Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory

Abstract

This book studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. The results, not previously published in book form, show that Hopf algebras play a natural role in local Galois module theory. Included in this work are expositions of short exact sequences of Hopf algebras; Hopf Galois structures on separable field extensions; a generalization of Noether's theorem on the Galois module structure of tamely ramified extensions of local fields to wild extensions acted on by Hopf algebras; connections between tameness and being Galois for algebras acted on by a Hopf algebra; constructions by Larson and Greither of Hopf orders over valuation rings; ramification criteria of Byott and Greither for the associated order of the valuation ring of an extension of local fields to be Hopf order; the Galois module structure of wildly ramified cyclic extensions of local fields of degree p and p 2 ; and Kummer theory of formal groups. Contents: Introduction; Hopf algebras and Galois extensions; Hopf Galois structures on separable field extensions; Tame extensions and Noether's theorem; Hopf algebras of rank p; Larson orders; Cyclic extensions of degree p; Non-maximal orders; Ramification restrictions; Hopf algebras of rank p 2 ; Cyclic Hopf Galois extensions of degree p 2 ; Formal groups; Principal homogeneous spaces and formal groups; Bibliography; Index.

Galois extensions of structured ring spectra. Stably dualizable groups

“…We say that H coacts on B over A if there is a coassociative and counital map Note that there is no finiteness/dualizability condition on H in this definition. See [Chi00] for a recent text on Hopf-Galois extensions in the algebraic setting.…”

Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups

“…induced from the action of G on L making L into a Galois extension of K (see [Ch00] for an exposition). By results of Greither and Pareigis [GP87] and Byott [By96], there is a 1-1 map from equivalence classes of regular embeddings β of G into Hol(G) to Hopf Galois structures on L/K.…”

Monogenic Hopf algebras over discrete valuation rings with low ramification

“…Applying the results of [15] when the smooth resolution is given by one-dimensional formal groups we know m/i H (m) ∼ = PH(H ). By [3, 39.5] We conclude by giving a result concerning extensions of K. For further applications to finding maximal orders in Hopf algebras, see [3]. Corollary 6.2.…”