Topological Realisations of Absolute Galois Groups

  • Robert A. Kucharczyk
  • Peter ScholzeEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 245)


Let F be a field of characteristic 0 containing all roots of unity. We construct a functorial compact Hausdorff space \(X_F\) whose profinite fundamental group agrees with the absolute Galois group of F, i.e. the category of finite covering spaces of \(X_F\) is equivalent to the category of finite extensions of F. The construction is based on the ring of rational Witt vectors of F. In the case of the cyclotomic extension of \(\mathbb {Q}\), the classical fundamental group of \(X_F\) is a (proper) dense subgroup of the absolute Galois group of F. We also discuss a variant of this construction when the field is not required to contain all roots of unity, in which case there are natural Frobenius-type automorphisms which encode the descent along the cyclotomic extension.


Galois groups Fundamental groups Witt vectors 

2010 Mathematics Subject Classification

12F10 11R32 14F35 



Part of this work was done while the second author was a Clay Research Fellow. All of it was done while the first author was supported by the Swiss National Science Foundation. The authors wish to thank Lennart Meier for asking a very helpful question, Markus Land and Thomas Nikolaus for a discussion about Proposition 7.10, and Eric Leichtnam for pointing out some typographical errors in an earlier version.


  1. 1.
    Fontaine, J.-M., Wintenberger, J.-P.: Extensions algébrique et corps des normes des extensions APF des corps locaux. C. R. Acad. Sci. Paris Sér. A-B 288(8), A441–A444 (1979). MR 527692zbMATHGoogle Scholar
  2. 2.
    Weinstein, J.: \(\text{Gal}(\bar{\mathbb{Q}}_p/{\mathbb{Q}}_p)\) as a geometric fundamental group (2014). arXiv:1404.7192
  3. 3.
    Scholze, P.: Perfectoid spaces. Publ. Math. Inst. Hautes Études Sci. 116, 245–313 (2012). MR 3090258MathSciNetCrossRefGoogle Scholar
  4. 4.
    Almkvist, G.: Endomorphisms of finitely generated projective modules over a commutative ring. Ark. Mat. 11, 263–301 (1973). MR 0424786MathSciNetCrossRefGoogle Scholar
  5. 5.
    Voevodsky, V.: On motivic cohomology with \(\varvec {Z}/l\) -coefficients. Ann. Math. 174(1), 401–438 (2011). MR 2811603MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bost, J.-B., Connes, A.: Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Math. (N.S.) 1(3), 411–457 (1995). MR 1366621MathSciNetCrossRefGoogle Scholar
  7. 7.
    Steen, L.A., Arthur Seebach, J.: Counterexamples in Topology, 2nd edn. Springer, New York, Heidelberg (1978). MR 507446CrossRefGoogle Scholar
  8. 8.
    Fabel, P.: Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology). Bull. Pol. Acad. Sci. Math. 59(1), 77–83 (2011). MR 2810974MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brazas, J.: The fundamental group as a topological group. Topol. Appl. 160(1), 170–188 (2013). MR 2995090MathSciNetCrossRefGoogle Scholar
  10. 10.
    Brazas, J.: Semicoverings: a generalization of covering space theory. Homol. Homotopy Appl. 14(1), 33–63 (2012). MR 2954666MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bhatt, B., Scholze, P.: The pro-étale topology for schemes. Astérisque (369), 99–201 (2015). MR 3379634Google Scholar
  12. 12.
    Klevdal, C.: A correspondence, Galois, with generalized covering spaces. Undergraduate honors Thesis. University of Colorado, Boulder (2015)Google Scholar
  13. 13.
    Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 3, Société Mathématique de France, Paris, 2003, Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic geometry seminar of Bois Marie 1960–61], Directed by Grothendieck, A. With two papers by Raynaud, M. Updated and annotated reprint of the 1971 original. Lecture Notes in Mathematics, vol. 224, Springer, Berlin; MR0354651 (50 #7129). MR 2017446Google Scholar
  14. 14.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002). MR 1867354zbMATHGoogle Scholar
  15. 15.
    Munkres, J.R.: Topology, 2nd edn. Prentice Hall, Upper Saddle River (2000)zbMATHGoogle Scholar
  16. 16.
    Rudin, W.: Fourier Analysis on Groups. Interscience tracts in pure and applied mathematics, vol. 12. Interscience Publishers (a division of John Wiley and Sons), New York, London (1962). MR 0152834Google Scholar
  17. 17.
    Stein, K.: Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem. Math. Ann. 123, 201–222 (1951). MR 0043219MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shelah, S.: Infinite abelian groups, whitehead problem and some constructions. Isr. J. Math. 18, 243–256 (1974). MR 0357114MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shelah, S.: A compactness theorem for singular cardinals, free algebras, whitehead problem and transversals. Isr. J. Math. 21(4), 319–349 (1975). MR 0389579MathSciNetCrossRefGoogle Scholar
  20. 20.
