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WITT GROUPS OF THE PUNCTURED SPECTRUM OF A 3-DIMENSIONAL REGULAR LOCAL RING AND A PURITY THEOREM

Published online by Cambridge University Press:  01 April 1999

M. OJANGUREN ,
R. PARIMALA ,
R. SRIDHARAN  and
V. SURESH
M. OJANGUREN
Affiliation:
Université de Lausanne, Section de Mathématiques, CH-1015 Lausanne-Dorigny, Switzerland
R. PARIMALA
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
R. SRIDHARAN
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
V. SURESH
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India
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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[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 AP and k(P) = AP/PAP. For this choice of a parameter πP, 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 πP, its kernel and cokernel do not. We have a homomorphism

formula here

A part of the so-called Gersten conjecture is the following question on ‘purity’. Is the sequence

formula here

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.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 59 , Issue 2 , April 1999 , pp. 521 - 540
Copyright
The London Mathematical Society 1999

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