of class c and FI a finitely generated abelian group. The homological techniques ... The two-sided analogue of Sn will be SeII = Sn*®z Su. If IL, Il ... T-» II e |(S, II)|, then DeY is generated as a right S
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 170, August 1972
HOMOLOGYIN VARIETIES OF GROUPS. IV BY
C. R. LEEDHAM-GREEN AND T. C. HURLEY ABSTRACT.
The study
of homology
groups
=ß„(II, A), 'S a variety,
II a group
in ii, and A a suitable Il-module, is continued. A 'Tor' is constructed which a better (but imperfect) approximation to these groups than a Tor previously sidered, ii (II, Z) is calculated for various varieties 3i and groups II.
Introduction.
We continue
the study
of homology
in the variety
3} and
[31], here
after
to as [H i], [H II] and [H III].
referred
were compared two theories
gives
consideration
a better
group.
II as above,
in the two cases
be exact esting
33 (n, A), which
neither
The principal
case
of metabelian
conventions
and notations
(s; ?2., - • • , 72 ) if it is the direct
2 < t, and
generated
one of order
r = s + t. A cyclic
Received
by the editors
much
group
abelian
above
enables
product
generated
agrees
of s infinite
Orzech
to
inter-
that
of trivial
action,
theory. [H III]
will
to be of rank
cyclic
n . divides
by the element
with
and all centre-
is particularly
be said
i, where
of class
the results
the calculations
of Grace
will
used
In §3 we repeat
of [H I], [H II] and
72. for each
FI a finitely
always
action.
for
as those
groups
as in the classical
group
from the
and was prompted
c and
from 3} (JJ, Z) in the case group
a different
Comparing
groups
the
approximately
of all metabelian
from a theorem
type
groups,
mentioned
with trivial
it follows
in force.
cyclic
Tor
that
arises
of all nilpotent is harder.
The variety
as an obstruction
A finitely
theory,
groups
shown
are the same
in the homology
main
finite
here
A is a module
can be calculated
may be interpreted
theory
A refinement
cases.
in this
that
when
theory
of class
techniques
for 3] the two varieties
groups.
in these
in that
that are nilpotent
the group
we deduce
calculations,
by-metabelian
groups
homology
(n, Z) is calculated
3} (n, Z) for 5} the variety
3] (II, A) if 72= 2, even
this
as in the Hochschild
The homological
though
These
EÍ
^251 and
1. In § 1 we introduce
to 3} (n, A);
In §2,3}
33 (IT, A), where as in [2ll,
[H II], and it was
in dimension
modules
of metabelian
abelian
in [H III] to calculate
the above
agree
of Barry Mitchell's.
3} the variety
Fl-module,
in [H i] and
approximation
of two-sided
by a remark
c and
Tor
do not always
Tor which
generated
with a certain
A is a suitable
groups
is a group
gives con-
a will
groups, 72
for
rer and
and
t
1
