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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Some $K$-theoretic properties of the kernel of a locally nilpotent derivation on $k[X_1, \dots , X_4]$
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by S. M. Bhatwadekar, Neena Gupta and Swapnil A. Lokhande PDF
Trans. Amer. Math. Soc. 369 (2017), 341-363 Request permission

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
  • 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
  • Received by editor(s): July 8, 2014
  • Received by editor(s) in revised form: December 24, 2014
  • Published electronically: March 1, 2016
  • © Copyright 2016 American Mathematical Society
  • 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
  • MathSciNet review: 3557776