On algebras which are locally $\mathbb {A}^{1}$ in codimension-one

Abstract:

Let $R$ be a Noetherian normal domain. Call an $R$-algebra $A$ “locally $\mathbb {A}^{1}$ in codimension-one” if $R_P \otimes _R A$ is a polynomial ring in one variable over $R_P$ for every height-one prime ideal $P$ in $R$. We shall describe a general structure for any faithfully flat $R$-algebra $A$ which is locally $\mathbb {A}^{1}$ in codimension-one and deduce results giving sufficient conditions for such an $R$-algebra to be a locally polynomial algebra. We also give a recipe for constructing $R$-algebras which are locally $\mathbb {A}^{1}$ in codimension-one. When $R$ is a normal affine spot (i.e., a normal local domain obtained by a localisation of an affine domain), we give criteria for a faithfully flat $R$-algebra $A$, which is locally $\mathbb {A}^{1}$ in codimension-one, to be Krull and a further condition for $A$ to be Noetherian. The results are used to construct intricate examples of faithfully flat $R$-algebras locally $\mathbb {A}^{1}$ in codimension-one which are Noetherian normal but not finitely generated.