Profinite groups of finite cohomological dimension

Thomas Weigel; Pavel Zalesskii

doi:10.1016/j.crma.2003.12.022

Group Theory

Profinite groups of finite cohomological dimension

Presented by: Jean-Pierre Serre

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

Comptes Rendus. Mathématique, Volume 338 (2004) no. 5, pp. 353-358.

Abstract
Résumé

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:=cd_p(G)<∞ and if the order of $H^{k} (N, 𝔽_{p})$ is finite for k:=cd_p(N), the virtual cohomological p-dimension of G/N equals n−k.

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:=cd_p(G) et si l'ordre du groupe $H^{k} (N, 𝔽_{p})$ est fini pour k:=cd_p(N), la p-dimension cohomologique virtuelle de G/N est égale à n−k.

Received: 2003-10-11
Accepted: 2003-12-01
Published online: 2004-02-10

DOI: 10.1016/j.crma.2003.12.022

Author's affiliations:

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

@article{CRMATH_2004__338_5_353_0,
     author = {Thomas Weigel and Pavel Zalesskii},
     title = {Profinite groups of finite cohomological dimension},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {353--358},
     publisher = {Elsevier},
     volume = {338},
     number = {5},
     year = {2004},
     doi = {10.1016/j.crma.2003.12.022},
     language = {en},
}

TY  - JOUR
AU  - Thomas Weigel
AU  - Pavel Zalesskii
TI  - Profinite groups of finite cohomological dimension
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 353
EP  - 358
VL  - 338
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2003.12.022
LA  - en
ID  - CRMATH_2004__338_5_353_0
ER  -

%0 Journal Article
%A Thomas Weigel
%A Pavel Zalesskii
%T Profinite groups of finite cohomological dimension
%J Comptes Rendus. Mathématique
%D 2004
%P 353-358
%V 338
%N 5
%I Elsevier
%R 10.1016/j.crma.2003.12.022
%G en
%F CRMATH_2004__338_5_353_0

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/

References
Cited by

[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

Cited by Sources:

Comments - Policy