A generalized Vaserstein symbol

Abstract

Let $R$ be a ring with $2 \in R^{\times}$. Then the usual Vaserstein symbol is a map from the orbit space of unimodular rows of length $3$ under the action of the group $E_3 (R)$ to the elementary symplectic Witt group. Now let $P_0$ be a projective module of rank $2$ with trivial determinant. Then we provide a generalized symbol map which is defined on the orbit space of the set of epimorphisms $P_0 \oplus R \rightarrow R$ under the action of the group of elementary automorphisms of $P_0 \oplus R$. We also generalize results by Vaserstein and Suslin on the surjectivity and injectivity of the Vaserstein symbol. Finally, we use local-global principles for transvection groups in order to deduce that the generalized Vaserstein symbol is an isomorphism if $R$ is a regular Noetherian ring of dimension $2$ or a regular affine algebra of dimension $3$ over a field $k$ with $c.d.(k) \leq 1$ and $6 \in k^{\times}$.