Vanishing cycles, ramification of valuations, and class field theory. (With the collaboration of Takeshi Saito.) (Q1115928)

Let A be an excellent henselian discrete valuation ring with field of fractions k and with algebraically closed residue field E. Let B be the henselization of A[T] with respect to the maximal ideal \(m_ AA[T]+TA[T]\). Let \(U\subset \text{Spec}(B\otimes_ Ak)\) be a non-empty open subset, u the corresponding inclusion, K a field of positive characteristic with \(\text{char}(K)\neq \text{char}(E)\), and F a K-module in the étale site \(U_{et}\) which is locally constant of finite rank. Let x be the closed point of \(\text{Spec}(B)\) and p the generic point of \(\text{Spec}(B\otimes_ AE).\) Then the problem is to evaluate the dimension of \(R^ 1_ x(u_ F)= H^ 1_{et}(\text{Spec}(B\otimes_ A\bar k),u_(F)).\) In this sense Deligne had computed the dimension of this space under the hypothesis that F is unramified with respect to the valuation ring A by giving an explicit formula. The main purpose of the paper under review is to prove a similar formula in general, i.e. when F may be ramified with respect to \(A_ p\). This formula contains some terms involving the Swan conductor, and these are related to the 2-dimensional class field theory.