The low-dimensional homology of projective linear group of degree two

In this article we study the low-dimensional homology of the projective linear group \mathrm{PGL}_{2}(A) over a commutative ring A . In particular, we prove a Bloch–Wigner type exact sequence over local domains. As application we prove that H_{2}\big(\mathrm{PGL}_{2}(A),\mathbb{Z}\big[\tfrac{1}{2}\big]\big)\simeq\mathrm{K}_{2}(A)\big[\tfrac{1}{2}\big]\quad\text{and}\quad H_{3}\big(\mathrm{PGL}_{2}(A),\mathbb{Z}\big[\tfrac{1}{2}\big]\big)\simeq\mathrm{K}_{3}^{\mathrm{ind}}(A)\big[\tfrac{1}{2}\big], provided |A/\mathfrak{m}_{A}|\neq 2,3,4,8 .