On the functorial homology of abelian groups

We construct a functorial filtration on the integral homology of an abelian group B, for which 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). It is shown that L-i/Lambda(j)(B, 0) is isomorphic to the group of elements in Tor(i)(B,..(j).,B) which are anti-invariant under the natural action of the symmetric group Sigma(j). A presentation is given for the groups L(j-1)Lambda(j)(B, 0), which are of particular interest, since they are natural generalizations of the group Omega B introduced by Eilenberg-Mac Lane [8]. This is illustrated by a functorial description of the integral homology in degrees less than or equal to 5 of an abelian group B, and sheds new light on the computations by Hamsher of H-i(B) for all i [12]. (C) 1999 Elsevier Science B.V. All rights reserved.