Geodesic
Cohomological Dimension and Schreier's Formula in Galois Cohomology
Canadian mathematical bulletin, Tome 50 (2007) no. 4, pp. 588-593

Voir la notice de l'article provenant de la source Cambridge University Press

Let $p$ be a prime and $F$ a field containing a primitive $p$ -th root of unity. Then for $n\,\in \,\mathbb{N}$ , the cohomological dimension of the maximal pro- $p$ -quotient $G$ of the absolute Galois group of $F$ is at most $n$ if and only if the corestriction maps ${{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)\,\to \,{{H}^{n}}\left( G,\ {{\mathbb{F}}_{p}} \right)$ are surjective for all open subgroups $H$ of index $p$ . Using this result, we generalize Schreier's formula for ${{\dim}_{{{\mathbb{F}}_{p}}}}\,{{H}^{1}}\,\left( H,\ {{\mathbb{F}}_{p}} \right)$ to ${{\dim}_{{{\mathbb{F}}_{p}}}}{{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)$ .
DOI : 10.4153/CMB-2007-056-3
Mots-clés : 12G05, 12G10, cohomological dimension, Schreier's formula, Galois theory, p-extensions, pro-p-groups 
Labute, John; Lemire, Nicole; Mináč, Ján; Swallow, John. Cohomological Dimension and Schreier's Formula in Galois Cohomology. Canadian mathematical bulletin, Tome 50 (2007) no. 4, pp. 588-593. doi: 10.4153/CMB-2007-056-3
@article{10_4153_CMB_2007_056_3,
     author = {Labute, John and Lemire, Nicole and Min\'a\v{c}, J\'an and Swallow, John},
     title = {Cohomological {Dimension} and {Schreier's} {Formula} in {Galois} {Cohomology}},
     journal = {Canadian mathematical bulletin},
     pages = {588--593},
     year = {2007},
     volume = {50},
     number = {4},
     doi = {10.4153/CMB-2007-056-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2007-056-3/}
}
TY  - JOUR
AU  - Labute, John
AU  - Lemire, Nicole
AU  - Mináč, Ján
AU  - Swallow, John
TI  - Cohomological Dimension and Schreier's Formula in Galois Cohomology
JO  - Canadian mathematical bulletin
PY  - 2007
SP  - 588
EP  - 593
VL  - 50
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2007-056-3/
DO  - 10.4153/CMB-2007-056-3
ID  - 10_4153_CMB_2007_056_3
ER  - 
%0 Journal Article
%A Labute, John
%A Lemire, Nicole
%A Mináč, Ján
%A Swallow, John
%T Cohomological Dimension and Schreier's Formula in Galois Cohomology
%J Canadian mathematical bulletin
%D 2007
%P 588-593
%V 50
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2007-056-3/
%R 10.4153/CMB-2007-056-3
%F 10_4153_CMB_2007_056_3

[K] Koch, H., Galois theory of p-extensions. Translated from the 1970 German original by Franz Lemmermeyer. With a postscript by the author and Lemmermeyer. Springer-Verlag, Berlin, 2002. Google Scholar

[LMS1] Lemire, N., Mináč, J., and Swallow, J., Galois module structure of Galois cohomology and partial Euler-Poincaré characteristics. To appear in J. Reine Angew. Math. ArXiv:math.NT/0409484, 2004. Google Scholar

[LMS2] Lemire, N., Mináč, J., and Swallow, J., When is Galois cohomology free or trivial. New York J. Math. 11(2005), 291–302. Google Scholar

[NSW] Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of number fields. Grundlehren der Mathematischen Wissenschaften 323, Springer-Verlag, Berlin, 2000. Google Scholar

[S1] Serre, J.-P., Galois cohomology. Springer-Verlag, Heidelberg, 1997. Google Scholar

[S2] Serre, J.-P., Sur la dimension cohomologique des groupes profinis. Topology 3(1965), 413–420. Google Scholar

[SJ] Suslin, A. and Joukhovitski, S., Norm varieties. J. Pure Appl. Algebra 206(2006), no. 1–2, 245–276. Google Scholar

[V1] Voevodsky, V., Motivic cohomology with ℤ/2-coefficients. Publ. Math. Inst. Hautes études Sci. no. 98 (2003), 59–104. Google Scholar

[V2] Voevodsky, V., On motivic cohomology with ℤ/l coefficients. K-theory preprint archive 639, 2003. http://www.math.uiuc.edu/K-theory/0639/. Google Scholar

Cité par Sources :