Motivic spectral sequence for relative homotopy K-theory

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 ${\text{CH}}^i(X\mathrm{\mid }D)$ to the relative motivic cohomology groups $H^{2i}(X\mathrm{\mid }D,\mathbb{Z}(i))$ appearing in the above spectral sequence. This map is shown to be an isomorphism when $X$ is affine and $i=\dim (X)$.