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

Bhatwadekar, S.; Dutta, Amartya

doi:10.1090/S0002-9947-97-01946-6

Kernel of locally nilpotent $R$-derivations of $R[X,Y]$
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by S. M. Bhatwadekar and Amartya K. Dutta

Trans. Amer. Math. Soc. 349 (1997), 3303-3319

DOI: https://doi.org/10.1090/S0002-9947-97-01946-6

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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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