The low dimensional homology of projective linear group of rank two

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