Galois actions on homotopy groups of algebraic varieties

Pridham, Jonathan P

doi:10.2140/gt.2011.15.501

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Galois actions on homotopy groups of algebraic varieties

Jonathan P Pridham

Geometry & Topology 15 (2011) 501–607

DOI: 10.2140/gt.2011.15.501

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 $π_{n}^{ét} (X_{\bar{K}}) \otimes_{\hat{ℤ}} ℚ_{ℓ}$ .

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/

Volume 15, issue 1 (2011)