Algebraic Geometry of Hopf - Galois Extensions

We continue the investigation of Hopf-Galois extensions with central invariants started in [30]. Our objective is not to imitate algebraic geometry using Hopf-Galois extension but to understand their geometric properties. Let H be a finite-dimensional Hopf algebra over a ground field k. Our main object of study is an H-Galois extension U ⊇ O such that O is a central subalgebra of U . Let us briefly discuss geometric properties of the object. By Kreimer-Takeuchi theorem the module UO is projective. Thus, it defines a vector bundle of algebras on the spectrum of O by Serre theorem. The fibers carry a structure of Frobenius algebra. A similar structure was of interest to geometers for a while because commutative Frobenius algebras naturally arise in the study of symmetric Poisson brackets of hydro-dynamical type [1]. More recently, a concept of Frobenius manifold was introduced [17]; it is a manifold such that tangent spaces carry a structure of a commutative Frobenius algebra which multiplication has a generating function. Our set-up is different: we have a vector bundle rather than the tangent bundle and our algebras are not necessarily commutative. However, we have more structure involved: a Hopf-Galois extension may be regarded as a “quantum” principal bundle [31]. (We should point out that the notion of a quantum principal bundle in non-commutative geometry is more involved but, nevertheless, quantum principal bundles with universal differential calculus are the same as Hopf-Galois extensions [10, 16].) If H is commutative (i.e. an algebra of functions on a finite group scheme G) then a commutative H-Galois extension U ⊇ O is a G-principal bundle on the spectrum of O. Finally, we emphasize that centrality of invariants is a crucial property for a geometrical treatment of Hopf-Galois extensions. We will illustrate this claim throughout the paper. Date: September 24, 1997. 1991 Mathematics Subject Classification. Primary 16W30; Secondary 14F05.