The third homology of projective special linear group of degree two

In this paper we investigate the third homology of the projective special linear group ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(A)$. As a result of our investigation we prove a projective refined Bloch-Wigner exact sequence over certain class of rings. The projective Bloch-Wigner exact sequence over an algebraically closed field of characteristic zero is a classical result and has many application in algebra, number theory and geometry.