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 \mathbb{Z}(r)_{X\vert D,\mathrm {Nis}} of the pair (X,D) to a shift of the relative Milnor K-sheaf \mathcal {K}^M_{r,X\vert D,\mathrm {Nis}} of (X,D). We show that this map induces an isomorphism H^{i+r}_{\mathcal {M},\mathrm {Nis}}(X\vert D,\mathbb{Z}(r))\cong H^i(X_{\mathrm {Nis}}, \mathcal {K}^M_{r, X\vert D,\mathrm {Nis}}), for all i\ge \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 (\mathbb{A}^1_k, (m+1)\{0\}).