A note on rigidity and triangulability of a derivation

TL;DR

This paper investigates the properties of derivations over algebraic domains, establishing conditions under which rigidity and triangulability are preserved or characterized, especially in the context of polynomial extensions.

Contribution

It extends known results on derivation properties to more general algebraic domains and clarifies conditions for rigidity and triangulability in polynomial rings.

Findings

Rigidity of D_K implies rigidity of D.

Triangulability over A is equivalent to triangulability over A[X] when n=3, r=2.

Results generalize Daigle's work from fields to algebraic domains.

Abstract

Let A be a $\mathfrak Q$-domain, K=frac(A), B=A^{[n]} and D\in \lnd_A(B). Assume rank D= rank D_K=r, where D_K is the extension of D to K^{[n]}. Then we show that (i) If D_K is rigid, then D is rigid. (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.