A note on rigidity and triangulability of a derivation. (English) Zbl 1327.14211

Let \(A\) be an integral domain containing \(\mathbb{Q}\) and \(A^{[n]}=A[X_1,\dots, X_n]=B\). An \(A\)-derivation \(D:B\to B\) is locally nilpotent if for each \(x\in B\), \(D^s(x)=0\) for some \(s>0\). For such a\(D\), its rank is defined as the smallest \(r\) such that \(A[X_1,\dots, X_{n-r}]\subset \mathrm{ker} D\) (after a change of variables). Such a derivation is triangulable if for a suitable change of variables, one has \(D(X_i)\in A[X_1,\dots, X_{i-1}]\) for \(1\leq i\leq n\). A derivation \(D\) of rank \(r\) is rigid, if for any two choices of variables \(X_i,X_i'\) with \(A_1=A[X_1,\dots, X_{n-r}]\) and \(A_2=A[X_1',\dots,X_{n-r}']\) both contained in \(\mathrm{ker} D\), then \(A_1=A_2\).
D. Daigle proved that when \(A\) is a field and \(n=3\), all locally nilpotent derivations are rigid [J. Pure Appl. Algebra 113, No. 3, 297–305 (1996; Zbl 0874.13021)]. The authors prove in the same vein for an arbitrary \(A\) as above, if \(D\) is such a derivation and \(D\) has the same rank when localized at the fraction field of \(A\) and rigid after localization, then \(D\) itself is rigid. A similar generalization of a result of Daigle on triangulability is also proved.


14L30 Group actions on varieties or schemes (quotients)
13B25 Polynomials over commutative rings


Zbl 0874.13021
Full Text: DOI arXiv Euclid


[1] S.S. Abhyankar, P. Eakin and W. Heinzer, On the uniqueness of the coefficient ring in a polynomial ring , J. Algebra 23 (1972), 310-342. · Zbl 0255.13008
[2] S.M. Bhatwadekar and D. Daigle, On finite generation of kernels of locally nilpotent \(R\)-derivations of \(R[X,Y,Z]\) , J. Algebra 322 (2009), 2915-2916. · Zbl 1234.13027
[3] S. M. Bhatwadekar and Amartya K. Dutta, On residual variables and stably polynomial algebras , Comm. Algebra 21 (1993), 635-645. · Zbl 0778.13016
[4] D. Daigle, A necessary and sufficient condition for triangulability of derivations of \(k[X, Y,Z]\) , J. Pure Appl. Alg. 113 (1996), 297-305. · Zbl 0874.13021
[5] D. Daigle and G. Freudenburg, A counterexample to Hilbert’s fourteenth problem in dimension \(5\) , J. Algebra 221 (1999), 528-535. · Zbl 0963.13024
[6] G. Freudenburg, Algebraic theory of locally nilpotent derivations , Encyclopedia of Mathematical Sciences, vol. 136, Springer-Verlag, Berlin, 2006. \noindentstyle · Zbl 1121.13002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.