A purity theorem for the Witt group

Abstract

Let A be a regular local ring and K its field of fractions. We denote by W the Witt group functor that classifies quadratic spaces. We say that purity holds for A if W(A) is the intersection of all W(Ap) ⊂ W(K), as p runs over the height-one prime ideals of A. We prove purity for every regular local ring containing a field of characteristic ≠ 2. The question of purity and of the injectivity of W(A) into W(K) for arbitrary regular local rings is still open. © Elsevier, Paris.