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 P0 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 P0 R R under the action of the group of elementary automorphisms of P0 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) 1 and 6 k×.

Keywords
generalized Vaserstein symbol
Mathematical Subject Classification 2010
Primary: 13C10, 14F42, 19A13, 19G38
Milestones
Received: 13 April 2018
Revised: 17 April 2019
Accepted: 2 May 2019
Published: 10 January 2020
Authors
Tariq Syed
Fakultät für Mathematik
Universität Duisburg-Essen
Essen
Germany