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

Keshari, Manoj K.; Lokhande, Swapnil A.

J. Commut. Algebra 6, No. 1, 95-100 (2014).

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.

Reviewer: N. Mohan Kumar (St. Louis)

MSC:

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

Keywords:

locally nilpotent derivations; rigid derivations; triangulable derivations

Citations:

Zbl 0874.13021

References:

[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