On the Rigidity of Some Extensions of Domains

Abstract

A ring is said to be rigid if it admits no nonzero locally nilpotent derivations, and an affine variety is rigid if its coordinate ring is rigid. In this paper, we improve some techniques for determining the rigidity of k-domains (affine varieties) over a field k of characteristic zero. First, we generalize the ABC theorem. Then we study locally nilpotent derivations of a simple algebraic extension $\mathit{R}[\mathit{z}]$ of a k-domain R, where $\mathit{r}{\mathit{z}}^{\mathit{n}}\in \mathit{R}$ for some nonzero $\mathit{r}\in \mathit{R}$ and some positive integer n. Subsequently, we study locally nilpotent derivations and rigidity of an extension $\mathit{R}[\mathit{x},\mathit{y}]$ of R such that ${\mathit{r}}_1{\mathit{x}}^{\mathit{m}}{\mathit{y}}^{\mathit{n}}\in \mathit{R}$ or ${\mathit{r}}_1{\mathit{x}}^{\mathit{m}}+{\mathit{r}}_2{\mathit{y}}^{\mathit{n}}\in \mathit{R}$ for some nonzero ${\mathit{r}}_1,{\mathit{r}}_2\in \mathit{R}$ and some positive integers $\mathit{m},\mathit{n}$. Finally, as applications of these general results, we prove the rigidity of some quadrinomial varieties and Pham–Brieskorn hypersurfaces.