A generalized Vaserstein symbol

Abstract

Let $R$ be a commutative ring. For any projective $R$-module $P_0$ of constant rank $2$ with a trivialization of its determinant, we define a generalized Vaserstein symbol on the orbit space of the set of epimorphisms $P_0\oplus R\to R$ under the action of the group of elementary automorphisms of $P_0\oplus R$, which maps into the elementary symplectic Witt group. We give criteria for the surjectivity and injectivity of the generalized Vaserstein symbol and deduce that it is an isomorphism if $R$ is a regular Noetherian ring of dimension $2$ or a regular affine algebra of dimension $3$ over a perfect field $k$ with $c.d.\left(k\right)\le 1$ and $6\in k^{\times }$.