Pfaffians and quadratic forms
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.
scientific article; zbMATH DE number 4071002