A note on relative Vaserstein symbol

Abstract

In an unpublished work of Fasel–Rao–Swan the notion of the relative Witt group [Formula: see text] is defined. In this paper, we will give the details of this construction. Then we study the injectivity of the relative Vaserstein symbol [Formula: see text]. We establish injectivity of this symbol if [Formula: see text] is a non-singular affine algebra of dimension [Formula: see text] over a perfect [Formula: see text]-field and [Formula: see text] is a local complete intersection ideal of [Formula: see text]. It is natural to ask whether the Vaserstein symbol is injective for a singular [Formula: see text]-dimensional affine algebra. At the end of the paper, we will give an example of a singular [Formula: see text]-dimensional algebra over a perfect [Formula: see text]-field for which the Vaserstein symbol is injective.