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 PDF

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

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.

References

Shreeram S. Abhyankar, William Heinzer, and Paul Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310–342. MR 306173, DOI 10.1016/0021-8693(72)90134-2
Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR 0274461, DOI 10.1007/BFb0060932
H. Bass, E. H. Connell, and D. L. Wright, Locally polynomial algebras are symmetric algebras, Invent. Math. 38 (1976/77), no. 3, 279–299. MR 432626, DOI 10.1007/BF01403135
S. M. Bhatwadekar and Amartya K. Dutta, On residual variables and stably polynomial algebras, Comm. Algebra 21 (1993), no. 2, 635–645. MR 1199695, DOI 10.1080/00927879308824585
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
H. L. Manocha and J. B. Srivastava (eds.), Algebra and its applications, Lecture Notes in Pure and Applied Mathematics, vol. 91, Marcel Dekker, Inc., New York, 1984. Papers from the international symposium held at New Delhi, December 21–25, 1981. MR 750836
D. Daigle and G. Freudenburg, Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of $k[X_1, \cdots , X_n]$, Preprint.
José M. Giral, Krull dimension, transcendence degree and subalgebras of finitely generated algebras, Arch. Math. (Basel) 36 (1981), no. 4, 305–312. MR 623141, DOI 10.1007/BF01223706
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Nobuharu Onoda, Subrings of finitely generated rings over a pseudogeometric ring, Japan. J. Math. (N.S.) 10 (1984), no. 1, 29–53. MR 884429, DOI 10.4099/math1924.10.29
D. Rees, On a problem of Zariski, Illinois J. Math. 2 (1958), 145–149. MR 95843, DOI 10.1215/ijm/1255380843
Rudolf Rentschler, Opérations du groupe additif sur le plan affine, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A384–A387 (French). MR 232770
Peter Russell and Avinash Sathaye, On finding and cancelling variables in $k[X,\,Y,\,Z]$, J. Algebra 57 (1979), no. 1, 151–166. MR 533106, DOI 10.1016/0021-8693(79)90214-X