A local-global principle for symplectic $\mathrm K_2$

TL;DR

This paper establishes a local-global principle for the symplectic K_2 group, showing that an element is trivial if and only if it becomes trivial under all maximal localizations, thus linking local properties to global triviality.

Contribution

It introduces a new local-global criterion for symplectic K_2, connecting triviality of elements to their behavior under maximal localization homomorphisms.

Findings

Proves the equivalence of triviality and local triviality under maximal localizations.

Provides a new tool for analyzing symplectic K_2 elements.

Enhances understanding of the structure of symplectic Steinberg groups.

Abstract

We prove that an element of the symplectic Steinberg group is trivial if and only if its image under any maximal localisation homomorphism is trivial.