Quadratic spaces with trivial Arf invariant
type: Item
instance of: scholarly article
title: Quadratic spaces with trivial Arf invariant
zbMATH Open document ID: 0625.16003
DOI: 10.1016/0021-8693(85)90160-7
author: Antonio Paques
author: M.-A. Knus
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: In an effort to unify the constructions of \textit{S. Parimala} [Bull. Am. Math. Soc. 82, 962-964 (1976; Zbl 0353.15035)] of non-extended rank 4 quadratic spaces and the constructions of \textit{M.-A. Knus} and \textit{M. Ojanguren} [Proc. Am. Math. Soc. 66, 223-226 (1977; Zbl 0374.16011)] of non-extended rank 3 quadratic spaces over polynomial rings, a classification of quadratic spaces of ranks 3 and 4 of trivial discriminant over domains R (in which 2 is invertible) in terms of projective modules over Azumaya R-algebras of rank 4 was given by \textit{M.-A. Knus, \textit{M. Ojanguren}} and \textit{R. Sridharan} [J. Reine Angew. Math. 303/304, 231-248 (1978; Zbl 0385.16001)]. The aim of this paper is to get rid of the assumption that R is a domain in which 2 is invertible and obtain a classification of rank 4 quadratic spaces of trivial Arf invariant over an arbitrary commutative ring R in terms of modules over rank 4 Azumaya R-algebras. The main step in the proof is to show that a quadratic space of rank 4 has trivial Arf invariant if and only if its Clifford algebra has an idempotent ``defining its gradation''.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 4014921
zbMATH Keywords: rank 4 quadratic spaces
zbMATH Keywords: projective modules
zbMATH Keywords: Azumaya R-algebras
zbMATH Keywords: trivial Arf invariant
zbMATH Keywords: Clifford algebra
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90160-7
OpenAlex ID: W2076496295
cites work: Quadratic forms over semilocal rings
cites work: Q5562621
cites work: Clifford Algebras and Spinor Norms Over a Commutative Ring
cites work: Q3285650
cites work: Forms over rings with involution
cites work: Théorie de la descente et algèbres d'Azumaya
cites work: Q4165471
cites work: Q3909169
cites work: Q3876003
cites work: Sur les algèbres de Clifford
cites work: Q5665299
cites work: Projective modules and Hermitian matrices
cites work: Quadratic and Hermitian forms in additive and Abelian categories
cites work: Sur les deux premiers invariants d'une forme quadratique
Recommended article: Indecomposable quadratic spaces over the affine plane
Recommended article: On rank 4 quadratic spaces with given Arf and Witt invariants
Recommended article: Espaces adéliques quadratiques
Recommended article: Quadratic spaces
Recommended article: Q3677860
Recommended article: A purity theorem for quadratic spaces
Recommended article: Almost multi-quadratic mappings in non-Archimedean spaces
Recommended article: A generalization of the space of complete quadrics
description: scientific article; zbMATH DE number 4014921