# Rost nilpotence and \'etale motivic cohomology

## TL;DR

This paper proves an étale motivic analogue of the Rost nilpotence principle for all smooth projective schemes over perfect fields, offering new proofs for Rost nilpotence in specific cases like surfaces and certain threefolds.

## Contribution

It introduces an étale motivic analogue of Rost nilpotence applicable to all smooth projective schemes over perfect fields, simplifying proofs in key cases.

## Key findings

- Étale motivic analogue of Rost nilpotence holds generally.
- Provides an elegant proof of Rost nilpotence for surfaces.
- Establishes Rost nilpotence for birationally ruled threefolds in characteristic 0.

## Abstract

A smooth projective scheme $X$ over a field $k$ is said to satisfy the Rost nilpotence principle if any endomorphism of $X$ in the category of Chow motives that vanishes on an extension of the base field $k$ is nilpotent. We show that an \'etale motivic analogue of the Rost nilpotence principle holds for all smooth projective schemes over a perfect field. This provides a new approach to the question of Rost nilpotence and allows us to obtain an elegant proof of Rost nilpotence for surfaces, as well as for birationally ruled threefolds over a field of characteristic $0$.