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Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified class field theory over finite fields

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

Published online by Cambridge University Press:  07 May 2018

Uwe Jannsen ,
Shuji Saito  and
Yigeng Zhao
Uwe Jannsen
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany email [email protected]
Shuji Saito
Affiliation:
Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 Okayama, Meguro, Tokyo 152-8551, Japan email [email protected]
Yigeng Zhao
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany email [email protected]
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Abstract

In order to study $p$-adic étale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wildly ramified class field theory for the open subvariety $U$.

Keywords

logarithmic de Rham–Witt sheavesclass field theorywild ramificationétale dualityquasi-algebraic groups

MSC classification

Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies
Secondary: 14F35: Homotopy theory; fundamental groups 11R37: Class field theory 14G17: Positive characteristic ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 6 , June 2018 , pp. 1306 - 1331
Copyright
© The Authors 2018 

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References

Artin, M., Algebraic approximation of structures over complete local rings , Publ. Math. Inst. Hautes Études Sci. 36 (1969), 23–58.CrossRefGoogle Scholar
Bass, H. and Tate, J., The Milnor ring of a global field , in Algebraic K-Theory II: Classical algebraic K-theory, and connections with arithmetic (Springer, Berlin, Heidelberg, 1973), 347–446.Google Scholar
Berthelot, P., Le théorème de dualité plate pour les surfaces (d’après J.S. Milne) , in Surfaces algébriques (Springer, Berlin, Heidelberg, 1981), 203–237.CrossRefGoogle Scholar
Bloch, S. and Kato, K., p-adic etale cohomology , Publ. Math. Inst. Hautes Études Sci. 63 (1986), 107–152.CrossRefGoogle Scholar
Colliot-Thélène, J., Sansuc, J. and Soulé, C., Torsion dans le groupe de Chow de codimension deux , Duke Math. J. 50 (1983), 763–801.CrossRefGoogle Scholar
Geisser, T. and Hesselholt, L., The de Rham–Witt complex and p-adic vanishing cycles , J. Amer. Math. Soc. 19 (2006), 1–36.CrossRefGoogle Scholar
Geisser, T. and Levine, M., The K-theory of fields in characteristic p , Invent. Math. 139 (2000), 459–493.CrossRefGoogle Scholar
Gros, M. and Suwa, N., La conjecture de Gersten pour les faisceaux de Hodge–Witt logarithmique , Duke Math. J. 57 (1988), 615–628.CrossRefGoogle Scholar
Hyodo, O. and Kato, K., Semi-stable reduction and crystalline cohomology with logarithmic poles, in Périodes p-adiques, Bures-sur-Yvette, 1988 , Astérisque vol. 223 (Société Mathématique de France, Paris, 1994), 221–268; MR 1293974.Google Scholar
Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline , Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501–661.CrossRefGoogle Scholar
Kato, K., Galois cohomology of complete discrete valuation fields , in Algebraic K-theory, Lecture Notes in Mathematics, vol. 967, ed. Dennis, R. K. (Springer, Berlin, Heidelberg, 1982), 215–238.CrossRefGoogle Scholar
Kato, K., Duality theories for the p-primary étale cohomology I , in Algebraic and topological theories (Kinokuniya-shoten, Tokyo, 1985), 127–148.Google Scholar
Kato, K., Existence theorem for higher local fields , in Invitation to higher local fields (Geometry & Topology Publications, Coventry, 2000), 165–195.Google Scholar
Kerz, M., Milnor K-theory of local rings with finite residue fields , J. Algebraic Geom. 19 (2010), 173–191.CrossRefGoogle Scholar
Lorenzon, P., Logarithmic Hodge–Witt forms and Hyodo–Kato cohomology , J. Algebra 249 (2002), 247–265; MR 1901158.CrossRefGoogle Scholar
Milne, J. S., Duality in the flat cohomology of a surface , Ann. Sci. Éc. Norm. Supér. (4) 9 (1976), 171–201; MR 0460331.CrossRefGoogle Scholar
Milne, J. S., Values of zeta functions of varieties over finite fields , Amer. J. Math. 108 (1986), 297–360.CrossRefGoogle Scholar
Pépin, C., Dualité sur un corps local de caractéristique positive à corps résiduel algébriquement clos, Preprint (2014), arXiv:1411.0742.Google Scholar
Rost, M., Chow groups with coefficients , Doc. Math. 1 (1996), 319–393.CrossRefGoogle Scholar
Rülling, K. and Saito, S., Higher Chow groups with modulus and relative Milnor K-theory , Trans. Amer. Math. Soc. 370 (2018), 987–1043.CrossRefGoogle Scholar
Saito, S., A global duality theorem for varieties over global fields , in Algebraic K-theory: connections with geometry and topology (Kluwer, Dordrecht, 1989), 425–444.CrossRefGoogle Scholar
P. Deligne avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier, Cohomologie étale (SGA 4½), Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, Heidelberg, 1977).CrossRefGoogle 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