Induced Hopf Galois structures and their local Hopf Galois modules (Q2075302)

In the paper under review the authors prove the following result which is similar to that of the classical case in which \(L/K\) is a finite Galois extension of fields whose Galois group is a direct product of subgroups. Theorem 1.3. Let \(L/K\) be a finite Galois extension with Galois group \(G=J\rtimes G'\). Let \(E=L^{G'}\) and \(F=L^J\). Then: \begin{itemize} \item[(i)] \(E/K\) and \(F/K\) are Hopf Galois extensions. \item[(ii)] \(L=EF\) and \(E\cap F=K\). \item[(iii)] \(E/K\) and \(F/K\) are linearly disjoint. \end{itemize} Let \(E/K\) be \(H_1\)-Galois and let \(L/E\) be \(H_2\)-Galois. We consider the corresponding induced Hopf Galois structure of \(L/K\). Let \(H\) be its associated Hopf algebra. Then: \begin{itemize} \item[(iv)] \(H=H_1\otimes_K \overline{H}\), where \(\overline{H}\) is the Hopf algebra of the Hopf Galois structure of \(F/K\) such that \(\overline{H}\otimes_K E=H_2\). \item[(v)] The Hopf action of \(H\) on \(L\) is the Kronecker product of the Hopf actions of \(H_1\) on \(E\) and \(\overline{H}\) on \(F\). \end{itemize} Now suppose that \({\mathcal O}_N\) is a principal ideal domain with field of fractions \(N\). Let \(M/N\) be a separable Hopf Galois extension of degree \(n\) and let \({\mathcal O}_M\) be the integral closure of \({\mathcal O}_N\) in \(M\). If \(H\) is the Hopf algebra of a Hopf Galois structure of \(M/N\), then \[{\mathfrak A}_H=\{\alpha \in H \mid \alpha\cdot x \in {\mathcal O}_M\ \text {for every} \ x\in {\mathcal O}_M \} \] is the associated \({\mathcal O}_N\)-order to \({\mathcal O}_M\) in \(H\). As analogs to Lemma 5 of [\textit{N. P. Byott} and \textit{G. Lettl}, J. Théor. Nombres Bordx. 8, No. 1, 125--141 (1996; Zbl 0859.11059)] the authors of the present paper prove the following theorems. In the statement of Theorem 1.5, the assumption that the structure is an induced one means the structure is the one described in Theorem 1.3. Theorem 1.5. Let \(K\) be the quotient field of a principal ideal domain \({\mathcal O}_K\), \(L/K\) a finite separable Hopf Galois extension, and \({\mathcal O}_L\) the integral closure of \({\mathcal O}_K\) in \(L\). Assume that the structure is an induced one and its Hopf algebra is \(H=H_1\otimes_K \overline{H}\). If \(E/K\) and \(F/K\) are arithmetically disjoint, then the following statements hold: \begin{itemize} \item[(i)] \({\mathfrak A}_H={\mathfrak A}_{H_1}\otimes_{{\mathcal O}_K} {\mathfrak A}_{\overline H}\). \item[(ii)] If \({\mathcal O}_E\) is \({\mathfrak A}_{H_1}\)-free and \({\mathcal O}_F\) is \({\mathfrak A}_{\overline H}\)-free, then \({\mathcal O}_L\) is \({\mathfrak A}_H\)-free. Moreover, an \({\mathfrak A}_H\)-generator of \({\mathcal O}_L\) is the product of an \({\mathfrak A}_{H_1}\)-generator of \({\mathcal O}_E\) and an \({\mathfrak A}_{\overline{H}}\)-generator of \({\mathcal O}_F\). \end{itemize} Since the extensions \(E/K\) and \(F/K\) of Theorem 1.5 are linearly disjoint, it follows that for every Hopf Galois structure \(H_1\) of \(E/K\), \(H_1\otimes_K F\) is a Hopf Galois structure of \(L/F\). The authors of the present paper prove Theorem 1.6. Under the same hypothesis, \begin{itemize} \item[(i)] \({\mathfrak A}_{H_1\otimes_K F}={\mathfrak A}_{H_1}\otimes_{{\mathcal O}_K} {\mathcal O}_F\). \item[(ii)] If \({\mathcal O}_E\) is \({\mathfrak A}_{H_1}\)-free, then \({\mathcal O}_L\) is \({\mathfrak A}_{H_1\otimes_K F}\)-free. \end{itemize} The authors include many examples in the paper that illustrate the tools used in the proofs of the above results, or the results themselves.