Bloch Groups of Rings

TL;DR

This paper introduces a generalized definition of Bloch groups for commutative rings, linking them to algebraic K-theory and homology, and computes these groups for specific rings like finite fields and integers.

Contribution

It extends the concept of Bloch groups to general rings, aligning with classical definitions in certain cases, and explores their relation to algebraic K-theory and homology.

Findings

Defined Bloch groups for general rings consistent with classical cases

Connected Bloch groups to third homology of SL_2 and K_3

Computed Bloch groups for specific rings like finite fields and integers

Abstract

We give a definition of (refined) Bloch groups of general commutative rings which agrees with the standard definition in the case of local rings whose residue field has at least $4$ elements. Under appropriate conditions on a ring $A$, satisfied by any field or local ring, these groups are closely related to third homology of $\mathrm{SL}_2(A)$ and to indecomposable $K_3$ of $A$. We analyze these conditions. We calculate the Bloch groups of $\mathbb{F}_2,\mathbb{F}_3,\mathbb{Z}$ and $\mathbb{Z}[\frac{1}{2}]$.