  In this article we study the low dimensional homology of the projective
linear group $\textrm{PGL}_2(A)$ over a $\textrm{GE}_2$-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
$H_3(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq
K_3^{\textrm{ind}}(A)\left[\frac{1}{2}\right]$.
