Bloch Groups of Rings

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 $\mathbb{Z}[\frac{1}{2}]$.

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