# Rigidity in equivariant algebraic $K$-theory

## TL;DR

This paper establishes a rigidity property for equivariant algebraic K-theory spectra under certain conditions, showing that reduction mod n preserves the K-theory spectrum in the presence of a finite group action.

## Contribution

It extends the non-equivariant rigidity results to the equivariant setting by revisiting and adapting recent proofs, under specific algebraic conditions.

## Key findings

- Reduction map of mod-n equivariant K-theory spectra is an equivalence.
- Rigidity holds for henselian pairs with finite group actions under certain coprimality conditions.
- The proof revisits and adapts recent non-equivariant rigidity proofs to the equivariant context.

## Abstract

If $(R,I)$ is a henselian pair with an action of a finite group $G$ and $n\ge 1$ is an integer coprime to $|G|$ and such that $n\cdot |G|\in R^*$, then the reduction map of mod-$n$ equivariant $K$-theory spectra \[ K^G(R)/n\stackrel{\simeq}{\longrightarrow} K^G(R/I)/n\] is an equivalence. We prove this by revisiting the recent proof of non-equivariant rigidity by Clausen, Mathew, and Morrow.