Galois actions on homotopy groups of algebraic varieties

Abstract

We study the Galois actions on the $\ell$–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 $\ell$–adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever $\ell$ is not equal to the residue characteristic $p$ of $K$. For quasiprojective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When $\ell =p$, 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 ${\pi }_n^{\text{ ét}}\left(X_{\bar{K}}\right){\otimes }_{\widehat{\mathbb{Z}}}{\mathbb{Q}}_{\ell }$.