Comptes Rendus
no. 5
Group Theory
Profinite groups of finite cohomological dimension
[Des groupes profinis de dimension cohomologique finie]

Présenté par : Jean-Pierre Serre

Thomas Weigel1 ; Pavel Zalesskii2
1 Università degli studi di Milano-Bicocca, Dipartimento di Matematica e Applicazioni, Via Bicocca degli Arcimboldi, 8, 20126 Milano, Italy
2 Department of Mathematics, University of Brasilia, 70910-900 Brasilia-DF, Brazil
Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 353-358.

Soit 1→N→G→G/N→1 une suite exacte courte de groupes profinis, et soit p un nombre premier. Nous montrons que si G a p-dimension cohomologique finie n:=cdp(G) et si l'ordre du groupe H k (N,𝔽 p ) est fini pour k:=cdp(N), la p-dimension cohomologique virtuelle de G/N est égale à n−k.

Let 1→N→G→G/N→1 be a short exact sequence of profinite groups, and let p be a prime number. We prove that if G is of finite cohomological p-dimension n:=cdp(G)<∞ and if the order of H k (N,𝔽 p ) is finite for k:=cdp(N), the virtual cohomological p-dimension of G/N equals n−k.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2003.12.022
Thomas Weigel 1 ; Pavel Zalesskii 2

1 Università degli studi di Milano-Bicocca, Dipartimento di Matematica e Applicazioni, Via Bicocca degli Arcimboldi, 8, 20126 Milano, Italy
2 Department of Mathematics, University of Brasilia, 70910-900 Brasilia-DF, Brazil 
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     title = {Profinite groups of finite cohomological dimension},
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Thomas Weigel; Pavel Zalesskii. Profinite groups of finite cohomological dimension. Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 353-358. doi : 10.1016/j.crma.2003.12.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.12.022/

[1] D.J. Benson Representations and Cohomology, II. Cohomology of Groups and Modules, Cambridge Stud. Adv. Math., vol. 31, Cambridge University Press, 1991

[2] R. Bieri On groups of cohomological dimension 2, topol. and algebra, Proc. Colloq. in Honor of B. Eckmann, Zürich, 1977, 1978, pp. 55-62

[3] A. Brumer Pseudocompact algebras, profinite groups and class formations, J. Algebra, Volume 4 (1966), pp. 442-470

[4] A. Engler, D. Haran, D. Kochloukova, P.A. Zalesskii, Normal subgroups of profinite groups of finite cohomological dimension, J. London Math. Soc., in press

[5] W.N. Herford, P.A. Zalesskii, Virtually free pro-p groups, ANUM preprint No. 132/01, University of Technology, Vienna, 2001

[6] J. Neukirch; A. Schmidt; K. Wingberg Cohomology of Number Fields, Grundlehren Math. Wiss., vol. 323, Springer, Berlin, 1999

[7] J.-P. Serre Cohomologie Galoisienne, Lecture Notes in Math., Springer-Verlag, Berlin, 1994

[8] P. Symonds; T. Weigel Cohomology of p-adic analytic groups (M. du Sautoy; D. Segal; A. Shalev, eds.), New Horizons in pro-p-Groups, Progr. Math., vol. 184, Birkhäuser, 2000, pp. 349-410

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