Skip to main content Accessibility help

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

  • Uwe Jannsen (a1), Shuji Saito (a2) and Yigeng Zhao (a3)


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 $U$ .



Hide All
[Art69] Artin, M., Algebraic approximation of structures over complete local rings , Publ. Math. Inst. Hautes Études Sci. 36 (1969), 2358.
[BT73] 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), 347446.
[Ber81] Berthelot, P., Le théorème de dualité plate pour les surfaces (d’après J.S. Milne) , in Surfaces algébriques (Springer, Berlin, Heidelberg, 1981), 203237.
[BK86] Bloch, S. and Kato, K., p-adic etale cohomology , Publ. Math. Inst. Hautes Études Sci. 63 (1986), 107152.
[CSS83] Colliot-Thélène, J., Sansuc, J. and Soulé, C., Torsion dans le groupe de Chow de codimension deux , Duke Math. J. 50 (1983), 763801.
[GH06] Geisser, T. and Hesselholt, L., The de Rham–Witt complex and p-adic vanishing cycles , J. Amer. Math. Soc. 19 (2006), 136.
[GL00] Geisser, T. and Levine, M., The K-theory of fields in characteristic p , Invent. Math. 139 (2000), 459493.
[GS88] Gros, M. and Suwa, N., La conjecture de Gersten pour les faisceaux de Hodge–Witt logarithmique , Duke Math. J. 57 (1988), 615628.
[HK94] 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), 221268; MR 1293974.
[Ill79] Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline , Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501661.
[Kat82] 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), 215238.
[Kat85] Kato, K., Duality theories for the p-primary étale cohomology I , in Algebraic and topological theories (Kinokuniya-shoten, Tokyo, 1985), 127148.
[Kat00] Kato, K., Existence theorem for higher local fields , in Invitation to higher local fields (Geometry & Topology Publications, Coventry, 2000), 165195.
[Ker10] Kerz, M., Milnor K-theory of local rings with finite residue fields , J. Algebraic Geom. 19 (2010), 173191.
[Lor02] Lorenzon, P., Logarithmic Hodge–Witt forms and Hyodo–Kato cohomology , J. Algebra 249 (2002), 247265; MR 1901158.
[Mil76] Milne, J. S., Duality in the flat cohomology of a surface , Ann. Sci. Éc. Norm. Supér. (4) 9 (1976), 171201; MR 0460331.
[Mil86] Milne, J. S., Values of zeta functions of varieties over finite fields , Amer. J. Math. 108 (1986), 297360.
[Pép14] Pépin, C., Dualité sur un corps local de caractéristique positive à corps résiduel algébriquement clos, Preprint (2014), arXiv:1411.0742.
[Ros96] Rost, M., Chow groups with coefficients , Doc. Math. 1 (1996), 319393.
[RS18] Rülling, K. and Saito, S., Higher Chow groups with modulus and relative Milnor K-theory , Trans. Amer. Math. Soc. 370 (2018), 9871043.
[Sai89] Saito, S., A global duality theorem for varieties over global fields , in Algebraic K-theory: connections with geometry and topology (Kluwer, Dordrecht, 1989), 425444.
[SGA4½]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).
[Zha16] Zhao, Y., Duality for relative logarithmic de Rham-Witt sheaves on semistable schemes over $\mathbb{F}_{q}[[t]]$ , Preprint (2016), arXiv:1611.08722.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

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

  • Uwe Jannsen (a1), Shuji Saito (a2) and Yigeng Zhao (a3)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed