Higher Chow groups with modulus and relative Milnor K-theory

TL;DR

This paper establishes an explicit cycle map linking motivic complexes and relative Milnor K-theory for smooth varieties with divisors, proving an isomorphism in cohomology that generalizes known cases and demonstrating a Zariski descent property.

Contribution

It constructs a cycle map for pairs (X,D) and proves it induces an isomorphism in motivic cohomology, extending classical results to divisors with normal crossings.

Findings

Established an explicit cycle map from motivic complexes to relative Milnor K-sheaves.

Proved the cycle map induces an isomorphism in motivic cohomology for certain degrees.

Demonstrated Zariski descent for motivic cohomology of pairs involving affine space.

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 r-th Nisnevich motivic complex of the pair (X,D) to a shift of the r-th relative Milnor K-sheaf of (X,D). We show that this map induces an isomorphism for all i greater or equal the dimension of X between the motivic Nisnevich cohomology of (X,D) in bidegree (i+r,r) and the i-th Nisnevich cohomology of the r-th relative Minor K-sheaf of (X,D). 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^1_k, (m+1){0}).