On the third invariant of a quadratic form
type: Item
instance of: scholarly article
title: On the third invariant of a quadratic form
zbMATH Open document ID: 0837.11022
DOI: 10.1007/BF00961666
published in: \(K\)-Theory
publication date: 1996-05-24T00:00:00Z
review text: In [J. Algebra 44, 434-456 (1977; Zbl 0354.16012)] \textit{C. Giffen} defined for any symmetric elementary matrix \(\alpha\) with coefficients in a commutative ring an invariant \(s_2 (\alpha)\), with values in a factor group of \(K_2 (A)\). This invariant is the obstruction to lift \(\alpha\) to a symmetric element of the Steinberg group of \(A\). Later \textit{C. Giffen} defined an invariant \(s_n (\alpha)\) for any \(n\) using homotopical techniques [Bull. Am. Math. Soc. 83, 1303-1305 (1977; Zbl 0373.18009)]. In this paper the authors define \(s_3 (\alpha)\) in a purely algebraic way. They first use cohomology of groups to define cohomological invariants \(s_n\). Applying the construction to the setting of quadratic forms, they show that \(s_1\) is a refinement of the signed discriminant. The fact that \(s_2\) is related to the Clifford invariant follows from results of \textit{M. Ojanguren}, \textit{R. Parimala} and \textit{R. Sridharan} [\(K\)-Theory 7, No. 6, 501-515 (1993; Zbl 0802.19004)]. Finally they give their nice alternative definition of \(s_3\) and describe relations with the Arason invariant for quadratic forms.
reviewed by: M.-A. Knus
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 19C99
Mathematics Subject Classification ID: 19D99
zbMATH DE Number: 790462
zbMATH Keywords: Giffen invariant
zbMATH Keywords: invariants of quadratic forms
zbMATH Keywords: higher \(K\)-theory
zbMATH Keywords: cohomological invariants
zbMATH Keywords: Arason invariant
author: Jean Barge
author: Manuel Ojanguren
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00961666
OpenAlex ID: W606798040
description: scientific article; zbMATH DE number 790462