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Demushkin groups of uncountable rank

Tamar Bar-On, Nikolay Nikolov

Abstract

We extend the theory of countably generated Demushkin groups to Demushkin groups of arbitrary rank. We investigate their algebraic properties and invariants, count their isomorphism classes and study their realization as absolute Galois group. At the end, we compute their profinite completion and conclude with some results on profinite completion of absolute Galois groups.

Demushkin groups of uncountable rank

Abstract

We extend the theory of countably generated Demushkin groups to Demushkin groups of arbitrary rank. We investigate their algebraic properties and invariants, count their isomorphism classes and study their realization as absolute Galois group. At the end, we compute their profinite completion and conclude with some results on profinite completion of absolute Galois groups.
Paper Structure (5 sections, 44 theorems, 4 equations)

This paper contains 5 sections, 44 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Algebraic properties of Demushkin groups of uncountable rank
  4. Demushkin groups of uncountable rank as absolute Galois groups
  5. Profinite completions of Demushkin groups

Key Result

Theorem 1

Let $q\ne 2$ be a prime power or equal to 0, and $\mu>\aleph_0$. For every nondegenerate skew symmetric bilinear form $(V,\varphi)$ of dimension $V$ there is a Demushkin group $G$ of rank $\mu$, with $q(G)=q, s(G)=0$, and whose cup-product bilinear form $H^1(G)\cup H^1(G)\to H^2(G)$ is isomorphic to

Theorems & Definitions (73)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • proof
  • Proposition 9
  • ...and 63 more