Page 1

JOURNAL OF ALGEBRA

85, 51-75 (1983)

n the Structure

Modules

Soluble

of Cert

over Group

Groups of Finite

IAN M. Musso~

Department of Mathematics, University of Wisconsin. Madison, Wisconsin 53706

Communicated by Graham Higmarz

Received November 18, 198 I

INTRODUCTION

Let k be a field of characteristic zero and G a finite extension of a torsion-

free soluble group of finite rank. The aim of this paper is to give a

description of the injective hull E,,(k)

various properties of this module. S. Donkin has, shown in [2] that if G is

poiycyclic-by-finite and V is a finite-dimensional

an artinian module whose endomorphism ring is Noetherian. These results

are deduced from the fact that the Hopf algebra .FO(G, k) of finitary

functions in quasi-affine [2, Theorem 1.6.5] and th.e main results of ] 3 1~

show that if G belongs to the more general class of groups indicated then

again .FO(G, k) is a quasi-affrne Hopf algebra (Theorem A). This ailows us to

draw the same conclusions concerning E&V).

For our other results we assume that G has a nilpotent normal subgroup N

such that G/N is tinitely generated free abelian.

In Theorem B we show that there is a tinitely

polynomial algebra k[X] on which

automorphisms, such that E&k) E k[X]

words for all 4,: 0, E k[X] and g E G we have

of the trivial module k and to study

kG-module,’ then E,,(V) is

generated (comm~tativc)

G acts as a group

as (right) I&modules.

of algebra

In other

In addition the generators of k[X] lie in a finite-dimensional

k[X] and so the module action on k[X] is determined by the action on such

a submoduie.

This description of E,,(k) is particularly

socle series. We introduce a weight function on monomials in k[X] such that

E, = socg’(k[X]) the tth term of the ascending secular series of k[%] is

sparmed by monomials of weight at most f (Theorem C). By counting

51

submodule of

well suited to the study of its

0021-8693/83 33.00

Copynght C 1983 by Academic

of reproduction

Press, Inc

Ali rights in any form reserved.

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52 IAN M.MUSSON

monomials of fixed weight we are able to compute the dimensions of the E,

as vector spaces. The weight function

fit = Fit(G) f? C,(E,)

which have

dimension subgroups.

In the first section we prove the nilpotent

involves extending work of Hall [6, Sect. 71 and Jennings [S] on filtrations

of the augmentation ideal to the case of nilpotent groups of finite rank. In

Sections 2 and 3 the general cases of Theorems B and C are proved together

by induction on the Hirsch number of G/N.

In Section 4 we study group theoretic properties of the subgroups o”, and

composition factors of E&k).

In general if r is a finite extension of a torsion free soluble group of finite

rank, then r has a subgroup G of finite index such that G/N is finitely

generated free abelian, where N is the Fitting subgroup of G. If E = E,,.(k),

then by Lemma 6.2 of [3] and induction on t socF’(E) = socl;“(E). Since the

restriction of E to kG is injective by Lemma 2.1 of ] 131, we deduce that E is

isomorphic to E,,(k) as a kG-module. Also we can apply Theorem C to

compute the dimensions of socr’(E).

is determined by the subgroups

certain features in common with

case of Theorem B. This

1. TORSION-FREE NILPOTENT GROUPS OF FINITE RANK

If G is a torsion-free nilpotent group of finite rank, then the augmentation

ideal g has the AR (Artin-Rees) property by Theorem C of [ 191. Thus if

E = E,,(k), then E = lJ E,, where E, = ann, g’. Moreover E, is the injective

hull of k as a kG/g’-module.

We claim that E, z (kG/g’)* as kG-modules.

(kG/g’)** s kG/g’

is a free kG/g’-module

submodule. It is also easy to see that Et is the tth term of the ascending

secular series of E.

If G is finitely generated the dimensions of the augmentation quotients

Llt/!f+

were computed by Hall [6, p. 561, see also [ 16, Theorem 3.4.101. We

develop a similar theory for the finite rank case. This is perhaps the most

appropriate setting for such a theory, since if G is any group, then n, gt = 0

if and only if G is residually (torsion-free nilpotent) by Theorem VI.2.26 of

[ 151. Hence if G is torsion free nilpotent

filtration of kG given by the powers of g. However, if we require also that

g/g* g G/G’ @ pk has finite dimension, then G/G’ has finite rank and conse-

quently so does G by Theorem 2.26 of [ 171.

