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

Abstract:

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 $k[X_1,\dots ,X_4]$ is that $F$ defines a polynomial curve.