Right exact localizations of groups. (English) Zbl 1476.20053
A localization in a category \(\mathcal{C}\) is an endofunctor \(L : \mathcal{C}\to \mathcal{C}\) together with a natural transformation \(\eta : \mbox{Id}\to L\) such that \(\eta L = L\eta : L\to L^2\) is an isomorphism. An object \(C\in\mathcal{C}\) is local with respect to a localization \(L\) if the morphism \(\eta_C : C \to LC\) is an isomorphism.
The authors study localizations in the category of groups and the most interesting class for them is the class of localizations which coincide with their zero derived functors called right exact (in the sense of F. Keune [J. Algebra 54, 159–177 (1978; Zbl 0403.18009)]). It is proved that a right exact localization \(L\) preserves the class of nilpotent groups and that for a finite \(p\)-group \(G\) the map \(\eta_G : G \to LG\) is an epimorphism. Furthermore, it is shown that G. Baumslag’s \(P\)-localization with respect to a set of primes \(P\) [Commun. Pure Appl. Math. 18, 25–30 (1965; Zbl 0136.01204)], A. K. Bousfield’s \(HR\)-localization [Homological localization towers for groups and \(\Pi\)-modules. Providence, RI: American Mathematical Society (AMS) (1977; Zbl 0364.20058)] and others are right exact.
At the end, a conjecture of E. Dror Farjoun about Nikolov-Segal maps [N. Nikolov and D. Segal, Invent. Math. 190, No. 3, 513–602 (2012; Zbl 1268.20031)] is discussed and a very special case of this conjecture is proved.
The authors study localizations in the category of groups and the most interesting class for them is the class of localizations which coincide with their zero derived functors called right exact (in the sense of F. Keune [J. Algebra 54, 159–177 (1978; Zbl 0403.18009)]). It is proved that a right exact localization \(L\) preserves the class of nilpotent groups and that for a finite \(p\)-group \(G\) the map \(\eta_G : G \to LG\) is an epimorphism. Furthermore, it is shown that G. Baumslag’s \(P\)-localization with respect to a set of primes \(P\) [Commun. Pure Appl. Math. 18, 25–30 (1965; Zbl 0136.01204)], A. K. Bousfield’s \(HR\)-localization [Homological localization towers for groups and \(\Pi\)-modules. Providence, RI: American Mathematical Society (AMS) (1977; Zbl 0364.20058)] and others are right exact.
At the end, a conjecture of E. Dror Farjoun about Nikolov-Segal maps [N. Nikolov and D. Segal, Invent. Math. 190, No. 3, 513–602 (2012; Zbl 1268.20031)] is discussed and a very special case of this conjecture is proved.
Reviewer: Marek Golasiński (Olsztyn)
MSC:
| 20J15 | Category of groups |
| 20J05 | Homological methods in group theory |
| 18C15 | Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads |
| 18G10 | Resolutions; derived functors (category-theoretic aspects) |
| 18E35 | Localization of categories, calculus of fractions |
| 55P60 | Localization and completion in homotopy theory |
Keywords:
\(HR\)-localization; nilpotent group; \(P\)-localization; \(p\)-local group; right exact localization functorReferences:
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