#### Vol. 4, No. 4, 2019

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A generalized Vaserstein symbol

### Tariq Syed

Vol. 4 (2019), No. 4, 671–706
##### 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 $6\in {k}^{×}$.

##### Keywords
generalized Vaserstein symbol
##### Mathematical Subject Classification 2010
Primary: 13C10, 14F42, 19A13, 19G38