##
**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.

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 |

### Citations:

Zbl 0874.13021
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\textit{M. K. Keshari} and \textit{S. A. Lokhande}, J. Commut. Algebra 6, No. 1, 95--100 (2014; Zbl 1327.14211)

### References:

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[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 |

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