Witt Groups of the Punctured Spectrum of a 3-Dimensional Regular Local Ring and a Purity Theorem

Let A be a regular local ring with quotient field K. Assumethat 2 is invertible in A. Let W(A)W(K) be the homomorphisminduced by the inclusion AK, where W( ) denotes the Witt groupof quadratic forms. If dim A4, it is known that this map isinjective 6, 7]. A natural question is to characterize theimage of W(A) in W(K). Let Spec¹(A) be the set of prime idealsof A of height 1. For PSpec¹(A), let _P be a parameter of thediscrete valuation ring A_P and k(P) = A_P/PA_P. For this choiceof a parameter _P, one has the second residue homomorphism _P:W(K)W(k(P))9, p. 209]. Though the homomorphism _P depends on the choiceof the parameter _P, its kernel and cokernel do not. We havea homomorphism A part of the so-called Gersten conjecture is the followingquestion on ‘purity’. Is the sequence exact? This question has an affirmative answer for dim(A)2 1;3, p. 277]. There have been speculations by Pardon and Barge-Sansuc-Vogelon the question of purity. However, in the literature, thereis no proof for purity even for dim(A) = 3. One of the consequencesof the main result of this paper is an affirmative answer tothe purity question for dim(A) = 3. We briefly outline our main result.