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The third homology of projective special linear group of degree two

Behrooz Mirzaii, Elvis Torres Pérez

TL;DR

This work determines the third homology group $H_3({\rm PSL}_2(A),\mathbb{Z})$ for suitable rings $A$ by developing a projective refined Bloch-Wigner exact sequence that links $H_3$ to refined scissors congruence groups $\mathcal{RP}_1(A)$ and refined Bloch groups $\mathcal{RB}(A)$. A central tool is a first-quadrant spectral sequence arising from a double complex tied to $\mathcal{PB}(A)$ and $\mathcal{PT}(A)$, which isolates low-degree terms and identifies them with $\mathcal{RP}_1'(A)$ and $\widetilde{\mu}_4(A)$, among others. Under hypotheses such as $A$ being a ${\rm GE}_2$-ring with appropriate residue-field conditions, the authors prove exact sequences of the form $0 \rightarrow {\rm Tor}_1^{\mathbb{Z}}(\widetilde{\mu}(A),\widetilde{\mu}(A)) \rightarrow H_3({\rm PSL}_2(A),\mathbb{Z}) \rightarrow \mathcal{RB}'(A) \rightarrow 0$, and extend these connections to the classical Bloch group via the algebraic closure of the quotient field. They also develop a refined description using monomial subgroups, yielding $H_3({\rm PSL}_2(A),{\rm PSM}_2(A);\mathbb{Z}) \cong \mathcal{RP}_1(A)/\mathbb{Z}[\mathcal{G}_A]\psi_1(-1)$ and an explicit filtration linking $H_3({\rm PSL}_2(A),\mathbb{Z})$ to $\mathcal{RB}(A)$ modulo distinguished torsion, with broad applicability to fields, local rings, finite fields, and number-theoretic rings.

Abstract

In this paper we investigate the third homology of the projective special linear group ${\rm PSL}_2(A)$. As a result of our investigation we prove a projective refined Bloch-Wigner exact sequence over certain class of rings. The projective Bloch-Wigner exact sequence over an algebraically closed field of characteristic zero is a classical result and has many application in algebra, number theory and geometry.

The third homology of projective special linear group of degree two

TL;DR

This work determines the third homology group for suitable rings by developing a projective refined Bloch-Wigner exact sequence that links to refined scissors congruence groups and refined Bloch groups . A central tool is a first-quadrant spectral sequence arising from a double complex tied to and , which isolates low-degree terms and identifies them with and , among others. Under hypotheses such as being a -ring with appropriate residue-field conditions, the authors prove exact sequences of the form , and extend these connections to the classical Bloch group via the algebraic closure of the quotient field. They also develop a refined description using monomial subgroups, yielding and an explicit filtration linking to modulo distinguished torsion, with broad applicability to fields, local rings, finite fields, and number-theoretic rings.

Abstract

In this paper we investigate the third homology of the projective special linear group . As a result of our investigation we prove a projective refined Bloch-Wigner exact sequence over certain class of rings. The projective Bloch-Wigner exact sequence over an algebraically closed field of characteristic zero is a classical result and has many application in algebra, number theory and geometry.
Paper Structure (4 sections, 24 theorems, 228 equations)

This paper contains 4 sections, 24 theorems, 228 equations.

Table of Contents

  1. The refined scissors congruence group
  2. The homology of projective special linear group of degree 2
  3. A projective refined Bloch-Wigner exact sequence
  4. Homological description of the refined scissors congruence group

Key Result

Proposition 1.2

(i) $H_0(X_\bullet(A^2))\overset{\bar{\epsilon}}{\simeq} \mathbb{Z}$ if and only if $A$ is a ${\rm GE}_2$-ring. (ii) $H_1(X_\bullet(A^2))=0$ if and only if ${\rm K}_2(2,A)/{\rm C}(2, A)$ is a perfect group.

Theorems & Definitions (60)

  • Definition 1.1
  • Proposition 1.2: Hutchinson hutchinson2022
  • proof
  • Example 1.3
  • Definition 1.4
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Definition 1.7
  • Remark 1.8
  • ...and 50 more