Cohomological Dimension and Schreier's Formula in Galois Cohomology

Labute, John ; Lemire, Nicole ; Mináč, Ján ; Swallow, John

Canadian mathematical bulletin, Tome 50 (2007) no. 4, pp. 588-593

Résumé

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 ${{\dim}_{{{\mathbb{F}}_{p}}}}{{H}^{n}}\left( H,\ {{\mathbb{F}}_{p}} \right)$ .

Détail

DOI : 10.4153/CMB-2007-056-3

Mots-clés : 12G05, 12G10, cohomological dimension, Schreier's formula, Galois theory, p-extensions, pro-p-groups