Kernel of locally nilpotent derivations of
Authors:
S. M. Bhatwadekar and Amartya K. Dutta
Journal:
Trans. Amer. Math. Soc. 349 (1997), 33033319
MSC (1991):
Primary 13B10; Secondary 13A30
MathSciNet review:
1422595
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper we study the kernel of a nonzero locally nilpotent derivation of the polynomial ring over a noetherian integral domain containing a field of characteristic zero. We show that if is normal then the kernel has a graded algebra structure isomorphic to the symbolic Rees algebra of an unmixed ideal of height one in , and, conversely, the symbolic Rees algebra of any unmixed height one ideal in can be embedded in as the kernel of a locally nilpotent derivation of . We also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general.
 [AEH]
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 0306173
(46 #5300)
 [AK]
Allen
Altman and Steven
Kleiman, Introduction to Grothendieck duality theory, Lecture
Notes in Mathematics, Vol. 146, SpringerVerlag, Berlin, 1970. MR 0274461
(43 #224)
 [BCW]
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 0432626
(55 #5613)
 [BD]
S.
M. Bhatwadekar and Amartya
K. Dutta, On residual variables and stably polynomial
algebras, Comm. Algebra 21 (1993), no. 2,
635–645. MR 1199695
(93k:13028), http://dx.doi.org/10.1080/00927879308824585
 [BH]
Winfried
Bruns and Jürgen
Herzog, CohenMacaulay rings, Cambridge Studies in Advanced
Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
(95h:13020)
 [C]
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
(85g:00023)
 [DF]
D. Daigle and G. Freudenburg, Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of , Preprint.
 [G]
José
M. Giral, Krull dimension, transcendence degree and subalgebras of
finitely generated algebras, Arch. Math. (Basel) 36
(1981), no. 4, 305–312. MR 623141
(82h:13008), http://dx.doi.org/10.1007/BF01223706
 [N]
Masayoshi
Nagata, Local rings, Interscience Tracts in Pure and Applied
Mathematics, No. 13, Interscience Publishers a division of John Wiley &
Sons New YorkLondon, 1962. MR 0155856
(27 #5790)
 [O]
Nobuharu
Onoda, Subrings of finitely generated rings over a pseudogeometric
ring, Japan. J. Math. (N.S.) 10 (1984), no. 1,
29–53. MR
884429 (88d:13024)
 [R]
D.
Rees, On a problem of Zariski, Illinois J. Math.
2 (1958), 145–149. MR 0095843
(20 #2341)
 [Rn]
Rudolf
Rentschler, Opérations du groupe additif sur le plan
affine, C. R. Acad. Sci. Paris Sér. AB 267
(1968), A384–A387 (French). MR 0232770
(38 #1093)
 [RS]
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
(80j:14030), http://dx.doi.org/10.1016/00218693(79)90214X
 [AEH]
 S.S. Abhyankar, P. Eakin and W. Heinzer, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310342. MR 46:5300
 [AK]
 A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, No. 146, SpringerVerlag, 1970. MR 43:224
 [BCW]
 H. Bass, E.H. Connell and D.L. Wright, Locally polynomial algebras are symmetric algebras, Invent. Math. 38 (1977), 279299. MR 55:5613
 [BD]
 S.M. Bhatwadekar and A.K. Dutta, On residual variables and stably polynomial algebras, Comm. Algebra 21(2) (1993), 635645. MR 93k:13028
 [BH]
 W. Bruns and J. Herzog, CohenMacaulay Rings, Cambridge University Press, 1993. MR 95h:13020
 [C]
 R.C. Cowsik, Symbolic powers and the number of defining equations, Algebra and its Applications, Lecture Notes in Pure and Applied Mathematics, No. 91, Marcel Dekker, New York, 1984, 1314. MR 85g:00023
 [DF]
 D. Daigle and G. Freudenburg, Locally nilpotent derivations over a UFD and an application to rank two locally nilpotent derivations of , Preprint.
 [G]
 J.M. Giral, Krull dimension, transcendence degree and subalgebras of finitely generated algebras, Arch. Math. 36 (1981), 305312. MR 82h:13008
 [N]
 M. Nagata, Local rings, Interscience, 1962. MR 27:5790
 [O]
 N. Onoda, Subrings of finitely generated rings over a pseudogeometric ring, Japan J. Math. 10(1) (1984), 2953. MR 88d:13024
 [R]
 D. Rees, On a problem of Zariski, Illinois J. Math. 2 (1958), 145149. MR 20:2341
 [Rn]
 R. Rentschler, Opérations du groupe additif sur le plan affine, C.R. Aca. Sc. Paris Sér. A 267 (1968), 384387. MR 38:1093
 [RS]
 P. Russell and A. Sathaye, On finding and cancelling variables in , J. Algebra 57 (1979), 151166. MR 80j:14030
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
13B10,
13A30
Retrieve articles in all journals
with MSC (1991):
13B10,
13A30
Additional Information
S. M. Bhatwadekar
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay400 005, India
Email:
smb@tifrvax.tifr.res.in
Amartya K. Dutta
Affiliation:
Stat  Math Unit, Indian Statistical Institute, 203, B.T. Road, Calcutta700 035, India
Email:
amartya@isical.ernet.in
DOI:
http://dx.doi.org/10.1090/S0002994797019466
PII:
S 00029947(97)019466
Keywords:
Locally nilpotent derivations,
inert subrings,
symbolic Rees algebra
Received by editor(s):
January 11, 1996
Article copyright:
© Copyright 1997 American Mathematical Society
