Volume 15, issue 1 (2011)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 –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 –adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever 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 = 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 πn ét(XK̄) ̂.

Keywords
étale homotopy
References
Publication
Received: 10 December 2009
Revised: 27 January 2011
Accepted: 20 December 2010
Published: 31 March 2011
Proposed: Haynes Miller
Seconded: Bill Dwyer, Paul Goerss
Authors
Jonathan P Pridham
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Wilberforce Road
Cambridge
CB3 0WB
UK
http://www.dpmms.cam.ac.uk/~jpp24/