Ramification theory of schemes in mixed characteristic case

We define generalizations of classical invariants of ramification, for coverings on a variety of arbitrary dimension over a local field of mixed characteristic. For an -adic sheaf, we define its Swan class as a 0-cycle class supported on the closed fiber. We present a formula for the Swan conductor of cohomology and its relative version. Let K be a complete discrete valuation field of characteristic 0 and F be the residue field of K. We assume F is a perfect field of characteristic p > 0. Let U be a separated smooth scheme purely of dimension d of finite type over K. Let f : V → U be a finite étale morphism. The goal of this talk is to introduce a map (( ,∆V )) log : Zd(V ×U V ) −−−→ F0G(VF )⊗ Q (0.1) and to show that this map gives generalizations of classical invariants of ramification. 0.1 source Since V → U is assumed finite étale, the fiber product V ×U V is also finite étale over U and hence is smooth of dimension d over K. Thus, Zd(V ×U V ) is the free abelian group generated by the classes of irreducible components of V ×U V . In particular, if U is connected and V is a Galois covering of Galois group G, it is identified with the free abelian group Z[G]. 0.2 target For a noetherian scheme X, the Grothendieck group of the category of coherent OX modules is denoted by G(X). Let FnG(X) ⊂ G(X) denote the topological filtration generated by the classes of modules of dimension of support at most n.