arXiv2015
Let k be an algebraically closed field of characteristic zero, D a locally nilpotent derivation on the polynomial ring k[X_1, X_2,X_3,X_4] and A the kernel of D. A question of M. Miyanishi asks whether projective modules over A are necessarily free. Implicit is a subquestion: whether the Grothendieck group K_0(A) is trivial. In this paper we shall demonstrate an explicit k[X_1]-linear fixed point free locally nilpotent derivation D of k[X_1,X_2, X_3, X_4] whose kernel A has an isolated singularity and whose Grothendieck group K_0(A) is not finitely generated; in particular, there exists an infinite family of pairwise non-isomorphic projective modules over the kernel A. We shall also show that, although Miyanishi's original question does not have an affirmative answer in general, suitably modified versions of the question do have affirmative answers when D annihilates a variable. For instance, we shall establish 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 is actually a polynomial ring over k; in particular, by the Quillen-Suslin theorem, Miyanishi's question has an affirmative answer. Our construction involves rings defined by the relation u^mv=F(z,t), where F(Z,T) is an irreducible polynomial in k[Z,T]. We shall show that a necessary and sufficient condition for such a ring to be the kernel of a k[X_1]-linear locally nilpotent derivation D of a polynomial ring k[X_1,...,X_4] is that F defines a polynomial curve.