A generalized Vaserstein symbol

TL;DR

This paper introduces a generalized Vaserstein symbol for projective modules over commutative rings, establishing conditions for its bijectivity and demonstrating its isomorphism in specific regular ring cases.

Contribution

It defines a new generalized Vaserstein symbol on epimorphism orbit spaces and provides criteria for its surjectivity, injectivity, and isomorphism in certain algebraic contexts.

Findings

The generalized Vaserstein symbol maps into the elementary symplectic Witt group.

Criteria for the symbol's surjectivity and injectivity are established.

The symbol is an isomorphism for regular Noetherian rings of dimension 2 or certain 3-dimensional affine algebras.

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 $P_0 \oplus R \rightarrow R$ 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 $c.d.(k) \leq 1$ and $6 \in k^{\times}$.