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 ⊆ K ⊗k A and R * K, then there exists an extension field L of k such that R ⊆ L ⊗k A and trdegk(L) < trdegk(R). This generalizes a result of Freudenburg, namely, the fact that this is true for A = k.