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

Authors:
S. M. Bhatwadekar, Neena Gupta and Swapnil A. Lokhande

Journal:
Trans. Amer. Math. Soc. **369** (2017), 341-363

MSC (2010):
Primary 13N15; Secondary 13A50, 13C10, 13D15

DOI:
https://doi.org/10.1090/tran/6649

Published electronically:
March 1, 2016

MathSciNet review:
3557776

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

**S. M. Bhatwadekar**

Affiliation:
Bhaskaracharya Pratishthana, 56/14 Erandavane, Damle Path, Off Law College Road, Pune 411 004, India

Email:
smbhatwadekar@gmail.com

**Neena Gupta**

Affiliation:
Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India

MR Author ID:
933477

Email:
neenag@isical.ac.in

**Swapnil A. Lokhande**

Affiliation:
Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India

Address at time of publication:
Indian Institute of Information Technology, Vadodara C/O, Block 9, Government Engineering College, sector 28, Gandhinagar, Gujarat – 382028, India

Email:
swaplokhande@gmail.com

Keywords:
Locally nilpotent derivation,
polynomial ring,
projective module,
Grothendieck group,
Picard group

Received by editor(s):
July 8, 2014

Received by editor(s) in revised form:
December 24, 2014

Published electronically:
March 1, 2016

Article copyright:
© Copyright 2016
American Mathematical Society