homology in varieties of groups. iv 293 - American Mathematical Society

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of class c and FI a finitely generated abelian group. The homological ... The two-sided analogue of Sn will be SeII = Sn*®z Su. ... SeII = DeY, ..... [12, Theorem ..... DEPARTMENT OF MATHEMATICS, IMPERIAL COLLEGE OF SCIENCE AND ...
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



^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