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[rarrhk ]K,
where W( ) denotes the Witt group of quadratic forms. If
dim A[les ]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
k(P) = AP/PAP.
For this choice of a parameter πP, one has the second residue homomorphism
[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
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)[les ]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.