Ga-actions and completions

Let delta be a locally nilpotent derivation on an affine domain B defined over the complex field C and let A = Ker delta. Let M be a maximal ideal of B and let m = A boolean AND M. Then delta extends to C-derivations delta(M) and (delta) over cap (M) on the local ring B-M and its M-adic completion (B-M) over cap. We shall show that Ker (delta) over cap (M) is not necessarily equal to the m-adic completion (A) over cap (M), though Ker delta(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 etale endomorphism phi of a smooth affine surface Y with a G(a)-action for which the induced endomorphism psi on the algebraic quotient X = Y // G(a) is ramified. (C) 2007 Elsevier Inc. All rights reserved.