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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Kernel of locally nilpotent $R$-derivations
of $R[X,Y]$


Authors: S. M. Bhatwadekar and Amartya K. Dutta
Journal: Trans. Amer. Math. Soc. 349 (1997), 3303-3319
MSC (1991): Primary 13B10; Secondary 13A30
MathSciNet review: 1422595
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Abstract: In this paper we study the kernel of a non-zero locally nilpotent $R$-derivation of the polynomial ring $R[X,Y]$ over a noetherian integral domain $R$ containing a field of characteristic zero. We show that if $R$ is normal then the kernel has a graded $R$-algebra structure isomorphic to the symbolic Rees algebra of an unmixed ideal of height one in $R$, and, conversely, the symbolic Rees algebra of any unmixed height one ideal in $R$ can be embedded in $R[X,Y]$ as the kernel of a locally nilpotent $R$-derivation of $R[X,Y]$. We also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general.


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Additional Information

S. M. Bhatwadekar
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400 005, India
Email: smb@tifrvax.tifr.res.in

Amartya K. Dutta
Affiliation: Stat - Math Unit, Indian Statistical Institute, 203, B.T. Road, Calcutta-700 035, India
Email: amartya@isical.ernet.in

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01946-6
PII: S 0002-9947(97)01946-6
Keywords: Locally nilpotent derivations, inert subrings, symbolic Rees algebra
Received by editor(s): January 11, 1996
Article copyright: © Copyright 1997 American Mathematical Society