Galois actions on homotopy groups of algebraic varieties

Pridham, Jonathan P¹

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

Geometry & topology, Tome 15 (2011) no. 1, pp. 501-607.

Résumé

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 ${\pi }_n^{\text{ ét}}\left(X_{\bar{K}}\right){\otimes }_{\widehat{\mathbb{Z}}}{\mathbb{Q}}_{\ell }$.

Détail

DOI : 10.2140/gt.2011.15.501

Keywords: étale homotopy

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