The low dimensional homology of projective linear group of rank two

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 $|A/\mathcal{m}_A|\neq 2,3,4,8$.