Module Structure of the K-Theory of Polynomial-Like Rings

Abstract

Suppose Γ is a submonoid of a lattice, not containing a line. In this note, we use the natural Γ-grading on the monoid algebra R[Γ] to prove structural results about the relative K-theory K(R[Γ],R). When R contains a field, we prove a decomposition indexed by the rays in Γ and a compatible action by the Witt vectors of R for each ℕ-grading of Γ. In characteristic zero, there is additionally an action by Witt vectors for the truncation set Γ. Finally, we apply this to get a ray-like description of K∗(R[x1,…,xn]) proposed by J. Davis.