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

TL;DR

This paper investigates the structure of the relative K-theory of polynomial-like rings using monoid algebra gradings, revealing decompositions and actions by Witt vectors, and applies these results to polynomial rings.

Contribution

It provides new structural results on the K-theory of monoid algebras and polynomial rings, including decompositions and Witt vector actions, extending previous understanding.

Findings

Decomposition of K-theory indexed by rays in the monoid

Witt vector actions on the K-theory for graded monoids

Application to a ray-like description of K-theory of polynomial 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 $K_*(R[x_1,...,x_n])$ proposed by J.\,Davis.