On rank 4 quadratic spaces with given Arf and Witt invariants
type: Item
instance of: scholarly article
title: On rank 4 quadratic spaces with given Arf and Witt invariants
zbMATH Open document ID: 0612.10014
DOI: 10.1007/BF01457069
author: R. Sridharan
author: M.-A. Knus
author: R. Parimala
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164094
review text: Let (M,q) be a rank 4 quadratic space over a commutative ring R, and let \(C=C_ 0+C_ 1\) be the Clifford algebra of (M,q). The center S of the algebra \(C_ 0\) is called the Arf invariant of (M,q). It turns out that \(C_ 0\) is a quaternion algebra over S and \(C_ 1\) is a projective \(C_ 0\)-module of rank one. Let [C]\(\in Br(R)\) be the Brauer class of the algebra C. Thus every quadratic space determines the triple \((C_ 0,C_ 1,[C]).\) The authors show that, for given S every triple (A,P,\({\mathfrak B})\), where A is a quaternion algebra over S, P is a projective A-module of rank one and \({\mathfrak B}\) is an element of the Brauer group of R satisfying some special conditions, is determined by a quadratic space over R of Arf invariant S and Witt invariant \({\mathfrak B}\). Moreover two quadratic spaces with given Arf invariant are equivalent if and only if the corresponding triples are equivalent. The general result is applied to the classification of quadratic spaces over fields and over the real sphere.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 13C99
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 3989442
zbMATH Keywords: Clifford algebra
zbMATH Keywords: quadratic space
zbMATH Keywords: Arf invariant
zbMATH Keywords: Brauer group
zbMATH Keywords: Witt invariant
reviewed by: Mieczysław Kula
MaRDI profile type: Publication
cites work: Q4205543
cites work: Separable algebras over commutative rings
cites work: Forms over rings with involution
cites work: Composition of binary quadratic forms
cites work: Q4165471
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OpenAlex ID: W1995949871
description: scientific article; zbMATH DE number 3989442