The structure of a Laurent polynomial fibration in n variables

Abstract

Bass, Connell and Wright have proved that any finitely presented locally polynomial algebra in n variables over an integral domain R is isomorphic to the symmetric algebra of a finitely generated projective R-module of rank n. In this paper we prove a corresponding structure theorem for a ring A which is a locally Lau�rent polynomial algebra in n variables over an integral domain R, viz., we show that A is isomorphic to an R-algebra of the form (Sym_R (Q ))[I ⁻¹], where Q is a direct sum of n finitely generated projective R-modules of rank one and I is a suitable invertible ideal of the symmetric algebra Sym_R (Q ). Further, we show that any faithfully flat algebra over a Noetherian normal domain R, whose generic and codimension-one fibres are Laurent polynomial algebras in n variables, is a locally Laurent polynomial algebra in n variables over R.