A note on relative Vaserstein symbol.

In an unpublished work of Fasel–Rao–Swan the notion of the relative Witt group W E (R , I) is defined. In this paper, we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol V R , I : U m 3 (R , I) / E 3 (R , I) → W E (R , I). We establish injectivity of this symbol if R is a non-singular affine algebra of dimension 3 over a perfect C 1 -field and I is a local complete intersection ideal of R. It is natural to ask whether the Vaserstein symbol is injective for a singular 3-dimensional affine algebra. At the end of the paper, we will give an example of a singular 3-dimensional algebra over a perfect C 1 -field for which the Vaserstein symbol is injective. [ABSTRACT FROM AUTHOR]