On Ramification Theory in the Imperfect Residue Field Case

Let K be a complete discrete valuation field with the residue field K, charK = p > 0. If K is a perfect field, there exists a beautiful theory of ramification in algebraic extensions of K. Given a finite Galois extension L/K with the Galois group G, one can introduce a canonical filtration (Gi) in G with quite a natural behavior with respect to subextensions in L/K. Namely, if H is a normal subgroup in G, one has Hi = Gi ∩H and (G/H) = (GH)/H. In the last relation we used upper numbering of ramification subgroups G = Gψ(j), where the Hasse-Herbrand function ψ = ψL/K can be easily calculated in terms of orders of Gi. Next, this “upper” filtration of G is compatible with class field theory. In particular, if L/K is abelian and the residue fields are finite (or quasi-finite), then θ(Uj) = G j for all j = 0, 1, . . . , where θ : K → G is the reciprocity map, and (Uj) is the filtration in K determined by the valuation. A comprehensive exposition of all these facts is given, e. g., in [S, Ch. IV, Ch.XV]. However, if K is not perfect, there exists no reasonable theory of upper numbering of ramification subgroups. The “lower” ramification subgroups can still be defined, however, the ramification filtration in the group G does not determine that in G/H. (Examples were given, e. g., in [L, H].) In the present article we treat the class of fields K with [K : K p ] = p. (In particular, this holds for a two-dimensional local field K.) In the case charK = p, we work with a relative situation K/k, when a complete subfield k in K with a perfect residue field is supposed to be fixed. (In the mixed characteristic case, i. e., when charK = 0, a subfield k can be chosen in a canonical way.) For a Galois extension L/K we introduce a new lower filtration on Gal(L/K) indexed by a special linearly ordered set I (see §1). Then a Hasse-Herbrand function ΨL/K : I → I can be defined with all the usual properties. Therefore, a theory of upper ramification groups, as well as the ramification theory of infinite extensions, can be developed. If we consider abelian extensions of exponent p, the ramification filtration determines a dual filtration on the additive (resp. multiplicative) group of K via Artin-Schreier (resp. Kummer) duality. In the case eK/k = 1, this dual filtration is described explicitly in §2. In §3, we consider a fieldK with a discrete valuation of rank two. (Main examples are provided by 2-dimensional local or local-global fields.) We introduce a new index