On the second partial Global Euler-Poincare characteristics for Galois cohomology
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916930956296192 |
|---|---|
| author | Luo, Yufan |
| author_facet | Luo, Yufan |
| contents | 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-Poincare characteristic $χ_{2}(G_{K,S},M)$ 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 $χ_{2}(G_{K,S},M)$. As an application, we investigate the presentation of the Galois group $G_{K,S}$. Furthermore, we construct counterexamples to the dimension conjecture for Galois deformation rings over all number fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03218 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the second partial Global Euler-Poincare characteristics for Galois cohomology Luo, Yufan Number Theory 11R34, 11R32 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-Poincare characteristic $χ_{2}(G_{K,S},M)$ 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 $χ_{2}(G_{K,S},M)$. As an application, we investigate the presentation of the Galois group $G_{K,S}$. Furthermore, we construct counterexamples to the dimension conjecture for Galois deformation rings over all number fields. |
| title | On the second partial Global Euler-Poincare characteristics for Galois cohomology |
| topic | Number Theory 11R34, 11R32 |
| url | https://arxiv.org/abs/2509.03218 |