A note on rigidity and triangulability of a derivation

Abstract

Let $A$ be a $\q$-domain, $K=\text{frac\,}(A)$, $B=A^{[n]}$ and $D\in \text{LND}_A(B)$. Assume rank $D=\text{rank\,}D_K=r$, where $D_K$ is the extension of $D$ to $K^{[n]}$. Then we show that

\hskip5pt(i) If $D_K$ is rigid, then $D$ is rigid.

\hskip2.5pt(ii) Assume $n=3$, $r=2$ and $B=A[X,Y,Z]$ with $DX=0$. Then $D$ is triangulable over $A$ if and only if $D$ is triangulable over $A[X]$. In case $A$ is a field, this result is due to Daigle.