On algebras which are locally 𝔸¹ in codimension-one

Abstract

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