For application in Section 3 we work in the more general context of No-

sequences adapting the arguments and notation of [ 161 as far as possible. A

sequence

G=G,zG,z...

This follows since

and has a unique maximal

it is of interest to study the

‘>Gd+,=l

(1.1)

Page 3

THE STRUCTURE OF CERTAIN INYECTWE MODULES

53

is an NO-sequence if G/G, is torsion free and (Gi, Gj) c Giij for all i and j.

For instance, the dimension subgroups of G form an NO-sequence by Lemma

3.32 of [I6].

For g E 6, g # 1 we write v(g) = m if g E G,\G,,

want to consider the filtration of g associated with (I. I )* For each k > 1, let

I, denote the subspace of kG spanned by elements

, and v(I) = co. We

(81 - l>(g* - 1) ... (8, - 1)

with 2 v(g,) > t.

Then I, is an ideal of kG since it is closed under multiplication

and 1 E kG.

If G is a torsion-free nilpotent group of finite rank, x E G and yp = x for

some p E L, y E 6, we can write y = xllp since an element of G has at most

one pth root. This identifies the isolator of (x> in G with a su

additive group of rationals. If H is a finitely generated subgroup of G we set

Hi=Gif7Hso that

by (g - 1)

is an NO-sequence of H. Using Lemma 34.4 of ] 16j we refine this to a

normal series

with ei/Ei+,

We choose an element xi in H whose image generates the factor L,/Li.

Then elements of H may be written uniquely in the form

infinite cyclic and central in H/Li, i .

,

x = X”,Xbz . . . xbn

1 2 n

with bj E 27.

We say the subgroup H is dense in G if every element of G may be written

uniquely in the form

x = Xblxbz . . .

1 2

X”,f?

(i.2)

for certain rational exponents b,. We adopt this terminology because if H is

represented as a group of unipotent matrices over H, then G is isomorphic to

a subgroup of the Zariski closure of H over Q.

LEMMA

has a g%niteZy generated dense subgroup.

1.1. If G is a torsion-free nilpotent group of$nite rank, then G

ProoJ:

isomorphic to additive subgroups of Q. Let Z be an isolated subgroup of the

centre of G having rank 1. By induction G/Z has a finitely generated dense

subgroup H,/Z generated by x,Z,..., x,- 1 Z. Let Hi be the subgroup of G

We use induction on the length of a central series with factors

Page 4

54

IAN M. MUSSON

generated by x, ,..., x,-, . If H, n Z # 1, this intersection is infinite cyclic

and we choose X, as a generator. If H, n Z = 1, let X, be any nontrivial

element of Z. It is now easy to see that H = (H,, xn) is a dense subgroup of

G.

We remark that if H is a finitely generated dense subgroup of G, and K a

finitely generated subgroup of G containing H, then a comparison of Hirsch

numbers shows that 1 K: HI < CO.

We use the representation (1.2) to show that 1,/I,+,

basis consisting of (images of) elements of the form

has a vector space

~(a,, a2,..., a,,) = (x, - 1)“’ (x2 - 1)“’ ... (x,, - l)“~?,

where the a, are nonnegative integers and wt(r) = Cr= 1 aiv(xi) = t.

LEMMA 1.2.

I /+l’

ProoJ:

in (1.2) with bi E Q.

Suppose bi =p/q, y = xpi and C = (x1’“). Then yy = x7 and

The elements q(a, ,..., a,,) with wt(q) < t span kG modulo

Every element of G may be written in the form x = x:1 ... xi’,11 as

(y9- l)=q(y- 1) modc2

--(xi - 1) mod c2.

Therefore (y - 1) =p/q(xi

us to express (y - 1) as a linear

(xi - 1)2,..., (xi - 1)’ mod cr+‘.

Now use this to expand x as a linear combination of q’s mod g”’

and delete from this expansion any r’s with wt(q) > t + 1.

- 1) mod c2 and repeating this argument allows

combination of elements (xi - l),

z I,, ,

For the proof of linear independence of the q’s we need the following

variant of [ 16, Lemma 3.4.121:

LEMMA 1.3.

nilpotent group K and suppose that K = K, 2 K, 2 1.. 2 Kd+ , = 1 is an NO-

sequence of K. Let Hi = Ki f7 H so that {Hi} is an NO-sequence of K, and let

{V,}, {IV,} be corresponding filtrations

kHf? W,= V,.

