On the functorial homology of Abelian groups (Q1124972)

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 \(\Omega_qB\).