On the second partial Global Euler-Poincare characteristics for Galois cohomology

2025/09/03 by Yufan Luo, Luo, Yufan
Mathematics · #11R32 #11R34 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.03218

openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field, let S be a finite set of primes of K containing all archimedean primes, and let G_K,S denote the Galois group of the maximal extension of K unramified outside S. In this paper, we study the second partial Euler--Poincaré characteristic χ₂(G_K,S,M) for a finite G_K,S-module M, without imposing the condition that the order of M is an S-unit. By adjoining a further finite set of primes of K, which can be chosen to be disjoint from any prescribed set of primes of density zero, we obtain an explicit formula for the corresponding second partial Euler--Poincaré characteristic. As an application, we investigate the presentation of the Galois group G_K,S. Furthermore, for any number field, we construct counterexamples to the dimension conjecture for Galois deformation rings.

Citations

• A structure theorem for subgroups of GL_n over complete local\n Noetherian rings with large residual image
• Über Pro-p-Fundamentalgruppen markierter arithmetischer Kurven

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