HIGHER CHOW GROUPS WITH MODULUS AND RELATIVE MILNOR K-THEORY

Let X be a smooth variety over a field k and D an effective divisor whose support has simple normal crossings. We construct an explicit cycle map from the Nisnevich motivic complex Z(r)(X vertical bar D, Nis) of the pair (X, D) to a shift of the relative Milnor K-sheaf K-r,X vertical bar D,Nis(M) of (X, D). We show that this map induces an isomorphism H-M,Nis(i+r) (X vertical bar D, Z(r)) congruent to H-i (X-Nis, K-r,K-X vertical bar D,(M)(Nis)), for all i >= dim X. This generalizes the well-known isomorphism in the case D = 0. We use this to prove a certain Zariski descent property for the motivic cohomology of the pair (A(k)(1), (m + 1){0}).