Vt cl kH n W, is trivial. Each Hi/H,+ 1 can be

regarded as a subgroup of the free abelian group Ki/Kj+,

both are equal. The fundamental theorem on abelian groups allows us to

write both as product of infinite cyclic groups in such a way that each

generator in Hi/Hi + 1 is a power of some generator of K,/K,+, .

Let H be a subgroup of finite index in a finitely generated

of lj and f, respectively. Then

ProoJ

The containment

and the ranks of

Page 5

THE STRUCTURE OF CERTAIN INJECTWE MODULES 55

IIn this way we obtain generators {y, ,..., y,) of

whose images form a basis of the appropriate quotie

K UCY,,+i f~or some s(i) > 1. (Note that if h E

h E K,, so the notation v(h) is unambiguous).

Now suppose that a E kH f? W,. Then we may write a as a sum of

products n= (x1 - 1)“” e.3 (x, - 1)“~ mod V, by Lemma 1.2.

Let ZI = mm{ n@(n) 1 7c E supp a mod V,}. We need to show that 2: > 1.

since this will give 71 E V,, so a E V, .

e have ySCi’g = xi for some g E KDCYi,+, .

then h E H, if and only ff

Xi - 1 = (Jp’g- 1)

s s(i)(yi - 1) mod WI.,,,,,+ I.

From this it follows that

7155 s(l)“1 *. * s(np

(y, - 1)“’ . I * (y,, - 1)“”

mod W,, ,

if wt(7r) = u.

Now write a = C b,rc(a, ,..., a,) mod V, and suppose 71 E supp a satisfies

wt(n) = v ( t, where 7r(a, ,..., a,) = (x:1 - 1)(,x, - 1)“’ .‘L (x, - 1)“j7. Then

v+l<t and

since

aE W,< WL>&]

s(n)On (yr - I)“’ ... (y, - 1)“” E W,, , . Hence by 116, Lemma 3.4.8 ]

we

have 2 b,s(l)“’ .~.

7c= (y, - 1)“’ **. (y,-

1pE w,,,,.

Therefore wt(n) > u + 1, a contradiction which proves the lemma.

THEOREM 1.4.~.

r = u(a, 5.S~)

kG mod I,.

With

OUY

previous flotation

wt(v) < t form

the

elements

a basis of

a,)= (x, - l)“l ... (x,_~)‘~

with

Proojr:

Suppose that a = C k, q(al ,..., a,) E I, where the sum is over q(a, ,..., a,,)

with wl(q) < t. Then we can write a as a sum of products of the form

z= (g, - 1) ... (g, - 1) with C v(g,) > t. Let K be the subgroup of G

generated by H and all the gi)s that occur here. Then M is finitely generated

and so by choice of H, 1 K: HI < co. Let { Wf}, { Vi} be the filtrations

with respect to the NO-sequences K, = G, n K and Hi = Gj n H of K and H,

respectively.

Then we have a = 2 k,v(a, ,..., a,) E W,.

f’ W,= %; by Lemma 1.3.

Since each PJ satisfies wt(q) < t this gives k, = 0 for all a = (CJ! ,..., a,,) by

j16, Lemma 3.4.81.

We need to change our notation somewhat in order to derive our results

By Lemma 1.2 these elements span I&/I,.

of t, h

owever, each y lies in

Page 6

56

IAN M.MUSSON

about injectives. First, it is clear from the structure of the group G that every

element x of G may be written uniquely in the form

x=xbn n .-. X;~X;I = x(b, , b, ,..., b,),

(1.3)

where the exponents bi are certain rational

generators xi in the reverse order to Eq. (1.2).

We consider the elements

numbers; that is, with the

e(a,, a2 ,..., a,) = (x,~ - 1)“” (x,-r - l)““-’ ... (x, - I)“’

and define wt(0) = C ai v(xi).

The result corresponding to Theorem 1.4.~ is as follows:

THEOREM 1.4.8. The elements 8 = B(a, ,..., a,,) with wt(r3) < t form a

basis of kG mod I,.

