Rigidity in equivariant algebraic $K$-theory

Abstract

If $\left(R,I\right)$ is a henselian pair with an action of a finite group $G$ and $n\ge 1$ is an integer coprime to $\left|G\right|$ such that $n\cdot \left|G\right|\in R^{\ast }$, then the reduction map of mod-$n$ equivariant $K$-theory spectra

$$K^G\left(R\right)∕n\underset{}{\overset{\simeq }{\to }}{K}^G\left(R∕I\right)∕n$$

is an equivalence. We prove this by revisiting the recent proof of nonequivariant rigidity by Clausen, Mathew, and Morrow.