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 P∈Spec1(A), let πP be a parameter of the discrete valuation ring AP and k(P) = AP/PAP. For this choice of a parameter πP, one has the second residue homomorphism ∂P:W(K)→W(k(P)) [9, p. 209]. Though the homomorphism ∂P depends on the choice of the parameter πP, its kernel and cokernel do not. We have a homomorphism ∂ = ( ∂ P ) : W ( K ) → ⊕ P ∈ S p e c 1 ( 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 ) → ∂ ⊕ P ∈ S p e c 1 ( 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.