Small embeddings of integral domains

Abstract

Let $A$ be a geometrically integral algebra over a field $k$. We prove that, for any affine $k$-domain $R$, if there exists an extension field $K$ of $k$ such that $R\subseteq K{\otimes }_{k}A$ and $R⊈K$, then there exists an extension field $L$ of $k$ such that $R\subseteq L{\otimes }_{k}A$ and ${trdeg}_k\left(L\right)<{trdeg}_k\left(R\right)$. This generalizes a result of Freudenburg, namely, the fact that this is true for $A=k^{\left[1\right]}$.