数とその余加群の無限大圏への一般化について研究を行った。
    2つのモノイダル構造をもち、一方のモノイダル構造が他方のモノイダル構造に関して、ラックスモノイダルになっている、あるいは同値であるが、一方のモノイダル構造が他方のモノイダル構造に関して、コラックスモノイダルになっているような圏をデュオイダル圏と呼ぶ。デュオイダル圏は双代数を定義できる最小の構造のみを備えた圏と考えることができる。今年度はデュオイダル圏の無限大圏への一般化について研究を行った。二つの無限大オペラッド上のモノイダル圏の構造をもち、一方のモノイダル構造が他方のモノイダル構造に関して、ラックスモノイダルになっている無限大圏を定式化した。また、無限大オペラッド上のモノイダル無限大バイカテゴリーに対して、ループ構成により、デュオイダル無限大圏が得られることを示した。また、デュオイダル無限大圏における双亜代数がホップ亜代数になるための条件をその余加群の無限大圏の性質により特徴づける研究を行った。
    また、研究集会「ホモトピー沖縄」、研究集会「空間の代数的・幾何的モデルとその周辺」、「非可換代数幾何学の大域的問題とその周辺」高知小研究集会、高知ホモトピー論談話会、福岡ホモトピー論セミナーなどに参加し、様々な研究者と研究課題について議論を行った。さらに、研究集会「ホモトピー沖縄」および高知ホモトピー論談話会ではデュオイダル無限大圏とホップ亜代数について講演を行った。

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  • Research on the stable homotopy category using quasi-categories

    Grant number:25400092  2013.04 - 2017.03

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)

    Torii Takeshi

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    Grant amount:\4940000 ( Direct expense: \3800000 、 Indirect expense:\1140000 )

    I have studied the stable homotopy category and its localizations by means of quasi-categories. Through spectral sequences, the stable homotopy category and its Bousfield localizations are considered to be related to the categories of representations for some groups and their derived categories. I gave a formulation of this relationship through the theories of model categories and quasi-categories. I have also constructed a functor between algebraic models of Bousfield localizations of the stable homotopy category via Morava K-theories. Furthermore, based on these, I have also studied more general moduli spaces of representations.

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  • Chromatic redshift and homotopical algebraic geometry

    Grant number:22540087  2010 - 2012

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)  Grant-in-Aid for Scientific Research (C)

    TORII Takeshi

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    Grant amount:\3120000 ( Direct expense: \2400000 、 Indirect expense:\720000 )

    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.

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  • 安定ホモトピー圏の大域的構造の研究

    Grant number:18740040  2006 - 2007

    日本学術振興会  科学研究費助成事業 若手研究(B)  若手研究(B)

    鳥居 猛

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    Grant amount:\1700000 ( Direct expense: \1700000 )

    安定ホモトピー圏の大域的構造の理解を目標とし、素ボルディズム関手および形式群を用いて安定ホモトピー圏の代数的モデルや数論的構造について研究を行った。特にクロマチックレベルが一つずれているMoravaK理論で局所化された安定ホモトピー圏の間の関係について調べた。
    Devinatz,Hopkins は Morava安定化群G_nの任意の閉部分群によるMoravaE理論E』のホモトピー固定点スペクトラムを構成し、特にMorava 安定化群全体によるホモトピー固定点スペクトラムが球面スペクトラムのMoravaK理論K(n)による局所化と一致することを示した。しかしながらDevinatz,Hopkins によるホモトピー固定点スペクトラムは、ホモトピー固定点スペクトラムが持つべき性質を満たすように技巧的に構成されている。Davis はこれを本来の固定点スペクトラムの観点から見直し、副有限群作用をもつ離散スペクトラムのモデル圏の構造を用いて連続スペクトラムのホモトピー固定点スペクトラムを定義した。さらに Davis はMorava E理論の Morava 安定化群によるホモトピー固定点スペクトラムの場合には、Devinatz,Hopkins のホモトピー固定点スペクトラムと一致することを示した。
    今年度の研究では Morava E理論 E_{n+1} のMorava K理論K(n)による局所化L_{K(n)}E_{n+1} のDavis の意味でのホモトピー固牢点スペクトラムについて考察し次のことを得た。
    (1) L_{K(n)}E_{n+1} は Morava 安定化群 G_{n+1} の作用に関して連続スペクトラムである。(2) Morava安定化群の任意の閉部分群に関する Davisの意味でのホモトピー固定点スペクトラムはDevinatz,Hopkinsの意味のホモトピー固定点スペクトラムのK(n)局所化と一致する。(3) L_{K(n)}E_{n+1} のホモトピー固定点スペクトル系列は K(n+1)局所 E_{n+1}-Adamsスペクトル系列のK(n)局所化に一致する。

