Galois actions on homotopy groups

Details Galois actions on homotopy groups Galois actions on homotopy groups

2007-12-06

arxiv.org/abs/0712.0928v4

Geom. Topol. 15:1 (2011), 501-607

Mathematics

Algebraic Geometry

72 pages; v2 corrections to Section 3; v3 references updated; v4 final version

Scientific paper

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.

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