Algebraic Geometry of Hopf

We continue the investigation of Hopf-Galois extensions with central invariants started in [29]. 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 hydrodynamical type [1]. More recently, a concept of Frobenius manifold was introduced [15]; 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 [30, 8]. 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. We introduce Hopf-Galois extensions with central invariants and discuss examples in Section 1. The notion of inverse image of a HopfGalois extension is discussed in Section 2. A geometric object should become trivial if one looks at it locally. We prove a weak version of this principle in Section 3. A Hopf-Galois extension starts enjoying being Date: July 15, 1997. 1991 Mathematics Subject Classification. Primary 16W30; Secondary 14F05.