A note on relative Vaserstein symbol

TL;DR

This paper explores the properties of the relative Vaserstein symbol in algebraic K-theory, establishing conditions for its injectivity in certain algebraic structures and providing a counterexample in the singular case.

Contribution

It details the construction of the relative Witt group and proves the injectivity of the relative Vaserstein symbol under specific conditions, including for singular 3-dimensional algebras.

Findings

Injectivity of the Vaserstein symbol is established for affine non-singular 3-folds over perfect C_1-fields.

An example of a singular 3-dimensional algebra with an injective Vaserstein symbol is provided.

The work extends understanding of the Vaserstein symbol beyond smooth cases.

Abstract

In an unpublished work of Fasel-Rao-Swan the notion of the relative Witt group $W_E(R,I)$ is defined. In this article we will give the details of this construction. Then we studied the injectivity of the relative Vaserstein symbol $V_{R,I}: Um_3(R,I)/E_3(R,I)\rightarrow W_E(R,I)$. We established injectivity of this symbol if $R$ is an affine non-singular algebra of dimension $3$ over a perfect $C_1$-field and $I$ is a local complete intersection ideal of $R$. It is believed that for a $3$-dimensional affine algebra non-singularity is not necessary for establishing injectivity of the Vaserstein symbol . At the end of the article we will give an example of a singular $3$-dimensional algebra over a perfect $C_1$-field for which the Vaserstein symbol is injective.