Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring
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.
scientific article; zbMATH DE number 4216043
scientific article