Galois objects with normal bases for free Hopf algebras of prime degree
Let p be a positive prime integer, let R be a commutative ring, let H be a free Hopf algebra over R of rank p, and let \(H^*\) denote the dual Hopf algebra. If R is an algebra over a certain subring of the p-adic integers; then \(H\cong R[x]/(x^ p-ax)\) and \(H^*\cong R[y]/(y^ p- by)\) as algebras for \(a,b\in R\), and H is characterized by b (or a). The author uses the characterization of H to describe the structure of an H- Galois algebra S over R. Assume that R contains a (p-1)st root \(\tilde b\) of b. It is shown that S has a normal basis if and only if there exists a unit z of S such that \(z\equiv 1 (mod \tilde b)\) and z is mapped to \(\tilde bz\) by the coset of y under the action of \(H^*\) on S. By assigning \(z^ p\) to S, the author obtains an isomorphism of the group of isomorphism classes of H-Galois algebras possessing normal bases (the second Harrison cohomology group of \(H^*)\) onto a subquotient \(G_ 1/G_ 2\) of the multiplicative group of units of R, where \(G_ 1\) consists of units which are congruent to 1 modulo \(\tilde b^ p=b\cdot \tilde b\) and \(G_ 2\) consists of p-th powers of units which are congruent to 1 modulo \(\tilde b.\) The exposition is very computational and sometimes minor misprints must be corrected to obtain valid equations. But the paper employs only the theory of Hopf-Galois objects except for reference to \textit{J. Tate} and \textit{F. Oort} [Ann. Sci. Ec. Norm. Supér., IV. Sér. 3, 1-21 (1970; Zbl 0195.508)] for the structure of H.
scientific article; zbMATH DE number 4008593
scientific article