We now return to the description of E = E,,(k) given at the beginning of

this section. We have E = U E,, where E, z (kG/g’)*. Defining weights of

elements of G with respect to the dimension subgroup sequence, we have that

kG/g’ is spanned by elements of the form O(a, ,..., a,) with wt(r3) < t. We

denote by B*(a, ,..., a,) the corresponding element of the dual basis. We can

regard the elements 8* as linear functions from kG to k vanishing on g’,

where t = wt(0). The module action is given by left translation, i.e., for

8* E E, g, h E G we have

(@“g)(h) = e*(gh).

We introduce functions X,, X, ,..., X,, as follows: if x E G is written as in

(1.3) we set X,(x) = bi. We regard Hom,(kG, k) as an algebra with pointwise

operations, i.e., for #1, & E Hom,(kG, k), g E G

($4 + h)(g) = i,(s) + h(g),

and as a right kG-module via left translation.

Denote by R (temporarily)

the Xi. For an indeterminate X and nonnegative integer a we write (z) for

X(X- 1) S-a (X-a + 1)/a!.

the subalgebra of Hom,(kG, k) generated by

LEMMA 1.5. We have

Q*(a,, a2,..., a,) = fi (2)

i=l I

as functions.

(1.4)

Page 7

THE STRUCTURE OF CERTAIN l[NJECTIVE MODULES

57

ProoJ:

as part of the dual basis of kG/g’ and so vanishes on gf. The proof involves

writing an element x as in (1.3) modulo 9’ as a linear c~rnbi~at~o~ of

elements B(c, ,*.., c,) and showing that the coefficient of @(a, ,..~) a,) can be

obtained by substituting bi for Xi in the product on the right of (1.4).

To do this we use induction on the number of nonzero ~1:s. Hf c&i = 0 for all

i, then QO, O,..., 0) = 1 and this case is easily dealt with.

Suppose that 0 = 19(a, ,..., a,) satisfies wt(i9) < t. Then 8* occurs

If only one ai is nonzero, we can simplify notation by su~~osi~g % is

isomorphic to an additive subgroup of Q, (x) is an infinite cyclic subgroup

of G and y = xb for b E Q.

We must find the coefficient of 6 = (X - I)” In the expansion of y mod a’,

ere satisfies wt(8) = a < t. The elements (x - l)j, 0 ,< i < t - 1, form a

basis for I&/Q’ by Theorem 1.4.8, so we can write

f-1

y = Xb = 1 /$(x - l)i + a,

i=O

where a E 9’.

(1.5)

We must find /I,. The idea is to differentiate this equation a times with

respect to x and then set all powers of x equal to 1.

Let b =p/q, H = (XI/~) and let K be a finitely generated subgroup of G

containing M such that a E f’. Clearly 01 E kN by (1.5) so a E 0’ by Lemma

1.3. Therefore by Lemma 3.4.5 of [ 151 01 is a sum of terms of the form

Using the product rule for differentiation

derivative of such terms vanishes at x’lq = 1.

Therefore

it is easy to see that the ~th

and this gives the correct value for A, .

To treat the general case suppose that ak # 0, but cli = 0 for t < k. Then

QO,..., 0, Qk,..~, a,) = 8(0 )...) ak+],..., a,) e(0 )...) Uk. o,..., 0).

bj = 0 for i < k and

e can assume

qo,..., 0, ak+ ] )~..) a,) f 1.

Write x = x(0 ,..., 0, b, ,..., b,) as x = x’xi,

andb=b,.Thenx-1=(x’-1)(x:-1)+(x’-1)+(x:-14).

Now the coefficient of 0(0 ,..., ak ,..., a,,) in the expansion of x is equal to

uct of the coefficient of $(O ,..., 0, akr, ,..., a,) in (x’ - I) and the

coefficient of B(0 ,..., 0, ak, 0 ,..., 0) in (x”, - 1). We have shown that (1.4)

holds.

where X’ = x(0,..., 0, 6,_ I ).*., b,)

Page 8

58

IAN M.MUSSON

Formula

element of R. Conversely we show how every element of R may be expressed

in terms of the 19*‘s.

The algebra R is spanned by monomials of the form

(1.4) shows how to express any function 19” E E,,(k) as an

m = m(a, ,..., a,) = Xzn ..a Xi*X;ll.

