The third homology of projective special linear group of degree two

Abstract: In this paper we investigate the third homology of the projective special linear group ${\rm PSL}_2(A)$. 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.