Localization and cohomology of nilpotent groups

Abstract

Cahiers de topologie et géométrie différentielle catégoriques, tome 14, n o 4 (1973), p. 341-369 <http://www.numdam.org/item?id=CTGDC_1973__14_4_341_0> © Andrée C. Ehresmann et les auteurs, 1973, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

On a theorem of Schur

“…Schur's theorem, and its converse, take on a particular significance in the localization theory of nilpotent groups [1]. For it is one of the main problems in that theory to calculate the Mislin genus Ᏻ(G) of an f g nilpotent group G and to identify its members.…”

“…In seeking conditions under which e 0 is Q-surjective-and again in proving Lorensen's theorem (Theorem 2.6), we need to apply a basic result in [1], namely, Theorem 6.1. We quote that result here as Lemma 2.2.…”

Localization and Genus in Group Theory

“…If up(x) = 1, pick n £ P' such that x" = a(y). Given a set of primes F containing (2. fit each group 67 in G into an exact sequence 1 -Nx ~ Gk -^ -1, where A^is the kernel of the F-localizing map 6^ -» LpGx-The directed diagram G yields a directed diagram of such short exact sequences, and the resulting sequence of colimits is exact (see [10]):…”

Various local global principles for abelian groups

“…For details, see [3], [4] A homomorphism of nilpotent groups e : G~H is said to P-localiz e if and only if (i) H is a P-local group ; i .e . for every prime p P, the functio n H h~--~ h P E H is a bijection ; (u) the kernel of e consists of all those torsion elements of G whose arder has no divisior in P ;…”