Galois objects with normal bases for free Hopf algebras of prime degree
type: Item
instance of: scholarly article
title: Galois objects with normal bases for free Hopf algebras of prime degree
zbMATH Open document ID: 0622.16005
DOI: 10.1016/0021-8693(87)90141-4
author: Susan Hurley
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 13B05
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 4008593
zbMATH Keywords: free Hopf algebra
zbMATH Keywords: dual Hopf algebra
zbMATH Keywords: H-Galois algebras
zbMATH Keywords: normal bases
zbMATH Keywords: Harrison cohomology group
zbMATH Keywords: Hopf-Galois objects
reviewed by: H. F. Kreimer
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90141-4
OpenAlex ID: W1980730290
cites work: Hopf algebras and Galois theory
cites work: Q3913495
cites work: Q5518497
cites work: Quadratic Hopf algebras and Galois extensions
cites work: Q5509718
cites work: Q5590089
cites work: Group schemes of prime order
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description: scientific article; zbMATH DE number 4008593