Stable cohomology of alternating groups

Fedor Bogomolov; Christian Böhning

Open Mathematics (2014)

  • Volume: 12, Issue: 2, page 212-228
  • ISSN: 2391-5455

Abstract

top
We determine the stable cohomology groups ( H S i 𝔄 n , ℤ 𝔄 n , ℤ p ℤ p ℤ of the alternating groups 𝔄 n for all integers n and i, and all odd primes p.

How to cite

top

Fedor Bogomolov, and Christian Böhning. "Stable cohomology of alternating groups." Open Mathematics 12.2 (2014): 212-228. <http://eudml.org/doc/269336>.

@article{FedorBogomolov2014,
abstract = {We determine the stable cohomology groups (\[H\_S^i \left( \{\{\{\mathfrak \{A\}\_n ,\mathbb \{Z\}\} \mathord \{\left\bad. \{\vphantom\{\{\mathfrak \{A\}\_n ,\mathbb \{Z\}\} \{p\mathbb \{Z\}\}\}\} \right. \hspace\{0.0pt\}\} \{p\mathbb \{Z\}\}\}\} \right)\] of the alternating groups \[\mathfrak \{A\}\_n\] for all integers n and i, and all odd primes p.},
author = {Fedor Bogomolov, Christian Böhning},
journal = {Open Mathematics},
keywords = {Stable cohomology; Alternating groups; Cohomological invariants; stable cohomology; alternating groups; cohomological invariants},
language = {eng},
number = {2},
pages = {212-228},
title = {Stable cohomology of alternating groups},
url = {http://eudml.org/doc/269336},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Fedor Bogomolov
AU - Christian Böhning
TI - Stable cohomology of alternating groups
JO - Open Mathematics
PY - 2014
VL - 12
IS - 2
SP - 212
EP - 228
AB - We determine the stable cohomology groups (\[H_S^i \left( {{{\mathfrak {A}_n ,\mathbb {Z}} \mathord {\left\bad. {\vphantom{{\mathfrak {A}_n ,\mathbb {Z}} {p\mathbb {Z}}}} \right. \hspace{0.0pt}} {p\mathbb {Z}}}} \right)\] of the alternating groups \[\mathfrak {A}_n\] for all integers n and i, and all odd primes p.
LA - eng
KW - Stable cohomology; Alternating groups; Cohomological invariants; stable cohomology; alternating groups; cohomological invariants
UR - http://eudml.org/doc/269336
ER -

References

top
  1. [1] Adem A., Milgram R.J., Cohomology of Finite Groups, 2nd ed., Grundlehren Math. Wiss., 309, Springer, Berlin, 2004 http://dx.doi.org/10.1007/978-3-662-06280-7 Zbl1061.20044
  2. [2] Bogomolov F.A., Stable cohomology of groups and algebraic varieties, Russian Acad. Sci. Sb. Math., 1993, 76(1), 1–21 http://dx.doi.org/10.1070/SM1993v076n01ABEH003398 Zbl0789.14022
  3. [3] Bogomolov F., Stable cohomology of finite and profinite groups, In: Algebraic Groups, Göttingen, June 27–July 13, 2005, Universitätsverlag Göttingen, Göttingen, 2007, 19–49 
  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, 57–76 
  5. [5] Bogomolov F., Petrov T., Unramified cohomology of alternating groups, Cent. Eur. J. Math., 2011, 9(5), 936–948 http://dx.doi.org/10.2478/s11533-011-0061-8 Zbl1236.20054
  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, 55–73 http://dx.doi.org/10.1007/978-0-8176-4934-0_3 
  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, 1–64 Zbl0834.14009
  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), 141–158 http://dx.doi.org/10.1007/BF01850658 Zbl0686.14050
  9. [9] Garibaldi S., Merkurjev A., Serre J.-P., Cohomological invariants in Galois cohomology, Univ. Lecture Ser., 28, American Mathematical Society, Providence, 2003 Zbl1159.12311
  10. [10] Kahn B., Relatively unramified elements in cycle modules, J. K-Theory, 2011, 7(3), 409–427 http://dx.doi.org/10.1017/is011003002jkt147 Zbl1235.14008
  11. [11] Kahn B., Sujatha R., Motivic cohomology and unramified cohomology of quadrics, J. Eur. Math. Soc. (JEMS), 2000, 2(2), 145–177 http://dx.doi.org/10.1007/s100970000015 
  12. [12] Mann B.M., The cohomology of the alternating groups, Michigan Math. J., 1985, 32(3), 267–277 http://dx.doi.org/10.1307/mmj/1029003238 Zbl0604.20053
  13. [13] Mùi H., Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 1975, 22(3), 319–369 Zbl0335.18010
  14. [14] Nguyen T.K.N., Modules de Cycles et Classes Non Ramifiées sur un Espace Classifiant, PhD thesis, Université Paris Diderot, 2010 
  15. [15] Nguyen T.K.N., Classes non ramifiées sur un espace classifiant, C. R. Math. Acad. Sci. Paris, 2011, 349(5–6), 233–237 http://dx.doi.org/10.1016/j.crma.2011.02.012 
  16. [16] Ore O., Theory of monomial groups, Trans. Amer. Math. Soc., 1942, 51(1), 15–64 http://dx.doi.org/10.2307/1989979 
  17. [17] Serre J.-P., Galois Cohomology, Springer Monogr. Math., Springer, Berlin, 2002 
  18. [18] Steenrod N.E., Cohomology Operations, Ann. of Math. Stud., 50, Princeton University Press, Princeton, 1962 
  19. [19] Tezuka M., Yagita N., The image of the map from group cohomology to Galois cohomology, Trans. Amer. Math. Soc., 2011, 363(8), 4475–4503 http://dx.doi.org/10.1090/S0002-9947-2011-05418-8 Zbl1310.11044

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.