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Let k be an algebraically closed field of characteristic zero, D a locallynilpotent derivation on the polynomial ring k[X_1, X_2,X_3,X_4] and A thekernel of D. A question of M. Miyanishi asks whether projective modules over Aare necessarily free. Implicit is a subquestion: whether the Grothendieck groupK_0(A) is trivial.  In this paper we shall demonstrate an explicit k[X_1]-linear fixed point freelocally nilpotent derivation D of k[X_1,X_2, X_3, X_4] whose kernel A has anisolated singularity and whose Grothendieck group K_0(A) is not finitelygenerated; in particular, there exists an infinite family of pairwisenon-isomorphic projective modules over the kernel A.  We shall also show that, although Miyanishi's original question does not havean affirmative answer in general, suitably modified versions of the question dohave affirmative answers when D annihilates a variable. For instance, we shallestablish that in this case the groups G_0(A) and G_1(A) are indeed trivial.Further, we shall see that if the above kernel A is a regular ring, then A isactually a polynomial ring over k; in particular, by the Quillen-Suslintheorem, Miyanishi's question has an affirmative answer.  Our construction involves rings defined by the relation u^mv=F(z,t), whereF(Z,T) is an irreducible polynomial in k[Z,T]. We shall show that a necessaryand sufficient condition for such a ring to be the kernel of a k[X_1]-linearlocally nilpotent derivation D of a polynomial ring k[X_1,...,X_4] is that Fdefines a polynomial curve.

Some K-theoretic properties of the kernel of a locally nilpotent derivation on k[X_1, \dots, X_4]