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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
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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.


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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
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

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