Motivic spectral sequence for relative homotopy K-theory

TL;DR

This paper develops a motivic spectral sequence for relative homotopy K-theory of schemes, linking it to cdh-hypercohomology and Chow groups with modulus, providing new tools for understanding algebraic cycles and K-theory.

Contribution

It constructs a new motivic spectral sequence for relative homotopy K-theory and establishes an isomorphism between Chow groups with modulus and relative motivic cohomology in certain cases.

Findings

Spectral sequence links K-theory and cdh-hypercohomology.

Cycle class map from relative motivic cohomology to K-theory.

Isomorphism between Chow groups with modulus and motivic cohomology for affine schemes.

Abstract

We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional cycles. Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme $X$ and a divisor $D \subset X$, we construct a canonical homomorphism from the Chow groups with modulus $\CH^i(X|D)$ to the relative motivic cohomology groups $H^{2i}(X|D, \Z(i))$ appearing in the above spectral sequence. This map is shown to be an isomorphism when $X$ is affine and $i = \dim(X)$.