Article contents
HIGHER IDELES AND CLASS FIELD THEORY
Part of:
(Co)homology theory
Algebraic number theory: global fields
Arithmetic algebraic geometry
Arithmetic problems. Diophantine geometry
Published online by Cambridge University Press: 02 October 2018
Abstract
We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.
- 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
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
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