If \(R\) is a smooth semi-local algebra of geometric type over an infinite field, it is proved that the Milnor \(K\)-group \(K^M_n(R)\) surjects onto Bloch’s higher Chow group \(CH^n(R,n)\) for all \(n\geq 0\). The proof actually gives an algorithm to change an admissible cycle into one that is clearly in the image. The arguments are quite technical. For some PID’s it is shown that the surjection is an isomorphism. All this generalizes earlier results of A. A. Suslin and Yu. P. Nesterenko [Izv. Akad. Nauk SSSR, Ser. Mat. 53, 121-146 (1989; Zbl 0668.18011)] and B. Totaro [K-Theory 6, 177-189 (1992; Zbl 0776.19003)] for the case where \(R\) is a field. One also gets a new proof of universal exactness of a Gersten resolution for the Milnor \(K\)-sheaf.