On the third invariant of a quadratic form
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.
scientific article; zbMATH DE number 790462