An arithmetic site for the rings of integers of algebraic number fields
type: Item
instance of: scholarly article
title: An arithmetic site for the rings of integers of algebraic number fields
zbMATH Open document ID: 0867.11072
author: Alexander Schmidt
published in: Inventiones Mathematicae
publication date: 1997-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/144360
review text: Let \(K\) be an algebraic number field and \(S\) a finite set of primes of \(K\) containing the infinite primes and the primes dividing a given prime number \(p\). Let \(G_S (p)\) be the Galois group of the maximal \(p\)-extension of \(K\) unramified outside \(S\). The author presents a geometric study of the group \(G_S (p)\) and generalizes results of \textit{K. Wingberg} [Invent. Math. 77, 557-584 (1984; Zbl 0563.12008)]. The basis of this study is the construction of a Grothendieck topology (called by the author positive topology) on \(\text{Spec}(O_K)\). There is a natural isomorphism of \(G_S(p)\) onto the pro-\(p\)-fundamental group of the open subscheme \(\text{Spec}(O_{K,S})\) of \(\text{Spec}(O_K)\) furnished with the positive topology. If \(K\) is an abelian number field containing the \(p\)th roots of unity, the author proves the following result, which in fact is valid in much greater generality: Let \(p\neq 2\) and \(G\) the positive pro-\(p\)-fundamental group of \(\text{Spec}(O_K)\). If the \(p\)-genus \(g_p(K)\) is greater than zero, then \(G\) is a Poincaré pro-\(p\)-group of dimension 3 with dualizing module \(\mu_{p^\infty}\). If \(G_p(K)=0\) then \(G\) is a free pro-\(p\)-group of rank \(r\) with \(1\leq r\leq 1+r_2(K)/2\), where \(r_2(K)\) is the number of complex places of \(K\).
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 922463
zbMATH Keywords: rings of integers
zbMATH Keywords: algebraic number field
zbMATH Keywords: Galois group of the maximal \(p\)-extension
zbMATH Keywords: Grothendieck topology
zbMATH Keywords: positive topology
zbMATH Keywords: pro-\(p\)-group
reviewed by: Helmut Koch
MaRDI profile type: Publication
OpenAlex ID: W4234694931
cites work: Q5609064
cites work: Q3972724
cites work: Explicit deformation of Galois representations
cites work: Q3922829
cites work: A Kurosh Subgroup Theorem for Free Pro--Products of Pro--Groups
cites work: On the cohomology of certain topological colimits of pro-C-groups
cites work: Q5602063
cites work: Q3208765
cites work: Q3871006
cites work: Q3754095
cites work: Local extensions associated with \(\ell\)-extensions of number fields with bounded ramification
cites work: Freie Produkte pro-endlicher Gruppen und ihre Kohomologie
cites work: ON $ p$-CLOSED ALGEBRAIC NUMBER FIELDS WITH RESTRICTED RAMIFICATION
cites work: Positively ramified extensions of algebraic number fields.
cites work: Cohomologie galoisienne. Cours au Collège de France, 1962--1963. Seconde édition.
cites work: Galois cohomology in unramified extensions of algebraic function fields
cites work: On Galois groups of p-closed algebraic number fields with restricted ramification.
cites work: On Galois groups of p-closed algebraic number fields with restricted ramification II.
cites work: Ein Analogon zur Fundamentalgruppe einer Riemann'schen Fläche im Zahlkörperfall
cites work: Positiv-zerlegte p-Erweiterungen algebraischer Zahlkörper.
DOI: 10.1007/S002220050041
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description: scientific article; zbMATH DE number 922463