Witt groups of the punctured spectrum of a 3-dimensional regular local ring and a purity theorem

Abstract

Let A be a regular local ring with quotient field K. Assume that 2 is invertible in A. Let W(A)→W(K) be the homomorphism induced by the inclusion A→K, where W( ) denotes the Witt group of quadratic forms. If dim A≤ 4, it is known that this map is injective [6,7]. A natural question is to characterize the image of W(A) in W(K). Let Spec¹(A) be the set of prime ideals of A of height 1. For PεSpec¹(A), let π_P be a parameter of the discrete valuation ring A_P and k(P)=A_P/PA_P. For this choice of a parameter π_P, one has the second residue homomorphism ∂_P:W(K)→W(k(P)) [9, p. 209]. Though the homomorphism π_P depends on the choice of the parameter π_P, its kernel and cokernel do not. We have a homomorphism ∂=∂_P:W(K)→⊕_PεSpec¹(A)W(k(P)) A part of the so-called Gersten conjecture is the following question on 'purity'. Is the sequence W(A)→W(K)→^∂⊕_PεSpec¹(A)W(k(P)) exact? This question has an affirmative answer for dim(A)≤2 [1; 3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogel on the question of purity. However, in the literature, there is no proof for purity even for dim(A)=3. One of the consequences of the main result of this paper is an affirmative answer to the purity question for dim(A)=3. We briefly outline our main result.