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 . 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 P ∈Spec 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 ∂ P depends on the choice of the parameter π P , 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.