A+ CATEGORY SCIENTIFIC UNIT

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

Paulo Roberto Brumatti, Marcelo Oliveira Veloso 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 $B$.

Authors

  • Paulo Roberto BrumattiIMECC-UNICAMP
    13083-970, Campinas, SP, Brazil
    e-mail
  • Marcelo Oliveira VelosoCAP-UFSJ
    36420-000, Ouro Branco, MG, Brazil
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image