Pfaffians and quadratic forms
type: Item
instance of: scholarly article
title: Pfaffians and quadratic forms
zbMATH Open document ID: 0656.10015
DOI: 10.1016/0001-8708(88)90065-5
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(R\) be a commutative ring and let \(A\) be an Azumaya \(R\)-algebra with an involution of symplectic type. The restriction of the reduced norm \(n_A\) to the \(R\)-module \(A_+\) of even elements of \(A\) has a square root \(n_+\) (this generalizes the Pfaffian). If \(\operatorname{rank}_R A=16\), then \((A_+,n_+)\) is a quadratic space of rank 6 with trivial Arf invariant and \(n_+\) represents 1. One of the principal results of the paper is the following classification theorem. Given any quadratic space \((M,q)\) over \(R\) of rank 6 with trivial Arf invariant and representing 1, there is an Azumaya \(R\)-algebra \(A\) of rank 16 with a symplectic involution and such that \((M,q)\) is isometric with \((A_+,n_+)\). A similar theory is developed for quadratic modules of rank 5. The case of rank 4 was treated earlier by the author and \textit{A. Paques} [J. Algebra 93, 267--291 (1985; Zbl 0625.16003)] and by the author, \textit{M. Ojanguren} and \textit{R. Sridharan} [J. Reine Angew. Math. 303/304, 231--248 (1978; Zbl 0385.16001)]. A section of the paper is devoted to computing the spin groups of forms of rank 6 and rank 5.
reviewed by: Kazimierz Szymiczek
Mathematics Subject Classification ID: 11E88
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 4071002
zbMATH Keywords: quadratic forms over commutative rings
zbMATH Keywords: Azumaya algebras
zbMATH Keywords: quadratic space of rank 6
zbMATH Keywords: trivial Arf invariant
zbMATH Keywords: symplectic involution
zbMATH Keywords: quadratic modules of rank 5
author: M.-A. Knus
MaRDI profile type: Publication
full work available at URL: 0001-8708(88)90065-5
OpenAlex ID: W2012771910
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description: scientific article; zbMATH DE number 4071002