Group actions, k-derivations and finite morphisms

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Let G be an affine algebraic group over an algebraically closed field k of characteristic zero. In this paper, 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 Fᵍ: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, Fᵍ is quasifinite but not finite. In the second one, Fᵍ is not even quasifinite.