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  • 安定ホモトピー圏と形式群のモジュライ

    Grant number:16740041  2004 - 2005

    日本学術振興会  科学研究費助成事業 若手研究(B)  若手研究(B)

    鳥居 猛

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    Grant amount:\1800000 ( Direct expense: \1800000 )

    前年度に引き続き、複素ボルディズム函手および形式群を用いた安定ホモトピー圏の代数化についての研究を行った。安定ホモトピー圏は複素ボルディズム函手を通して形式群のモジュライ空間上の層の圏と密接な関係にある。この関係を通して形式群のモジュライ空間の構造が安定ホモトピー圏の構造に関して非常に強い代数的制限を課していることが知られている。また、安定ホモトピー圏をMorava K-理論で局所化した圏は形式群を通して整数論の局所理論と深い関係にあることが知られている。Honda group lawと呼ばれる基本的な形式群の自己同型群はMorava stabilizer groupと呼ばれ、Morava E-理論の乗法的な一次作用素のなす群として現れる。この群のコホモロジーはMorava K-理論により局所化された安定ホモトピー圏の最も基本的な不変量である。今年度は前年度に得られた結果の応用として、異なる形式群の高さに対応するMorava E-理論の間についての比較定理を得た。これは高さ(n+1)のMorava E-cohomologyの係数環上の加群の構造と一次作用素の作用の様子から高さnのMorava E-理論の係数環上の加群の構造と一次作用素の作用の様子が得られることを主張している。また、Morava K-理論K(n)に関するBousfield局所化L_<K(n)>Xのホモトピー群に収束するスペクトル系列とL_<K(n)>L_<K(n+1)>Xのホモトピー群に収束するスペクトル系列の間に射を構成し、E_<2^->項における射をMorava stabilizer groupのコホモロジーの間のある種のinflation mapとして記述した。また、Rognesにより定式化されたstrictly commutative ring spectrumの間のガロア理論を用いて、これまでに構成した一般化されたChern指標の性質について研究し、そのE_<∞^->構造や新しい記述についての結果を得た。

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Class subject in charge

  • Homotopy Theory (2021academic year) Prophase  - 金7,金8

  • Stable Homotopy Theory (2021academic year) Prophase  - その他

  • Geometry II (2021academic year) 3rd and 4th semester  - 火3,火4

  • Seminar in Geometry (2021academic year) Year-round  - その他

  • Excercises in Geometry (2021academic year) 3rd and 4th semester  - 木7,木8

  • Advanced Geometry I (2021academic year) 1st and 2nd semester  - 火3,火4

  • Advanced Geometry Ia (2021academic year) 1st semester  - 火3,火4

  • Advanced Geometry Ib (2021academic year) Second semester  - 火3,火4

  • Geometry IIa (2021academic year) Third semester  - 火3,火4

  • Geometry IIb (2021academic year) Fourth semester  - 火3,火4

  • Seminars in Mathematics and Physics (2021academic year) Year-round  - その他

  • Advanced Study in Mathematics and Physics (2021academic year) Year-round  - その他

  • Group Study on Mathematical Sciences (2021academic year) 3rd and 4th semester

  • Homotopy Theory (2020academic year) Prophase  - 金7,金8

  • Stable Homotopy Theory (2020academic year) Prophase  - その他

  • Geometry II (2020academic year) 3rd and 4th semester  - 火3,火4

  • Seminar in Geometry (2020academic year) Year-round  - その他

  • Excercises in Geometry (2020academic year) 3rd and 4th semester  - 木7,木8

  • Advanced Geometry I (2020academic year) 1st and 2nd semester  - 火3,火4

  • Advanced Geometry Ia (2020academic year) 1st semester  - 火3,火4

  • Advanced Geometry Ib (2020academic year) Second semester  - 火3,火4

  • Geometry IIa (2020academic year) Third semester  - 火3,火4

  • Geometry IIb (2020academic year) Fourth semester  - 火3,火4

  • Seminars in Mathematics and Physics (2020academic year) Year-round  - その他

  • Advanced Study in Mathematics and Physics (2020academic year) Year-round  - その他

  • Invitation to Mathematical Sciences (2020academic year) 1st semester  - 金3,金4

  • Introduction to Mathematics IIIa (2020academic year) 1st semester  - 月7,月8

  • Introduction to Mathematics IIIb (2020academic year) Second semester  - 月7,月8

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