Chow cohomology groups of algebraic surfaces

For a normal algeraic surface we consider its higher Chow groups and Chow cohomology (the latter defined by the author). We study the canonical map from the Chow cohomology to the higher Chow group of the surface by relating it to the canonical map from the higher Chow group to Chow cohomology of the exceptional divisor of a desingularization of the surface.