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Bloch Groups of Rings

Rodrigo Cuitun Coronado, Kevin Hutchinson

TL;DR

This work generalizes the Bloch-group framework to all commutative rings by defining refined and pre-Bloch groups via a truncation of a clique complex on the projective line, tying these groups to $H_{3}(SL_{2}(A))$ and indecomposable $K_{3}$ under broad acyclicity conditions. The authors develop a hyperhomology spectral sequence $E(G,L^{\tau})$ whose $E^{\infty}_{0,3}$-term yields the (refined) Bloch groups and establish when these compute ordinary group homology, particularly for local rings. They introduce universal algebraic identities among constants $D_{A}$ and $C_{A}$ and cocycles $\\psi_{1}(u),\\psi_{2}(u)$, culminating in a fundamental identity $2\\langle\\langle u\\rangle\\rangle C_{A}=\\psi_{2}(u)-\\psi_{1}(u)$ that underpins calculations of $H_{3}(SL_{2}(A))$ and $K^{ind}_{3}(A)$. The paper then computes the refined Bloch groups for $\mathbb{F}_{2},\mathbb{F}_{3},\mathbb{Z},\mathbb{Z}[\tfrac{1}{2}]$, confirming expected relationships with indecomposable $K_{3}$ of these rings/fields and illustrating the interplay between refined, classical, and abelianized structures. These results pave the way for further analysis of local rings with small residue fields and connections to $K$-theory via spectral sequences.

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

Bloch Groups of Rings

TL;DR

This work generalizes the Bloch-group framework to all commutative rings by defining refined and pre-Bloch groups via a truncation of a clique complex on the projective line, tying these groups to and indecomposable under broad acyclicity conditions. The authors develop a hyperhomology spectral sequence whose -term yields the (refined) Bloch groups and establish when these compute ordinary group homology, particularly for local rings. They introduce universal algebraic identities among constants and and cocycles , culminating in a fundamental identity that underpins calculations of and . The paper then computes the refined Bloch groups for , confirming expected relationships with indecomposable of these rings/fields and illustrating the interplay between refined, classical, and abelianized structures. These results pave the way for further analysis of local rings with small residue fields and connections to -theory via spectral sequences.

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 elements. Under appropriate conditions on a ring , satisfied by any field or local ring, these groups are closely related to third homology of and to indecomposable of . We analyze these conditions. We calculate the Bloch groups of and .
Paper Structure (24 sections, 101 theorems, 153 equations)

This paper contains 24 sections, 101 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. The complexes $L_\bullet$ and $L^{\tau}_\bullet$ and pre-Bloch groups
  4. Bloch groups and the hyperhomology spectral sequence $E(G,L^{\tau})$
  5. Connection with $\mathrm{H}_{3}\left( \operatorname{SL}_{2}\left(A\right),\Bbb{Z} \right)$
  6. Classical Bloch group and classical refined Bloch group
  7. Review: The Bloch group and indecomposable $K_3$ of fields
  8. Special elements and algebraic identities in pre-Bloch groups
  9. The constant $D_{A}$
  10. The constant $D_{A}$ as an element of $\mathrm{H}_{3}\left( \operatorname{SL}_{2}\left(A\right),\Bbb{Z} \right)$
  11. The elements $\psi_{1}\left( u\right)$
  12. The elements $\psi_{2}\left( u\right)$ and the key identity
  13. The constant $C_{A}$
  14. The map $\lambda_1$ again
  15. The modules ${\mathcal{K}}^{\hbox{\tiny $(i)$}}_{A}$
  16. ...and 9 more sections

Key Result

Lemma 2.1

Theorems & Definitions (198)

  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Remark 2.9
  • ...and 188 more