On rank 4 quadratic spaces with given Arf and Witt invariants
Let (M,q) be a rank 4 quadratic space over a commutative ring R, and let \(C=C_ 0+C_ 1\) be the Clifford algebra of (M,q). The center S of the algebra \(C_ 0\) is called the Arf invariant of (M,q). It turns out that \(C_ 0\) is a quaternion algebra over S and \(C_ 1\) is a projective \(C_ 0\)-module of rank one. Let [C]\(\in Br(R)\) be the Brauer class of the algebra C. Thus every quadratic space determines the triple \((C_ 0,C_ 1,[C]).\) The authors show that, for given S every triple (A,P,\({\mathfrak B})\), where A is a quaternion algebra over S, P is a projective A-module of rank one and \({\mathfrak B}\) is an element of the Brauer group of R satisfying some special conditions, is determined by a quadratic space over R of Arf invariant S and Witt invariant \({\mathfrak B}\). Moreover two quadratic spaces with given Arf invariant are equivalent if and only if the corresponding triples are equivalent. The general result is applied to the classification of quadratic spaces over fields and over the real sphere.
scientific article; zbMATH DE number 3989442