Quadratic spaces with trivial Arf invariant
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''.
scientific article; zbMATH DE number 4014921
scientific article