The Milnor-Chow homomorphism revisited

Abstract

The aim of this note is to give a simplified proof of the surjectivity of the natural Milnor-Chow homomorphism $\rho :K_n^M(A)\to C{H}^n(A,n)$ between Milnor $K$-theory and higher Chow groups for essentially smooth (semi-)local $k$-algebras $A$ with $k$ infinite. 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.

8 pages