A note on Nakai's conjecture for the ring $K[X_1,\ldots,X_n] / (a_1X_1^m+\cdots+a_nX_n^m)$
Volume 123 / 2011
Colloquium Mathematicum 123 (2011), 277-283
MSC: 13N05, 13B10, 13N10, 13N15.
DOI: 10.4064/cm123-2-10
Abstract
Let $k$ be a field of characteristic zero, $k[X_1,\ldots, X_n]$ the polynomial ring, and $B$ the ring ${k[X_1,\ldots, X_n]}/{(a_1X_1^m+\cdots +a_mX_n^m)}$, $0\neq a_i\in k$ for all $i$ and $ m, n\in \mathbb{N}$ with $n\geq 2$ and $m\geq 1$. Let $\mathop{\rm Der} _k^2(B)$ be the $B$-module of all second order $k$-derivations of $B$ and $ \mathop{\rm der} _k^2(B)=\mathop{\rm Der} _k^{1}(B)+\mathop{\rm Der} _k^1(B)\mathop{\rm Der} _k^{1}(B)$ where $\mbox{Der}_k^{1}(B)$ is the $B$-module of $k$-derivations of $B$. If $m\geq 2$ we exhibit explicitly a second order derivation $D\in \mathop{\rm Der} _k^2(B)$ such that $D\notin \mathop{\rm der} _k^2(B)$ and thus we prove that Nakai's conjecture is true for the $k$-algebra $B$.
