Invariants of finite Hopf algebras

This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings. Suppose that H is a finite dimensional Hopf algebra and A a commutative algebra, say over a field K. Let δ : A → A ⊗H be an algebra homomorphism which makes A into a right H-comodule. In this case A is called an H-comodule algebra. The coaction of H on A corresponds to an action of the dual Hopf algebra H. For technical reasons all results in this paper are formulated in terms of coactions. The situation where H is commutative can be described geometrically by giving an action of the finite group scheme G = SpecH on the scheme X = SpecA. The subring of G-invariants A ⊂ A represents then the quotient scheme X/G. A fact of fundamental importance states that A is an integral extension of A. In case of ordinary finite groups this classical result goes back to the work of E. Noether. The question of whether a similar assertion is true for a noncommutative H was posed by Montgomery [20, 4.2.6]. Shortly afterwards Zhu [30] succeeded in verifying two special cases and constructing a counterexample in general. The first objective of the present article is to investigate the integrality of A over A more thoroughly (here A stands for the invariants of the given coaction). A part of the argument given in [7] and [22] for commutative H carries over without problems. Since A is commutative, A ⊗ H can be regarded as an A-algebra with respect to the action on the first tensorand. Since A⊗H is free of finite rank over A, with each element u ∈ A ⊗ H one can associate its characteristic polynomial PA⊗H/A(u, t) ∈ A[t]. Letting