K-theory and the Enriched Tits Building

TL;DR

This paper introduces simplicial complexes linked to rings that facilitate understanding algebraic K-theory, offering new insights into Suslin's theorem and potential generalizations.

Contribution

It defines new complexes associated with rings that connect to algebraic K-theory and provides a novel perspective on Suslin's theorem.

Findings

Exact sequence involving K-groups and the symbol map

New insights into Suslin's theorem on the Bloch group

Potential pathways for generalizations of existing theorems

Abstract

Motivated by the splitting principle, we define certain simplicial complexes associated to an associative ring $A$, which have an action of the general linear group $GL(A)$. This leads to an exact sequence, involving Quillen's algebraic K-groups of $A$ and the symbol map. Computations in low degrees lead to another view on Suslin's theorem on the Bloch group, and perhaps show a way towards possible generalizations.