Galois actions on homotopy groups of algebraic varieties

Research output: Contribution to journalArticlepeer-review


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 quasiprojective 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 π (X )⊗ℤℚl.
Original languageEnglish
Pages (from-to)501-607
Number of pages107
JournalGeometry and Topology
Issue number1
Publication statusPublished - 1 Jan 2011


  • étale
  • homotopy


Dive into the research topics of 'Galois actions on homotopy groups of algebraic varieties'. Together they form a unique fingerprint.

Cite this