## 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.

### MSC:

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

Zbl 0874.13021
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### References:

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