Ojanguren, M. ; Parimala, R. ; Sridharan, R. ; Suresh, V. (1999) Witt groups of the punctured spectrum of a 3dimensional regular local ring and a purity theorem Journal of the London Mathematical Society, 59 (2). pp. 521540. ISSN 00246107

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Official URL: http://jlms.oxfordjournals.org/content/59/2/521.ab...
Related URL: http://dx.doi.org/10.1112/S0024610799007103
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 Spec^{1}(A) be the set of prime ideals of A of height 1. For PεSpec^{1}(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 socalled 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 BargeSansucVogel 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.
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