Ramification theory for degree p extensions of arbitrary valuation rings in mixed characteristic (0, p )

Abstract

We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic p with perfect residue fields to the case of fields with more general valuations and residue fields. As seen in [VT16], the "defect" case gives rise to many interesting complications. In this paper, we present analogous results for degree p extensions of arbitrary valuation rings in mixed characteristic (0, p) in a more general setting. More specifically, the only assumption here is that the base field K is henselian. In particular, these results are true for defect extensions even if the rank of the valuation is greater than 1. A similar method also works in equal characteristic, generalizing the results of [VT16].

Upper Ramification Groups for Arbitrary Valuation Rings

“…Relation to the ramification theory in [Th16] and [Th18]. Assume that the residue field of A is of positive characteristic p. Our upper ramification groups match well with the ramification theories [Th16] and [Th18] of cyclic extensions L of K of degree p. In those papers, we considered an ideal H of A, which is a generalization of the classical Swan conductor and plays an important role in the ramification theory of L/K. Some of its crucial properties are :…”

“…In Section 6, we deduce the above Theorem 6.2 and other general results on extensions of valuation rings from the works [Th16] and [Th18].…”

“…6 Applications of the works [Th16], [Th18] In this section, we prove the following results as applications of the works [Th16] and [Th18].…”

“…As we will see in Section 6 (Theorem 6.2), for a finite Galois extension L of K and for the integral closure B of A, B is a filtered union of subrings B ′ of B over A which are finite flat over A and of complete intersection over A. This Theorem 6.2 is deduced from results in [Th16], [Th18]. In this paper, we obtain important results on the upper ramification groups as the "limit" of his ramification theories of B ′ /A.…”

“…In this paper, we do not write Gal( K/K) I nlog simply as Gal( K/K) I (we do not follow [AS02] concerning this point). Through the works [Th16] and [Th18], we have the impression that in the study of arbitrary valuation rings, the logarithmic ramification theory is more natural than the non-logarithmic theory, and therefore, we have the impression that it is the logarithmic upper ramification group (not the non-logarithmic one) that deserves the simpler notation.…”