Milnor K-theory of rings, higher Chow groups and applications

Abstract

If R is a smooth semi-local algebra of geometric type over an infinite field, we prove that the Milnor K-group K M n (R) surjects onto the higher Chow group CH n (R , n) for all n≥0. Our proof shows moreover that there is an algorithmic way to represent any admissible cycle in CH n (R , n) modulo equivalence as a linear combination of “symbolic elements” defined as graphs of units in R. As a byproduct we get a new and entirely geometric proof of results of Gabber, Kato and Rost, related to the Gersten resolution for the Milnor K-sheaf. Furthermore it is also shown that in the semi-local PID case we have, under some mild assumptions, an isomorphism. Some applications are also given.