A note on seminormal rings and \({\mathbb{A}}^ 1\)-fibrations (Q1081642)

Let R be a Noetherian ring. An \({\mathbb{A}}^ 1\)-fibration over R is an R- algebra A of finite type such that \(k(p)\otimes_ RA\simeq k(p)[X]\) for all \(p\in Spec R.\) This paper proves the following two results. Theorem 1: Let \(R\subset S\) be a finite extension of reduced Noetherian rings. Then R is seminormal in S if and only if for any flat \({\mathbb{A}}^ 1\)-fibration A over R, if \(S\otimes_ R A\) is a symmetric algebra of some \(P\in Pic(S)\), then A is a symmetric algebra of some \(Q\in Pic(R)\). - Theorem 2: Let R be Noetherian and let \(R_{red}\) have finite normalization. Then \(R_{red}\) is seminormal if and only if all flat \({\mathbb{A}}^ 1\)- fibrations over R are symmetric algebras.