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

Abstract

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