Central European Journal of Mathematics

, Volume 12, Issue 2, pp 212–228 | Cite as

Stable cohomology of alternating groups

  • Fedor BogomolovEmail author
  • Christian Böhning
Research Article


We determine the stable cohomology groups (\(H_S^i \left( {{{\mathfrak{A}_n ,\mathbb{Z}} \mathord{\left/ {\vphantom {{\mathfrak{A}_n ,\mathbb{Z}} {p\mathbb{Z}}}} \right. \kern-\nulldelimiterspace} {p\mathbb{Z}}}} \right)\) of the alternating groups \(\mathfrak{A}_n\) for all integers n and i, and all odd primes p.


Stable cohomology Alternating groups Cohomological invariants 


14E08 14F43 


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  1. [1]
    Adem A., Milgram R.J., Cohomology of Finite Groups, 2nd ed., Grundlehren Math. Wiss., 309, Springer, Berlin, 2004CrossRefzbMATHGoogle Scholar
  2. [2]
    Bogomolov F.A., Stable cohomology of groups and algebraic varieties, Russian Acad. Sci. Sb. Math., 1993, 76(1), 1–21MathSciNetCrossRefGoogle Scholar
  3. [3]
    Bogomolov F., Stable cohomology of finite and profinite groups, In: Algebraic Groups, Göttingen, June 27–July 13, 2005, Universitätsverlag Göttingen, Göttingen, 2007, 19–49Google Scholar
  4. [4]
    Bogomolov F., Böhning Chr., Isoclinism and stable cohomology of wreath products, In: Birational Geometry, Rational Curves, and Arithmetic, Simons Symposium ”Geometry Over Non-Closed Fields”, St. John, February 26–March 3, 2012, Springer, New York, 2013, 57–76Google Scholar
  5. [5]
    Bogomolov F., Petrov T., Unramified cohomology of alternating groups, Cent. Eur. J. Math., 2011, 9(5), 936–948MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Bogomolov F., Petrov T., Tschinkel Yu., Unramified cohomology of finite groups of Lie type, In: Cohomological and Geometric Approaches to Rationality Problems, Progr. Math., 282, Birkhäuser, Boston, 2010, 55–73CrossRefGoogle Scholar
  7. [7]
    Colliot-Thélène J.-L., Birational invariants, purity and the Gersten conjecture, In: K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Santa Barbara, July 6–24, 1992, Proc. Sympos. Pure Math., 58(1), American Mathematical Society, Providence, 1995, 1–64Google Scholar
  8. [8]
    Colliot-Thélène J.-L., Ojanguren M., Variétés unirationelles non rationelles: au-delà de l’exemple d’Artin et Mumford, Invent. Math., 1989, 97(1), 141–158MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Garibaldi S., Merkurjev A., Serre J.-P., Cohomological invariants in Galois cohomology, Univ. Lecture Ser., 28, American Mathematical Society, Providence, 2003zbMATHGoogle Scholar
  10. [10]
    Kahn B., Relatively unramified elements in cycle modules, J. K-Theory, 2011, 7(3), 409–427MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Kahn B., Sujatha R., Motivic cohomology and unramified cohomology of quadrics, J. Eur. Math. Soc. (JEMS), 2000, 2(2), 145–177MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Mann B.M., The cohomology of the alternating groups, Michigan Math. J., 1985, 32(3), 267–277MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Mùi H., Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 1975, 22(3), 319–369MathSciNetzbMATHGoogle Scholar
  14. [14]
    Nguyen T.K.N., Modules de Cycles et Classes Non Ramifiées sur un Espace Classifiant, PhD thesis, Université Paris Diderot, 2010Google Scholar
  15. [15]
    Nguyen T.K.N., Classes non ramifiées sur un espace classifiant, C. R. Math. Acad. Sci. Paris, 2011, 349(5–6), 233–237MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Ore O., Theory of monomial groups, Trans. Amer. Math. Soc., 1942, 51(1), 15–64MathSciNetCrossRefGoogle Scholar
  17. [17]
    Serre J.-P., Galois Cohomology, Springer Monogr. Math., Springer, Berlin, 2002Google Scholar
  18. [18]
    Steenrod N.E., Cohomology Operations, Ann. of Math. Stud., 50, Princeton University Press, Princeton, 1962zbMATHGoogle Scholar
  19. [19]
    Tezuka M., Yagita N., The image of the map from group cohomology to Galois cohomology, Trans. Amer. Math. Soc., 2011, 363(8), 4475–4503MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Versita Warsaw and Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew YorkUSA
  2. 2.Laboratory of Algebraic GeometryGU-HSEMoscowRussia
  3. 3.Fachbereich Mathematik der Universität HamburgHamburgGermany

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