Homology in varieties of groups. II

The study of (co-) homology groups Z,3(HX A), SnI3(H, A), 3 a variety, 11 a group in 3, and A a suitable 11-module, is pursued. They are compared with a certain Tor and Ext. The definition of the homology of an epimorphism due to Rinehart is shown to agree with that due to Barr and Beck (whenever both are defined). The edge effects of a spectral sequence are calculated. Introduction. In a previous paper [21], henceforth referred to as [HI], we discussed a homology and cohomology theory for a group HI with respect to a variety 3 containing HI, and showed that the theory does not always agree with the appropriate Tor and Ext. We pursue the matter with special attention to modules on which the group acts trivially. In dimension 1 the problem is related to the dimension problem (see the beginning of ?2). Two results of more general homological interest are produced: in ?1 we prove that Rinehart's definition of the homology of an epimorphism agrees with that of Barr and Beck (whenever both are defined), and in ?3 we calculate the edge effects of a well-known universal coefficient spectral sequence. Preliminaries. The principal conventions and definitions of [HI] will be used. In particular, Z denotes a variety containing the group HI; ZrI is a quotient ring of the integral group ring ZrI obtained by dividing out by the images of the Fox derivatives of the words defining 3, as in [HI, ?2]; if A is a rIH-module, then Zn(I, A) and 3n(fl, A) are the triple (co-) homology groups defined in [HI, ?3]; if I'-+ H E 1(, rI)I, Dr is a right ZrI-module which represents Der (F, -) as a functor from right ZrI-modules to Ab. In addition to the conventions listed in the preliminaries to [HI], ,C will denote the variety of nilpotent groups of class at most c, and e the variety of trivial groups. Thus [Y, Z] is the variety " centre by ". If U is a variety, U denotes the verbal subgroup of HI defined by U. 1. Exact sequences. If a: Fo -? F" is a surjection in (Z, HI) and A is a ZHImodule, then there is an exact sequence (1? Z1)O(F, A) -? Z(XF1, A) Mn -,(a, A) (1*1) ~ Mo(a, A) -? S0(F0, A) -? S0(F', A) O-0 Received by the editors September 10, 1970. AMS 1970 subject classifications. Primary 18H40, 20E10; Secondary 18C15, 18G10, 18H40, 18H10, 20J05.