Formula omitted]-actions and completions

Abstract

Let δ be a locally nilpotent derivation on an affine domain B defined over the complex field C and let A = Ker δ . Let M be a maximal ideal of B and let m = A ∩ M . Then δ extends to C -derivations δ M and δ ˆ M on the local ring B M and its M -adic completion B M ˆ . We shall show that Ker δ ˆ M is not necessarily equal to the m -adic completion A m ˆ , though Ker δ M = A m provided B is factorial. This gives a negative answer to a problem raised in [M. Miyanishi, Problems in Mathematisches Forschungsinstitut Oberwolfach Report No. 01/2007 (p. 70) on the workshop “Affine Algebraic Geometry”, 2007. [5]]. As a related result, we also give an example of a G a -equivariant, nonfinite étale endomorphism φ of a smooth affine surface Y with a G a -action for which the induced endomorphism ψ on the algebraic quotient X = Y / / G a is ramified.