In order to study p-adicétale 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 wild ramification class field theory for the open subvariety U .2010 Mathematics Subject Classification. 14F20, 14F35, 11R37, 14G17. Key words and phrases. logarithmic de Rham-Witt sheaves, class field theory, wild ramification,étale duality, quasi-algebraic groups.The authors are supported by the DFG through CRC 1085 Higher Invariants (Universität Regensburg). Relative logarithmic de Rham-Witt sheavesLet X be a smooth proper variety of dimension d over a perfect field k of characteristic p > 0, let D be an effective divisor such that Supp(D) is a simple normal crossing divisor on X, and let j : U := X − D ֒→ X be the complement of D.Proof. This follows from the fact that the sheaf W m Ω r X|D,log is the image of K M r,X|D under the map d log[−].The isomorphism (1.1.1) also has the following relative version.Theorem 1.1.5. The d log map induces an isomorphism ofétale sheavesProof. The assertion follows directly by the following commutative diagram K M r,X|D /(p m K M r,X ∩ K M r,X|D ) / / d log K M r,X /p m ∼ = d log W m Ω r X|D,log / / W m Ω r X,log .

We prove a twist formula for the ε \varepsilon -factor of a constructible sheaf on a projective smooth variety over a finite field in terms of characteristic class of the sheaf. This formula is a modified version of the formula conjectured by Kato and Saito in [Ann. of Math. 168 (2008), pp. 33–96]. We give two applications of the twist formula. First, we prove that the characteristic classes of constructible étale sheaves on projective smooth varieties over a finite field are compatible with proper push-forward. Secondly, we show that the two Swan classes in the literature are the same on proper smooth surfaces over a finite field.

We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.Dedicated to Professor Shuji Saito on the occasion of his 60th birthday

We prove a twist formula for the ε-factor of a constructible sheaf on a projective smooth variety over a finite field in terms of characteristic class of the sheaf. This formula is a modified version of the formula conjectured by Kato and Saito in [23, Ann. Math., 168 (2008):33-96, Conjecture 4.3.11].We give two applications of the twist formula. Firstly, we prove that the characteristic classes of constructible étale sheaves on projective smooth varieties over a finite field are compatible with proper push-forward. Secondly, we show that the two Swan classes in the literature are the same on proper smooth surfaces over a finite field.