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. INTRODUCTIONWe present a generalization and refinement of the classical ramification theory of complete discrete valuation rings to valuation rings satisfying either (I) or (II) (as explained in 0.2), in the case of Artin-Schreier extensions. The classical theory considers the case of complete discrete valued field extension L|K where the residue field k of K is perfect. In his paper [KK89], Kato gives a natural definition of the Swan conductor for complete discrete valuation rings with arbitrary (possibly imperfect) residue fields. He also defines the refined Swan conductor rsw in this case using differential 1-forms and powers of the maximal ideal m L . The generalization we present is a further refinement of this definition. Moreover, we can deal with the extensions with defect, a case which was not treated previously. 0.1. Invariants of Ramification Theory. Let K be a valued field of characteristic p > 0 with henselian valuation ring A, valuation v K and residue field k. Let L = K(α) be the Artin-Schreier extension definedSince A is henselian, it follows that B is a valuation ring. Let v L be the valuation on L that extends v K and let l denote the residue field of L. Let Γ := v K (K × ) denote the value group of K. The Galois group Gal(L|K) = G is cyclic of order p, generated by σ : α → α + 1.Let A = {f ∈ K × | the solutions of the equation α p − α = f generate L over K}. Consider the ideals J σ and H, of B and A respectively, defined as below:Our first result compares these two invariants via the norm map N L|K = N , by considering the ideal N σ of A generated by the elements of N (J σ ). We also consider the idealThe ideals I σ and J σ play the roles of i(σ) and j(σ) (the Lefschetz numbers in the classical case, as explained in 2.1), respectively, in the generalization. 0.2. Main Results. We did not make any assumptions regarding the rank or defect in these definitions. Now consider two special cases of the scenario described above:(I) (Defectless) In this case, we assume that L|K is defectless. For Artin-Schreier extensions L|K considered in this paper, it means that either v L (L × )/v L (K × ) has order p and the residue extension l|k is trivial or the residue extension l|k is of degree p and L has the same value group Γ as K. (II) (Rank 1) The value group Γ of K is isomorphic to a subgroup of R as an ordered group. We will prove the following results: 1

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].

No abstract

No abstract

No abstract