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On extensions of number fields with given quadratic algebras and cohomology

Oussama Hamza

TL;DR

The paper develops a graph-based framework to compute cohomology of pro-$p$ groups and to realize non-analytic mild quotients with higher cohomological dimension. It leverages Grad, Right Angled Artin Algebras, and Koszul duality to relate the graded algebra ${\mathcal E}(G)$ to the cohomology ring $H^{\bullet}(G)$, with ${\mathcal E}(G)$ often identified with a graph-derived algebra ${\mathcal E}(\Gamma)$. For suitable graph decompositions $(\Gamma_{A},\Gamma_{B})$, the cohomology $H^{\bullet}(G)$ is governed by clique counts $c_n(\Gamma)$, and the cohomological dimension equals $\max(2, n_{\Gamma_B})$. The authors construct (non-analytic) Galois quotients with prescribed ramification and cohomological data, providing explicit arithmetic realizations of pro-$p$ groups with controlled cohomology via Galois theory and RAAG techniques. Overall, the work connects graph combinatorics, graded-algebra methods, and arithmetic Galois theory to realize pro-$p$ groups with targeted cohomology and dimension profiles.

Abstract

We introduce a criterion on the presentation of finitely presented pro-$p$ groups which allows us to compute their cohomology groups and infer quotients of mild groups of cohomological dimension strictly larger than two, from (non-free) mild groups. We interpret these groups as Galois groups over $p$-rational fields with prescribed ramification and splitting.

On extensions of number fields with given quadratic algebras and cohomology

TL;DR

The paper develops a graph-based framework to compute cohomology of pro- groups and to realize non-analytic mild quotients with higher cohomological dimension. It leverages Grad, Right Angled Artin Algebras, and Koszul duality to relate the graded algebra to the cohomology ring , with often identified with a graph-derived algebra . For suitable graph decompositions , the cohomology is governed by clique counts , and the cohomological dimension equals . The authors construct (non-analytic) Galois quotients with prescribed ramification and cohomological data, providing explicit arithmetic realizations of pro- groups with controlled cohomology via Galois theory and RAAG techniques. Overall, the work connects graph combinatorics, graded-algebra methods, and arithmetic Galois theory to realize pro- groups with targeted cohomology and dimension profiles.

Abstract

We introduce a criterion on the presentation of finitely presented pro- groups which allows us to compute their cohomology groups and infer quotients of mild groups of cohomological dimension strictly larger than two, from (non-free) mild groups. We interpret these groups as Galois groups over -rational fields with prescribed ramification and splitting.
Paper Structure (15 sections, 3 theorems, 59 equations)

This paper contains 15 sections, 3 theorems, 59 equations.

Table of Contents

  1. Preliminaries on Right Angled Artin Algebras (RAAA)
  2. Introductory results on graphs
  3. Gradation and RAAAs
  4. Proof of Theorem \ref{['gradgroup']}
  5. Decomposition
  6. Monomial analysis
  7. Structure of $\widehat{\mu_{{\mathbf{B}}}}$.
  8. Conclusion
  9. Example
  10. Applications to pro-$p$ groups with quadratic presentation
  11. Proof of Proposition \ref{['compcoho']}
  12. Free pro-$p$ groups
  13. Mild quadratic pro-$p$ group
  14. Pro-$p$ Right Angled Artin Groups
  15. Prescribed and restricted ramification

Key Result

Theorem A

Let $\Gamma:=\Gamma_{{\mathbf{A}}}\sqcup \Gamma_{{\mathbf{B}}}$ be a graph where $\Gamma_{{\mathbf{A}}}$ is bipartite. Then, there exist a totally imaginary field $K$ and a set $T$ of primes in $K$ such that $G_K^T$ is (minimally) presented by generators $\{x_1;\dots;x_d\}$ indexed on the vertices o

Theorems & Definitions (11)

  • Theorem A
  • Proposition 1
  • Theorem B
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 1 more