Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T20:08:07.039Z Has data issue: false hasContentIssue false
  • English
  • Français

HIGHER IDELES AND CLASS FIELD THEORY

Part of: Arithmetic problems. Diophantine geometry (Co)homology theory Algebraic number theory: global fields Arithmetic algebraic geometry

Published online by Cambridge University Press:  02 October 2018

MORITZ KERZ  and
YIGENG ZHAO
MORITZ KERZ
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany email [email protected]
YIGENG ZHAO
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.

MSC classification

Primary: 11G45: Geometric class field theory
Secondary: 14F20: Étale and other Grothendieck topologies and (co)homologies 14F35: Homotopy theory; fundamental groups 11R37: Class field theory 14G17: Positive characteristic ground fields
Type
Article
Information
Nagoya Mathematical Journal , Volume 236: Celebrating the 60th Birthday of Shuji Saito , December 2019 , pp. 214 - 250
Copyright
© 2018 Foundation Nagoya Mathematical Journal  

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The authors are supported by the DFG through CRC 1085 Higher Invariants (Universität Regensburg).

References

Bloch, S. and Kato, K., p-adic etale cohomology , Publ. Math. Inst. Hautes Études Sci. 63 (1986), 107–152.10.1007/BF02831624Google Scholar
Deligne, P., “Cohomologie étale (SGA 4 $\frac{1}{2}$ )”, in Avec la collaboration de J.-F. Boutot, A. Grothendieck, L. Illusie et J.-L. Verdier, Lecture Notes in Mathematics 569, Springer, New York, 1977.10.1007/BFb0091518Google Scholar
Forré, P., The kernel of the reciprocity map of varieties over local fields , J. Reine Angew. Math. 2015(698) (2015), 55–69.10.1515/crelle-2012-0122Google Scholar
Geisser, T., Motivic cohomology over Dedekind rings , Math. Z. 248(4) (2004), 773–794.10.1007/s00209-004-0680-xGoogle Scholar
Geisser, T., Duality via cycle complexes , Ann. Math. 172(2) (2010), 1095–1126.10.4007/annals.2010.172.1095Google Scholar
Grothendieck, A. and Hartshorne, R., Local Cohomology: A Seminar, Lecture Notes in Mathematics 41 , Springer, New York, 1967.Google Scholar
Geisser, T. and Levine, M., The K-theory of fields in characteristic p , Invent. Math. 139(3) (2000), 459–493.10.1007/s002220050014Google Scholar
Geisser, T. and Levine, M., The Bloch–Kato conjecture and a theorem of Suslin–Voevodsky , J. Reine Angew. Math. 530 (2001), 55–104.Google Scholar
Gros, M. and Suwa, N., La conjecture de Gersten pour les faisceaux de Hodge–Witt logarithmique , Duke Math. J. 57(2) (1988), 615–628.10.1215/S0012-7094-88-05727-4Google Scholar
Hiranouchi, T., Class field theory for open curves over local fields, preprint, 2016, arXiv:1412.6888v2.Google Scholar
Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline , Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501–661.10.24033/asens.1374Google Scholar
Jannsen, U. and Saito, S., Kato homology of arithmetic schemes and higher class field theory over local fields , Documenta Math. Extra Volume: Kazuya Kato’s Fiftieth Birthday (2003), 479–538.Google Scholar
Jannsen, U., Saito, S. and Sato, K., Étale duality for constructible sheaves on arithmetic schemes , J. Reine Angew. Math. 688 (2014), 1–65.10.1515/crelle-2012-0043Google Scholar
Jannsen, U., Saito, S. and Zhao, Y., Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified class field theory over finite fields , Compos. Math. 154(6) (2018), 1306–1331.10.1112/S0010437X1800711XGoogle Scholar
Kunz, E., Cox, D. and Dickenstein, A., Residues and Duality for Projective Algebraic Varieties, University Lecture Series, 47 , American Mathematical Society, Providence, 2008.10.1090/ulect/047Google Scholar
Kerz, M., The Gersten conjecture for Milnor K-theory , Invent. Math. 175(1) (2009), 1–33.10.1007/s00222-008-0144-8Google Scholar
Kerz, M., Ideles in higher dimension , Math. Res. Lett. 18(4) (2011), 699–713.10.4310/MRL.2011.v18.n4.a9Google Scholar
Kato, K. and Saito, S., Unramified class field theory of arithmetical surfaces , Ann. of Math. (2) 118 (1983), 241–275.10.2307/2007029Google Scholar
Kato, K. and Saito, S., “ Global class field theory of arithmetic schemes ”, in Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I, American Mathematical Society, Providence, 1986, 255–331.10.1090/conm/055.1/862639Google Scholar
Kerz, M. and Saito, S., Cohomological Hasse principle and motivic cohomology for arithmetic schemes , Publ. Math. Inst. Hautes Études Sci. 115(1) (2012), 123–183.10.1007/s10240-011-0038-yGoogle Scholar
Kerz, M. and Saito, S., Chow group of 0-cycles with modulus and higher dimensional class field theory , Duke Math. J. 165(15) (2016), 2811–2897.10.1215/00127094-3644902Google Scholar
Matsumi, P., Class field theory for F q[[X 1, X 2, X 3]] , J. Math. Sci. Univ. Tokyo 9(4) (2002), 689–749.Google Scholar
Milne, J. S., Values of zeta functions of varieties over finite fields , Amer. J. Math. 108 (1986), 297–360.10.2307/2374676Google Scholar
Rülling, K. and Saito, S., Higher Chow groups with modulus and relative Milnor K-theory , Trans. Amer. Math. Soc. 370 (2018), 987–1043.10.1090/tran/7018Google Scholar
Saito, S., Class field theory for curves over local fields , J. Number Theory 21(1) (1985), 44–80.10.1016/0022-314X(85)90011-3Google Scholar
Saito, S., Class field theory for two dimensional local rings , Adv. Stud. Pure Math. 12 (1987), 343–373.10.2969/aspm/01210343Google Scholar
Saito, S., “ A global duality theorem for varieties over global fields ”, in Algebraic K-Theory: Connections with Geometry and Topology, Kluwer Academic Publisher, Dordrecht, 1989, 425–444.10.1007/978-94-009-2399-7_14Google Scholar
Sato, K., Logarithmic Hodge–Witt sheaves on normal crossing varieties , Math. Z. 257 (2007), 707–743.10.1007/s00209-006-0033-zGoogle Scholar
Sato, K., ℓ-adic class field theory for regular local rings , Math. Ann. 344(2) (2009), 341–352.10.1007/s00208-008-0309-1Google Scholar
Shiho, A., On logarithmic Hodge–Witt cohomology of regular schemes , J. Math. Sci. Univ. Tokyo 14 (2007), 567–635.Google Scholar
Zhao, Y., Duality for relative logarithmic de Rham–Witt sheaves on semistable schemes over $\mathbb{F}_{q}[[t]]$ , preprint, 2016, arXiv:1611.08722.Google Scholar