Group Actions,k-Derivations,and Finite Morphisms

Let G be an affine algebraic group over an algebraically closed field k of characteristic zero. In this article, we consider finite G-equivariant morphisms F:X → Y of irreducible affine G-varieties. First we determine under which conditions on Y the induced map F ^G :X//G → Y//G of quotient varieties is also finite. This result is reformulated in terms of kernels of derivations on k-algebras A ? B such that B is integral over A. Second we construct explicitly two examples of finite G-equivariant maps F. In the first one, F ^G is quasifinite but not finite. In the second one, F ^G is not even quasifinite.