Advertisement

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)

Abstract

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.

Keywords

Fundamental Group Hopf Algebra Spectral Sequence Group Scheme Homotopy Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AI09.
    F. Andreatta and A. Iovita. Comparison isomorphisms for smooth formal schemes. 2009, www.mathstat.concordia.ca/faculty/iovita/research.html.Google Scholar
  2. AM69.
    M. Artin and B. Mazur. Etale homotopy. Number 100 in Lecture Notes in Mathematics. Springer, Berlin, 1969.Google Scholar
  3. Del71.
    P. Deligne. Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math., 40:5–57, 1971.Google Scholar
  4. Del75.
    P. Deligne. Poids dans la cohomologie des variétés algébriques. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pages 79–85. Canad. Math. Congress., Montreal, Que., 1975.Google Scholar
  5. Del80.
    P. Deligne. La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math., 52:137–252, 1980.Google Scholar
  6. DMOS82.
    P. Deligne, J. S. Milne, A. Ogus, and K. Y. Shih. Hodge cycles, motives, and Shimura varieties, volume 900 of Lecture Notes in Mathematics. Springer, Berlin, 1982.Google Scholar
  7. Fal89.
    G. Faltings. Crystalline cohomology and p-adic Galois-representations, pages 25–80. Johns Hopkins Univ. Press, Baltimore, MD, 1989.Google Scholar
  8. Fri82.
    E. M. Friedlander. Étale homotopy of simplicial schemes, volume 104 of Annals of Mathematics Studies. 1982.Google Scholar
  9. Hai93.
    R. M. Hain. Completions of mapping class groups and the cycle C − C  − . In Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), volume 150 of Contemp. Math., pages 75–105. Amer. Math. Soc., Providence, RI, 1993.Google Scholar
  10. Hai97.
    R. Hain. Infinitesimal presentations of the Torelli groups. J. Amer. Math. Soc., 10(3):597–651, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  11. Hai98.
    R. M. Hain. The Hodge de Rham theory of relative Malcev completion. Ann. Sci. École Norm. Sup. (4), 31(1):47–92, 1998.Google Scholar
  12. Haz78.
    M. Hazewinkel. Formal groups and applications, volume 78 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.Google Scholar
  13. HM03a.
    R. Hain and M. Matsumoto. Tannakian fundamental groups associated to Galois groups. In Galois groups and fundamental groups, volume 41 of Math. Sci. Res. Inst. Publ., pages 183–216. Cambridge Univ. Press, Cambridge, 2003.Google Scholar
  14. HM03b.
    R. Hain and M. Matsumoto. Weighted completion of Galois groups and Galois actions on the fundamental group of \({\mathbb{P}}^{1} -\{ 0, 1,\infty \}\). Compositio Math., 139(2):119–167, 2003.Google Scholar
  15. HM09.
    R. Hain and M. Matsumoto. Relative pro- completions of mapping class groups. J. Algebra, 321(11):3335–3374, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  16. Kan58.
    Daniel M. Kan. On homotopy theory and c.s.s. groups. Ann. of Math. (2), 68:38–53, 1958.Google Scholar
  17. Ked06.
    K. S. Kedlaya. Fourier transforms and p-adic ‘Weil II’. Compos. Math., 142(6):1426–1450, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  18. KPT09.
    L. Katzarkov, T. Pantev, and B. Toën. Algebraic and topological aspects of the schematization functor. Compos. Math., 145(3):633–686, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  19. Laf02.
    L. Lafforgue. Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math., 147(1):1–241, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  20. Mor78.
    J. W. Morgan. The algebraic topology of smooth algebraic varieties. Inst. Hautes Études Sci. Publ. Math., 48:137–204, 1978.CrossRefzbMATHGoogle Scholar
  21. Ols11.
    M. C. Olsson. Towards non-abelian p-adic Hodge theory in the good reduction case. Mem. Amer. Math. Soc., 220(990), 2011.Google Scholar
  22. Pri07.
    J. P. Pridham. The pro-unipotent radical of the pro-algebraic fundamental group of a compact Kähler manifold. Ann. Fac. Sci. Toulouse Math. (6), 16(1):147–178, 2007.Google Scholar
  23. Pri08.
    J. P. Pridham. Pro-algebraic homotopy types. Proc. London Math. Soc., 97(2):273–338, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  24. Pri09.
    J. P. Pridham. Weight decompositions on étale fundamental groups. Amer. J. Math., 131(3):869–891, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  25. Pri10.
    J. P. Pridham. Formality and splitting of real non-abelian mixed Hodge structures. 2010, arXiv:math.AG/0902.0770v2. submitted.Google Scholar
  26. Pri11.
    J. P. Pridham. Galois actions on homotopy groups. Geom. Topol., 15:1:501-607, 2011.Google Scholar
  27. Toë06.
    B. Toën. Champs affines. Selecta Math. (N.S.), 12(1):39–135, 2006.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

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

Personalised recommendations