On -adic Pro-algebraic and Relative Pro- Fundamental Groups

  • Jonathan P. Pridham
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 2)


We recall -adic relative Malcev completions and relative pro- completions of pro-finite groups and homotopy types. These arise when studying unipotent completions of fibres or of normal subgroups. Several new properties are then established, relating to -adic analytic moduli and comparisons between relative Malcev and relative pro- completions. We then summarise known properties of Galois actions on the pro-ℚℓ-algebraic geometric fundamental group and its big Malcev completions. For smooth varieties in finite characteristics different from , these groups are determined as Galois representations by cohomology of semisimple local systems. Olsson’s non-abelian étale-crystalline comparison theorem gives slightly weaker results for varieties over -adic fields, since the non-abelian Hodge filtration cannot be recovered from cohomology.


Fundamental Group Hopf Algebra Spectral Sequence Group Scheme Homotopy Type 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.DPMMS, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

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