© 1999 by London Mathematical Society
© The London Mathematical Society
Witt Groups of the Punctured Spectrum of a 3-Dimensional Regular Local Ring and a Purity Theorem
Université de Lausanne, Section de Mathématiques CH-1015 Lausanne-Dorigny, Switzerland
School of Mathematics, Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005, India
Received 15 August 1996.
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 
P be a parameter of the discrete valuation ring AP and k(P) = AP/PAP. For this choice of 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 choice of the parameter P, its kernel and cokernel do not. We have a homomorphism
 |
A part of the so-called Gersten conjecture is the following question 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-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.