Rings with trivial FML-invariant

TL;DR

This paper investigates the structure of certain algebraic rings with trivial FML-invariant over fields that are not algebraically closed, showing they are locally embeddable into polynomial rings over a dense subset of their spectrum.

Contribution

It extends the understanding of FML-invariant implications from algebraically closed fields to more general fields, establishing local polynomial embedding properties.

Findings

If FML$(B)=k$, then a dense open subset of Spec$(B)$ has prime ideals where the localized algebra embeds into a polynomial ring.

The result generalizes the unirationality condition to non-algebraically closed fields.

Provides a geometric interpretation of the FML-invariant condition in terms of local embeddings.

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 $\kappa(\mathfrak{p}) = B_{\mathfrak{p}}/{\mathfrak{p}}B_{\mathfrak{p}}$.