# Notes on equivariant higher Chow groups

## TL;DR

This paper establishes a comparison between equivariant and ordinary higher Chow groups for varieties with finite group actions, demonstrating the degeneration of the equivariant motivic spectral sequence and deriving a Riemann-Roch theorem for equivariant algebraic K-theory.

## Contribution

It introduces a comparison theorem linking equivariant and ordinary higher Chow groups and proves the rational degeneration of the equivariant motivic spectral sequence.

## Key findings

- Comparison theorem between equivariant and ordinary higher Chow groups
- Rational degeneration of the equivariant motivic spectral sequence
- Riemann-Roch theorem for equivariant algebraic K-theory

## Abstract

In this short note, we prove a comparision theorem between Levine-Serp\'e's equivariant higher Chow groups of an algebraic variety equipped with an action of a finite group and ordinary higher Chow groups of its fixed points. As a consequence, we show that the equivariant motivic spectral sequence degenerates rationally. This yields a Riemann-Roch Theorem for equivariant algebraic $K$-theory.