Ramification theory of schemes over a local field

We introduce the Swan class of an -adic etale sheaf on a variety over a local field. It is a generalization of the classical Swan conductor measuring the wild ramification and is defined as a 0-cycle class supported on the reduction. We establish a Riemann-Roch formula for the Swan class. Let K be a complete discrete valuation field of characteristic 0. We assume that the residue field F is a perfect field of characteristic p > 0. Let U be a separated scheme of finite type over K and F be an -adic sheaf on U where is a prime different from p. The etale cohomology H∗ c (UK̄ ,F) with compact support defines an -adic representation of the absolute Galois group GK = Gal(K̄/K). We will give a formula for the alternating sum Sw H∗ c (UK̄ ,F) of the Swan conductor. In the case where U is smooth over K and F is a smooth sheaf on U , it takes the form Sw H∗ c (UK̄ ,F)− rank F × Sw H∗ c (UK̄ ,Q ) = deg SwF . We will have a relative version of the formula for an arbitrary sheaf F and an arbitrary morphism U → V . The general version will be formulated by introducing a map SwU : K0(U,F ) −−−→ F0G(UF ) . Here K0(U,F ) denotes the Grothendieck group of constructible F -sheaves on U and F0G(UF ) denotes the dimension 0-part of the Grothendieck group of coherent modules on the reduction of U whose precise definition will be given later. Note that the reduction modulo defines a natural map K0(U,Q ) → K0(U,F ). In the case U = Spec K, we have K0(Spec K,F ) = K0(RepGK (F )), F0G(Spec KF ) = G(F ) = Q and, for an F -representation V of GK , we have SwSpec K(V ) = Sw(V ). The main result in this talk is the following.