Unstable operations in \'etale and motivic cohomology

TL;DR

This paper classifies all étale cohomology operations and introduces new motivic cohomology operations, providing a comprehensive understanding of their structure and differences from existing operations.

Contribution

It fully classifies étale cohomology operations and constructs new motivic cohomology operations, offering insights into their structure and potential generalizations.

Findings

All étale cohomology operations are constructed by Epstein.

New operations P^a on motivic cohomology differ from Voevodsky's.

Complete classification of motivic cohomology operations on H^{p,1} and H^{1,q}.

Abstract

We classify all \'etale cohomology operations on $H_\et^n(-,\muell{i})$, showing that they were all constructed by Epstein. We also construct operations $P^a$ on the mod-$\ell$ motivic cohomology groups $H^{p,q}$, differing from Voevodsky's operations; we use them to classify all motivic cohomology operations on $H^{p,1}$ and $H^{1,q}$ and suggest a general classification.