Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring
type: Item
instance of: scholarly article
title: Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring
zbMATH Open document ID: 0737.11038
DOI: 10.1007/BF02571782
author: Cornelius Greither
published in: Mathematische Zeitschrift
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174387
review text: L. N. Childs has determined all Kummer extensions \(L/K\) of degree \(p\) with \(K\) a \(p\)-adic field such that the corresponding rings of integers give a Hopf Galois extension, i.e. \(\text{Spec}({\mathcal O}_L)\) is a principal homogeneous space over \(\text{Spec}({\mathcal O}_K)\) under a finite \({\mathcal O}_K\)-group. Here we extend his results in several ways: first to cyclic extensions \(L/K\) of degree \(p\) (i.e. \(\zeta_p\in K\) is no longer required), second to arbitrary (non-normal) extensions of degree \(p\), and lastly to cyclic extensions of order \(p^2\). In the second step, it is also shown that all \({\mathcal O}_K\)-groups of order \(p\) do occur as Hopf Galois groups. In the third step, some information about certain \({\mathcal O}_K\)-groups of order \(p^2\) is needed. This amounts to the calculation of some Ext groups, which is done in Part I of the paper. The extensions thus obtained are surprisingly complicated, and their principal homogeneous spaces are discussed. It is no longer true for order \(p^ 2\) that all \({\mathcal O}_K\)-groups occur as Hopf Galois groups. Another difference is the following: For \([L:K]=p\), it suffices to know the ramification number in order to decide whether \({\mathcal O}_L\) is Hopf Galois, but for \([L:K]=p^2\) the knowledge of the two ramification numbers does not suffice for this.
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 4216043
zbMATH Keywords: Hopf Galois extension
zbMATH Keywords: cyclic extension
MaRDI profile type: Publication
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OpenAlex ID: W2160498749
description: scientific article; zbMATH DE number 4216043