The different and differentials of local fields with imperfect residue fields

Abstract

Let K be a complete field with respect to a discrete valuation and let L be a finite Galois extension of K. If the residue field extension is separable then the different of L/K can be expressed in terms of the ramification groups by a well-known formula of Hilbert. We will identify the necessary correction term in the general case, and we give inequalities for ramification groups of subextensions L'/K in terms of those of L/K. A question of Krasner in this context is settled with a counterexample. These ramification phenomena can be related to the structure of the module of differentials of the extension of valuation rings. For the case that, where p is the residue characteristic, this module is shown to determine the correction term in Hilbert's formula.1991 Mathematics subject classification: 11S15, 13N05.

Congruences of models of elliptic curves

“…, the proposition also holds for L/K. As O K is complete, O L /O K can be decomposed into successive Galois monogeneous (cyclic) extensions (see for instance the explanations in [9], proof of Theorem 4.1). Thus we are reduced to the case O L is monogeneous…”

p −1-Linear Maps in Algebra and Geometry

“…Step 3 is somewhat involved, see for example [Mor53,SS75,dS97]. On the other hand, see [ST10a] for a complete proof.…”

A monogenic Hasse-Arf theorem

“…In [4], De Smit shows that most of the classical ramification-theoretic properties of residually separable extensions B/A hold in the slightly more general, "monogenic" case where we require only that B is generated as an A-algebra by one element. The purpose of this note is to show that the Hasse-Arf theorem also holds in this context.…”