The low dimensional homology of projective linear group of rank two

TL;DR

This paper investigates the low-dimensional homology of the projective linear group PGL_2 over a commutative ring, establishing exact sequences and isomorphisms with algebraic K-theory groups under certain conditions.

Contribution

It proves a Bloch-Wigner type exact sequence for PGL_2 over local domains and links homology groups to algebraic K-theory groups, providing new insights into their structure.

Findings

H_2(PGL_2(A), Z[1/2]) is isomorphic to K_2(A)[1/2]

H_3(PGL_2(A), Z[1/2]) is isomorphic to K_3^{ind}(A)[1/2]

Established a Bloch-Wigner type exact sequence for local domains.

Abstract

In this article we study the low dimensional homology of the projective linear group $\textrm{PGL}_2(A)$ over a commutative ring $A$. In particular, we prove a Bloch-Wigner type exact sequence over local domains. As applications we prove that $H_2(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq K_2(A)\left[\frac{1}{2}\right]$ and $H_3(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq K_3^{\textrm{ind}}(A)\left[\frac{1}{2}\right]$ provided $|A/\mathcal{m}_A|\neq 2,3,4,8$.