Ramified class field theory and duality over finite fields

Abstract

We prove a duality theorem for the p-adic étale motivic cohomology of a variety U which is the complement of a divisor on a smooth projective variety over F p . This extends the duality theorems of Milne and Jannsen-Saito-Zhao. The duality introduces a filtration on H 1 ét (U, Q Z). We identify this filtration to the classically known Matsuda filtration when the reduced part of the divisor is smooth. We prove a reciprocity theorem for the idele class groups with modulus introduced by Kerz-Zhao and Rülling-Saito. As an application, we derive the failure of Nisnevich descent for Chow groups with modulus.

Hodge cohomology with a ramification filtration, I

“…Theorem 1.1 [Theorem 7.3] Let k be a field satisfying resolution of singularities and weak factorisation in the sense of Def.A.4, e.g., a field of characteristic zero, Cor.A.6, Thm.A. 7.…”

“…where O(X ∞ −|X ∞ |) is the line bundle associated to the divisor X ∞ −|X ∞ |. 2 Krishna and Gupta have shown that Chow groups with modulus CH 0 (X ) cannot be isomorphic to the cohomology theory hom MDM eff k (M(X ), Z(d)[2d]) with d = dim X as they do not satisfy Nisnevich descent, [7,Thm. 1.5].…”

Suslin homology via cycles with modulus and applications

“…1.4], we obtain the following reciprocity theorem for Russell's relative Chow group. Let π abk 1 (X, D) be the log version (see [21,Defn. 9.6] or [4,Defn.…”

Suslin homology via cycles with modulus and applications

“…1.4], we obtain the following reciprocity theorem for Russell's relative Chow group. Let π abk 1 (X, D) be the log version (see [24,Defn. 9.6] or [4,Defn.…”