On the functorial homology of Abelian groups
type: Item
instance of: scholarly article
title: On the functorial homology of Abelian groups
zbMATH Open document ID: 0942.20038
DOI: 10.1016/S0022-4049(98)00140-6
author: Lawrence Breen
published in: Journal of Pure and Applied Algebra
publication date: 2000-06-05T00:00:00Z
review text: A functorial filtration on the integral homology of an Abelian group \(B\) such that the associated graded pieces are the values at \(B\) of the left derived functors \(L_i\Lambda^j(-,0)\) of the exterior algebra functors \(\Lambda^j\) is constructed. It is shown that \(L_i\Lambda^j(B,0)\) is isomorphic to the group of elements in \(\text{Tor}_i(B,\underset {j}\ldots,B)\) which are anti-invariant under the natural action of the symmetric group \(\Sigma_j\). A presentation of particular interest is given for the groups \(L_{i-1}\Lambda^j(B,0)\) which are natural generalizations of the group \(\Omega B\) introduced by Eilenberg-MacLane. Finally, this is illustrated by a functorial description of the integral homology in degrees \(\leq 5\) of an Abelian group \(B\) which sheds a new light on some computations of \(H_i(B)\) made by Hamsher for all \(i\). Here are some of the main results: Theorem 4.7. For each \(q>0\) and for all integers \(i\), the map \(L_ij_B^q\colon L_i\Lambda^qB\to\text{Tor}_i(B,\ldots,B)\) induced by \(L_ij_B^q\colon L_i\Lambda^qB\to B\otimes_L\cdots\otimes_LB\) is injective and determines an isomorphism \(L_i\Lambda^qB\to\text{Tor}_i(B,\ldots,B)^{\Sigma_q^\varepsilon}\) between \(L_i\Lambda^qB\) and that portion of \(\text{Tor}_i(B,\ldots,B)\) which is anti-invariant under the permutation action of the symmetric group \(\Sigma_q\). Theorem 5.12. For every integer \(q>0\), the map \(\lambda^q\) is an isomorphism onto the torsion part of \(L_{q-1}\Lambda^q(B,0)\). Furthermore, \(\lambda^q\) induces for every \(\nu\) an isomorphism \(\lambda^q\colon\lim_{J_k}\Gamma_q(_nB)\to_\nu\Omega_qB\) between the corresponding limit over the full subsystem \(J_\nu\) of \(J\) in which all integers \(s\) and \(n\) divide \(\nu\), and the \(\nu\)-torsion subgroup of \(\Omega_qB\).
Mathematics Subject Classification ID: 20K40
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 18G40
zbMATH DE Number: 1371470
zbMATH Keywords: integral homology groups
zbMATH Keywords: Künneth formula
zbMATH Keywords: Quillen spectral sequences
zbMATH Keywords: associated graded components of filtered groups
zbMATH Keywords: Bousfield-Quillen décalage formula
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
full work available at URL: s0022-4049(98)00140-6
OpenAlex ID: W2014505932
Wikidata QID: Q127098585
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description: scientific article; zbMATH DE number 1371470