Module structure of the $K$-theory of polynomial-like rings

Suppose $\mathrm{\Gamma }$ is a submonoid of a lattice, not containing a line. In this note, we use the natural $\mathrm{\Gamma }$-grading on the monoid algebra $R[\mathrm{\Gamma }]$ to prove structural results about the relative $K$-theory $K(R[\mathrm{\Gamma }],R)$. When $R$ contains a field, we prove a decomposition indexed by the rays in $\mathrm{\Gamma }$, and a compatible action by the Witt vectors of $R$ for each $\mathbf{N}$-grading of $\mathrm{\Gamma }$. In characteristic zero, there is additionally an action by Witt vectors for the truncation set $\mathrm{\Gamma }$. Finally, we apply this to get a ray-like description of $K_{\ast }(R[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n])$ proposed by J.,Davis.