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Bloch–Kato pro-p groups and locally powerful groups

  • Claudio Quadrelli EMAIL logo
Veröffentlicht/Copyright: 14. Februar 2012
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 26 Heft 3

Abstract.

A Bloch–Kato pro-p group G is a pro-p group with the property that the 𝔽p-cohomology ring of every closed subgroup of G is quadratic. It is shown that either such a pro-p group G contains no closed free pro-p groups of infinite rank, or there exists an orientationθ:G→ℤp× such that G is θ-abelian. In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d(G)=cd(G), and its 𝔽p-cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups. There are certain Galois-theoretical implications, since Bloch–Kato pro-p groups arise naturally as maximal pro-p quotients and pro-p Sylow subgroups of absolute Galois groups. Finally, we study certain closure operations of the class of Bloch–Kato pro-p groups, connected with the Elementary Type Conjecture.

Keywords: Galois cohomology; Bloch–Kato groups; powerful pro-p groups; absolute Galois groups; Tits alternative; Elementary Type Conjecture
MSC: 20E18; 12G05
Received: 2011-7-8
Revised: 2012-2-1
Published Online: 2012-2-14
Published in Print: 2014-5-1

© 2014 by Walter de Gruyter Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Smooth global solutions for the two-dimensional Euler Poisson system
  3. Primitive ideals in quantum Schubert cells: Dimension of the strata
  4. Group valued null sequences and metrizable non-Mackey groups
  5. The splitting relation for Fréchet spaces over non-archimedean fields
  6. Bloch–Kato pro-p groups and locally powerful groups
  7. On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds
  8. The Ext functor and self-sums
  9. Auslander–Reiten translations in monomorphism categories
  10. An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module
  11. Unitals over composition algebras
  12. Separation of bi-harmonic differential operators on Riemannian manifolds
https://doi.org/10.1515/forum-2011-0069
Schlagwörter für diesen Artikel
Galois cohomology; Bloch–Kato groups; powerful pro-p groups; absolute Galois groups; Tits alternative; Elementary Type Conjecture

Artikel in diesem Heft

  1. Frontmatter
  2. Smooth global solutions for the two-dimensional Euler Poisson system
  3. Primitive ideals in quantum Schubert cells: Dimension of the strata
  4. Group valued null sequences and metrizable non-Mackey groups
  5. The splitting relation for Fréchet spaces over non-archimedean fields
  6. Bloch–Kato pro-p groups and locally powerful groups
  7. On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds
  8. The Ext functor and self-sums
  9. Auslander–Reiten translations in monomorphism categories
  10. An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module
  11. Unitals over composition algebras
  12. Separation of bi-harmonic differential operators on Riemannian manifolds
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