Galois actions on homotopy groups

TL;DR

This paper investigates how Galois groups act on various homotopy groups of algebraic varieties, providing explicit descriptions and comparisons across different cohomological and homotopical frameworks.

Contribution

It offers new explicit characterizations of Galois actions on l-adic schematic and Artin-Mazur homotopy groups for varieties with good reduction, including cases where l equals the residue characteristic.

Findings

Galois actions on l-adic schematic homotopy groups are explicitly described for proper varieties.

A similar characterization is provided for quasi-projective varieties using the Gysin spectral sequence.

Comparison theorems relate these descriptions to Artin-Mazur homotopy groups under certain conditions.

Abstract

We study the Galois actions on the l-adic schematic and Artin-Mazur homotopy groups of algebraic varieties. For proper varieties of good reduction over a local field K, we show that the l-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever l is not equal to the residue characteristic p of K. For quasi-projective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When l=p, a slightly weaker result is proved by comparing the crystalline and p-adic schematic homotopy types. Under favourable conditions, a comparison theorem transfers all these descriptions to the Artin-Mazur homotopy groups.