Rings with trivial FML-invariant

Abstract

Let $k$ be a field of characteristic zero and $B$ a commutative integral domain that is also a finitely generated $k$-algebra. It is well known that if $k$ is algebraically closed and the "Field Makar-Limanov" invariant FML$(B)$ is equal to $k$, then $B$ is unirational over $k$. This article shows that, when $k$ is not assumed to be algebraically closed, the condition FML$(B)=k$ implies that there exists a nonempty Zariski-open subset $U$ of Spec$(B)$ with the following property: for each prime ideal $\mathfrak{p}\in U$, the $\kappa (\mathfrak{p})$-algebra $\kappa (\mathfrak{p}){\otimes }_{k}B$ can be embedded in a polynomial ring in $n$ variables over $\kappa (\mathfrak{p})$, where $n=\dim B$ and $\kappa (\mathfrak{p})=B_{\mathfrak{p}}/\mathfrak{p}{B}_{\mathfrak{p}}$.