Ramification theory for Artin–Schreier extensions of valuation rings

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. 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

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

“…We can easily modify the proof of Theorem 1.3 presented in 4.2 to fit the Artin-Schreier case. Similarly, we can imitate the proof of Lemma 6.13 and thus, the main results of [VT16] can be generalized to the higher rank defect case.…”

“…We have c = hpγ p−1 αz p−1 , c 0 = h 0 pγ p−1 0 α 0 z p−1 and hence, c 0 c = uγ 0 γ p−1 = a p−1 . We will verify that dα ′ 0 = adα ′ , the rest follows (see 6.3.3 [VT16]).…”

“…In the seventh section, we extend the main results to the non-Kummer case. The last section consists of some remarks about how the results of [VT16] can be generalized to Artin-Schreier defect extensions of higher rank valuations, in a similar fashion.…”

“…As in the case of Artin-Schreier extensions (see section 5.4 of [VT16]), it is enough to prove the following result:…”

Upper Ramification Groups for Arbitrary Valuation Rings

“…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.…”