Higher Chow groups with modulus and relative Milnor $K$-theory

Abstract:

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