Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring. (English) Zbl 0737.11038

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.


11S15 Ramification and extension theory
11S20 Galois theory
16T05 Hopf algebras and their applications
14L30 Group actions on varieties or schemes (quotients)
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