We prove the explicit characterization of the so-called "best f" for degree p Artin-Schreier and degree p Kummer extensions of Henselian valuation rings in residue characteristic p. This characterization is mentioned briefly in [Th16, Th18]. Existence of best f is closely related to the defect of such extensions and this characterization plays a crucial role in understanding their intricate structure. We also treat degree p Artin-Schreier defect extensions of higher rank valuation rings, extending the results in [Th16], and thus completing the study of degree p extensions that are the building blocks of the general theory.

Abstract: T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring A with field of fractions K and for a finite Galois extension L of K, the integral closure B of A in L is a filtered union of subrings of B which are of complete intersection over A. By this, we can obtain a ramification theory of Henselian valuation rings as the limit of the ramification theory of Saito. Our theory generalizes the ramification theory of complete discrete valuation rings of Abbes-Saito. We study "defect extensions" which are not treated in these previous works.

We provide an exposition of the canonical self-duality associated to a presentation of a finite, flat, complete intersection over a Noetherian ring, following work of Scheja and Storch.

Abstract:  It is conjectured that if k is an algebraically closed field of characteristic p > 0, then any branched G-cover of smooth projective k-curves where the "KGB" obstruction vanishes and where a p-Sylow subgroup of G is cyclic lifts to characteristic 0. Obus has shown that this conjecture holds given the existence of certain meromorphic differential forms on P_1^k with behavior determined by the ramification data of the cover. We give a more efficient computational procedure to compute these forms than was previously known. As a consequence, we show that all D_25- and D_27-covers lift to characteristic zero.

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 [Tha 16], 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 [Tha 16].

Abstract: The goal of this paper is to generalize and refine the classical ramification theory of complete discrete valuation rings to more general valuation rings, in the case of Artin–Schreier extensions. We define refined versions of invariants of ramification in the classical ramification theory and compare them. Furthermore, we can treat the defect case.

Consider a degree p Galois extension L/K of Henselian valued fields, with B/A the corresponding extension of valuation rings. In the classical case the different ideal D of B with respect to A equals the annihilator of the B-module Ω of relative differential 1-forms over A. However, we saw in [Tha 16] that this is not true in general. In this paper, we extend the relevant results of [Tha 16] to Artin-Schreier defect extensions with valuation of rank >1 and subsequently to the mixed characteristic case, in the spirit of the conductor-discriminant formula.