Galois actions on homotopy groups of algebraic varieties

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Abstract

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
Volume15
Issue number1
DOIs
Publication statusPublished - 1 Jan 2011

Keywords

  • étale
  • homotopy

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