The Milnor-Chow homomorphism revisited

The aim of this note is to give a simplified proof of the surjectivity of the natural Milnor–Chow homomorphism ρ: K_n^M(A) --> CHⁿ(A, n) between Milnor K–theory and higher Chow groups for essentially smooth (semi–)local k–algebras A with infinite residue fields. It implies the exactness of the Gersten resolution for Milnor K–theory at the generic point. Our method uses the Bloch-Levine moving technique and some properties of the Milnor K–theory norm for fields. Furthermore we give new applications.