Duality theorems for crossed products over rings. (Q555761)

Let \(H\) be a Hopf algebra with twisted antipode over a commutative Noetherian ring \(R\), \(A\#_\sigma H\) a right \(H\)-crossed product with invertible cocycle \(\sigma\), and \(U\subseteq H^*\) a right \(H\)-module subalgebra. Under suitable technical conditions, that include purity requirements for some inclusions, the author proves a duality theorem, namely, the existence of an isomorphism of algebras \((A\#_\sigma H)\#U\simeq A\otimes_R(H\#U)\). This duality theorem generalizes previous results, obtained in case \(R\) is a field or a Dedekind domain, due to \textit{R. J. Blattner} and \textit{S. Montgomery} [J. Algebra 95, 153-172 (1985; Zbl 0589.16010)], \textit{M. Koppinen} [J. Algebra 146, No. 1, 153-174 (1992; Zbl 0749.16021)] and \textit{C. Chen} [Commun. Algebra 21, No. 8, 2885-2903 (1993; Zbl 0781.16021)].