  The aim of this note is to give a simplified proof of the surjectivity of the
natural Milnor-Chow homomorphism $\rho: K^M_n(A) \to CH^n(A,n)$ 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.