    Godement, R.: Topologie algébrique et théorie des faisceaux. Hermann, Paris (1973). Troisième édition revue et corrigée, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XIII, Actualités Scientifiques et Industrielles, vol. 1252. MR 0345092Google Scholar
  21. 21.
    Eilenberg, S., Steenrod, N.: Foundations of Algebraic Topology. Princeton University Press, Princeton, New Jersey(1952). MR 0050886Google Scholar
  22. 22.
    The stacks project, website, available at
  23. 23.
    Hazewinkel, M.: Witt vectors. I. In: Handbook of Algebra, vol. 6, pp. 319–472. Elsevier/North-Holland, Amsterdam (2009). MR 2553661Google Scholar
  24. 24.
    Almkvist, G.: \(K\)-theory of endomorphisms. J. Algebra 55(2), 308–340 (1978). MR 523461MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kelley, J.L., Spanier, E.H.: Euler characteristics. Pac. J. Math. 26, 317–339 (1968). MR 0260842MathSciNetCrossRefGoogle Scholar
  26. 26.
    Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. 32, 5–361 (1967). MR 0238860CrossRefGoogle Scholar
  27. 27.
    Hilbert, D.: Die Theorie der algebraischen Zahlkörper. Jahresber. Deutsch. Math.-Verein. 4, 175–546 (1897)zbMATHGoogle Scholar
  28. 28.
    Neukirch, J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol. 322, Springer, Berlin (1999): Translated from the 1992 German original and with a note by Schappacher, N. With a foreword by Harder, G. MR 1697859CrossRefGoogle Scholar
  29. 29.
    Serre, J.-P.: Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier 6, 1–42 (1955–1956). MR 0082175 (18,511a)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lang, S.: Algebra, 3rd edn., Graduate Texts in Mathematics, vol. 211, Springer, New York (2002). MR 1878556CrossRefGoogle Scholar
  31. 31.
    Grothendieck, A.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents I. Inst. Hautes Études Sci. Publ. Math. 11, 5–167 (1961). MR0217085CrossRefGoogle Scholar
  32. 32.
    Pontrjagin, L.: The theory of topological commutative groups. Ann. Math. 35(2), 361–388 (1934). MR 1503168MathSciNetCrossRefGoogle Scholar
  33. 33.
    Fuchs, L., Loonstra, F.: On the cancellation of modules in direct sums over Dedekind domains. Nederl. Akad. Wetensch. Proc. Ser. A 74 = Indag. Math. 33, 163–169 (1971). MR 0289476MathSciNetCrossRefGoogle Scholar
  34. 34.
    May, W.: Unit groups of infinite abelian extensions. Proc. Am. Math. Soc. 25, 680–683 (1970). MR 0258786MathSciNetCrossRefGoogle Scholar
  35. 35.
    Cohen, J.M.: Homotopy groups of inverse limits. Proceedings of the advanced study institute on algebraic topology (Aarhus Univ., Aarhus, 1970), vol. I, Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 29–43. Various Publ. Ser., No. 13. MR 0346781Google Scholar
  36. 36.
    Hirschhorn, P.S.: The homotopy groups of the inverse limit of a tower of fibrations (2015), preprint.
  37. 37.
    Serre, J-P.: Cohomologie galoisienne, Cours au Collège de France, vol. 1962, Springer, Berlin, Heidelberg, New York (1962/1963). MR 0180551 (31#4785)Google Scholar
  38. 38.
    Schneider, P.: Equivariant homology for totally disconnected groups. J. Algebra 203(1), 50–68 (1998). MR 1620705MathSciNetCrossRefGoogle Scholar
  39. 39.
    Milne, J.S.: Étale Cohomology. Princeton mathematical series. Princeton University Press, Princeton (1980). MR 559531zbMATHGoogle Scholar
  40. 40.
    Milnor, J.: Algebraic \(K\) -theory and quadratic forms. Invent. Math. 9, 318–344 (1969/1970). MR 0260844Google Scholar
  41. 41.
    Matsumoto, H.: Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. Sci. École Norm. Sup. 2(4), 1–62 (1969). MR 0240214MathSciNetCrossRefGoogle Scholar
  42. 42.
    Bloch, S., Kato, K.: \(p\)-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. (63), 107–152 (1986). MR 849653Google Scholar
  43. 43.
    Adem, A., James Milgram, R.: Cohomology of finite groups, 2nd edn., Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol. 309, Springer, Berlin (2004). MR 2035696CrossRefGoogle Scholar
  44. 44.
    Artin, E., Schreier, O.: Algebraische Konstruktion reeller Körper. Abh. Math. Sem. Univ. Hambg. 5(1), 85–99 (1927). MR 3069467CrossRefGoogle Scholar
  45. 45.
    Jacobson, N.: Basic Algebra II, 2nd edn. W. H. Freeman and Company, New York (1989). MR 1009787zbMATHGoogle Scholar
  46. 46.
    Borger, J.: The basic geometry of Witt vectors, I: the affine case. Algebra Number Theory 5(2), 231–285 (2011). MR 2833791MathSciNetCrossRefGoogle Scholar
  47. 47.
    Deligne, P.: Cohomologie étale. Lecture Notes in Mathematics, vol. 569. Springer, Berlin, New York (1977)Google Scholar
  48. 48.
    Borger, J.: \(\Lambda \)-rings and the field with one element, (2009). arXiv:0906.3146

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département MathematikETH ZürichZürichSwitzerland
  2. 2.Mathematisches InstitutRheinische Friedrich-Wilhelms-Universität BonnBonnGermany

Personalised recommendations