On finite order elements in Hopf algebras over rings
by
Alexander Stolin
Department of Mathematics, University of Gothenburg, S-41296 Sweden
Coauthors: Lars Kadison

It has been known since papers by Larson, Sweedler and Pareigis that a finite rank Hopf algebra over a p.i.d. and more generally a finite projective Hopf algebra H over a commutative ring k of trivial Picard group, respectively, is a Frobenius algebra. With f a right norm in H^* and t a right norm in H such that ft equals the counit \epsilon, a Frobenius system was given recently by (f, S^-1(t₂), t₁) in [1] and in [2] for k a field. [1] shows that the Nakayama automorphism \eta has order at most 2 if H and H^* are unimodular, while [2] shows that S and \eta have order dividing 4dimH and 2dimH respectively, if k is a field.

The purpose in this paper is to generalize these results for finite projective Hopf algebras over Noetherian rings. To do this we need a sharp estimate on the order of a group-like element of H. Using a formula for S⁴ and a formula which relates the Nakayama automorphism \eta with S², we then obtain estimates for orders of S and \eta, which coincide in the case of fields with that of [2].

1. K. Beidar, Y. Fong, and A. Stolin, On antipodes and integrals in Hopf algebra over rings and the quantum Yang-Baxter equation, J. Algebra 194 (1997), 36-52.

2. D. Fischman, S. Montgomery, and H.-J. Schneider, Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math. Soc.  349 (1997), 4857-4895.

Date received: November 26, 1998