Abstract
In this paper we prove a formula, conjectured by Bloch and Srinivas [S2], which describes the Chow group of zero cycles of a normal quasi-projective surface $X$ over a field, as an inverse limit of relative Chow groups of a desingularization $\tilde X$ relative to multiples of the exceptional divisor. We then give several applications of this result — a relative version of the famous Bloch Conjecture on $0$-cycles, the triviality of the Chow group of $0$-cycles for any $2$-dimensional normal graded $\overline{\mathbb{Q}}$-algebra (analogue of the Bloch-Beilinson Conjecture), and the analogue of the Roitman theorem for torsion $0$-cycles in characteristic $p>0$ for normal varieties (including the case of $p$-torsion).
