Hopf algebras over number rings (Q579377)

Let n be a positive integer. In a series of papers, D. Radford, the reviewer and R. L. Wilson showed that for any field k, there exists a finite-dimensional Hopf algebra \(A_ k\) whose antipode has order 2n. For \(k=Q\), the rational numbers, \(A_ Q\) is obtained by Galois descent from \(A_{Q(q)}\), where q is a primitive n-th root of unity. By using a certain basis of \(A_ Q\), the author constructs a finite-rank Hopf algebra \(A_{Z[1/n]}\) whose antipode has order 2n. \{Reviewer's note: Recently, C. Greither has constructed an analogous example \(A_ Z\) over the ring Z of integers, and the author has used this to construct analogous examples \(A_ R\) over arbitrary commutative rings R. Both of these results will appear in Commun. Algebra.\}