#### Volume 15, issue 1 (2011)

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Galois actions on homotopy groups of algebraic varieties

### Jonathan P Pridham

Geometry & Topology 15 (2011) 501–607
##### 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 .

étale homotopy
##### Publication
Revised: 27 January 2011
Accepted: 20 December 2010
Published: 31 March 2011
Proposed: Haynes Miller