×

Witt rings of fields of quotients of two-dimensional regular local rings. (English) Zbl 0764.11023

Let \(R\) be a regular local ring of Krull dimension two, which contains \({1\over 2}\). Let \(\overline {R}\) be its field of quotients. The aim of this paper is to show that the following sequence of Witt groups is exact: \[ 0\to W(R)@>j^*>>W(\overline{R}) @>\oplus\partial_ p>> \bigoplus_{p\in P} W(\overline {R/p}) \to W(R/m)\to 0, \] where \(j^*\) is induced by the canonical imbedding \(j: R\to\overline {R}\) and \(P\) is the set of all prime ideals of the ring \(R\) of height 1.

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E08 Quadratic forms over local rings and fields
13H05 Regular local rings
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Bourbaki, N.: Commutative Algebra. Paris: Hermann 1972
[2] Fernández-Carmena, F.: On the Injectivity of the Map of the Witt Group of a Scheme into the Witt Group of its Function Field. Math. Ann.277, 453–468 (1987) · Zbl 0641.14003 · doi:10.1007/BF01458326
[3] Hatt-Arnold, D.: Le problème de Hasse-Minkowski pour les quadriques définies surC((X, Y)), Thèse 2181. Genéve: 1986 · Zbl 0623.10013
[4] Jaworski, P.: Witt rings of fields of formal power series in two variables. Ann. Math. Sil.2(14), 13–29. Katowice 1986 · Zbl 0594.10011
[5] Lam, T.Y.: Algebraic Theory of Quadratic Forms. Reading, Mass.: Benjamin 1973 · Zbl 0259.10019
[6] Milnor, J., Husemoller, D.: Symetric Bilinear Forms. Berlin Heidelberg New York: Springer 1973 · Zbl 0292.10016
[7] Ojanguren, M.: A splitting theorem for quadratic forms. Comment. Math. Helv.57, 145–157 (1982) · Zbl 0487.13005 · doi:10.1007/BF02565852
[8] Pardon, W.: A relation between Witt groups and zero cycles in a regular ring. In: Bak, A. (ed.) Algebraic K-Theory... (Lect. Notes Math., vol. 1046, pp. 261–328) Berlin Heidelberg New York Tokyo: Springer 1984 · Zbl 0531.10024
[9] Pardon, W.: A relation between Witt group of a regular local ring and the Witt groups of its residue class fields. (Preprint) · Zbl 0531.10024
[10] Scharlau, W.: Quadratic and Hermitian Forms. Berlin Heidelberg New York Tokyo: Springer 1985 · Zbl 0584.10010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.