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Injectivity theorems and algebraic closures of groups with coefficients
Author(s) -
Cha Jae Choon
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm033
Subject(s) - mathematics , isomorphism theorem , quotient , torsion (gastropod) , algebraic number , isomorphism (crystallography) , homology (biology) , pure mathematics , discrete mathematics , algebra over a field , mathematical analysis , medicine , biochemistry , crystal structure , chemistry , surgery , gene , crystallography
Recently, Cochran and Harvey defined torsion‐free derived series of groups and proved an injectivity theorem on the associated torsion‐free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain homology localization of groups. In order to give a concrete combinatorial description and existence proof of the necessary homology localization, we introduce a new version of algebraic closures of groups with coefficients by considering certain types of equations.