We set wt(m) = C aiv(Xi), where v(X,) = v(xi). Then we have to show that

the subspace of R spanned by 8*‘s with wt(0) < t is equal to the subspace

spanned by monomials m with wt(m) < t. This reduces to the case of an

additive subgroup of the rational numbers.

LEMMA 1.6. If G is an additive subgroup of the rational numbers, then

1,X ,..., Xf and 1,X,X(X-

1)/2 ,..., (X(X-

same subspace of Hom,(kG, k).

1) ..a (X-(t-

l)))/t!

span the

Proof. An easy induction.

Another consequence of Lemma 1.6 is that the monomials in X, ,..., X,, are

linearly independent elements of R, since the 8*‘s are linearly independent.

Therefore R = k[X] is just the polynomial

we have proved the following result:

algebra in X, ,..., X,, . In summary

THEOREM B (Nilpotent

Jinite rank, then E,,(k) E k[X] a finitely generated polynomial subalgebra of

Hom,(kG, k) on which G acts by left translation.

Case). If G is a torsion free nilpotent group of

It is easily checked that G acts as a group of algebra automorphisms on

WI.

Since we have identified

k[X]

k[X] = {f E Hom,(kG, k) / f(g’) = 0 f or some t}. We show in the next

section that k[X] carries a Hopf algebra structure in the nilpotent case.

with U, (kG/g’)*

it follows that

2. TORSION-FREE SOLUBLE GROUPS OF FINITE RANK

In this section we extend the results on the structure of E,,(k) for G

torsion-free nilpotent of finite rank, to the case where G is a finite extension

of a torsion-free soluble group of finite rank. Any such group G has a

normal nilpotent subgroup N such that G/N is finitely generated abelian-by-

finite (see the remarks [20, p. 251). Our strategy is to use induction on the

Hirsch number of G/N as in [2].

The first lemma extends a result of Brown, Section 3 of [ 11, see also 12,

Theorem 1.1.11. The proof given is due to R. L. Snider.

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THE STRUCTURE OF CERTAIN INJECTIVE MODULES

59

LEMMA 2. H. Let G be a finite extension of a torsionTree soluble group of

finite rank, k a field of characteristic zero and V a ~~~te-d~rne~~ion~~ kG-

module. Then E,,(V) is locally finite, that is, ezjery finitely generated

submod~le is finite dimensional.

BlsoS.

Once this has been established the result for E,JV)

follows: For a kG-module M there is a natural isomorphism of vector spaces

Hom,(M, V@ E) g Horn&M@

V*,E).

functors EIom,(-,

V 0 E) + Hom,(-

om,(--, E) are exact so Horn&,

V @ E) is exact. Therefore V @ E is

injective, and so contains a copy of E&V).

locally finite for any finite-dimensional

is locally finite.

Let M be a finitely generated submodule of E. y passing to a subgroup of

finite index we may assume G is torsion free. Th

of regular elements of kN. Moreover S is an Ore set in kN and S- ‘KN is

Noetherian. For the verifications of these statements we refer to Theorem C’,

Lemma 2.2 and Corollary Cl of [ 191. Since S is invariant under conjugation

by G it follows that S is an Ore set in kG. Also R = S-‘kG is an iterated

skew polynomial ring over S-‘kN and so is Noetherian.

Let t(E) = {e E E 1 es = 0 for some s E S}. Since S has the 0re condition

t(E) is a kG-submodule of E. If t(E) # 0, then t(E) n k f 0 which is clearly

impossible. Thus t(E) = 0 and divisibility

s E S there is a unique element fE E with fs = e. Hence E can be made into

an R-module. If 1= nR, then by adapting the proof of 116,

we can show that I has the AR property in R. Since

generated essential extension of k as an R-module M1’ (3 k _

t. Fence M1’ = 0 and M is a finitely generated submodule of E,(k), where

T = R/r”.

Finally

R/I z kG/N satisfies a polynomial

nilpotent so does T. Therefore by [7 1 M is artinian,

composition series of finite length. Again

composition factor is a simple kG/N-module

It is enough to show that E = E,,(k) is locally finite dimensional.

can be deduced as

This gives an isomorphism

@ V*, E). However,

of

- 6.J V* and

Thus to show that EkG(V) is

module V it is enough to show that E

the set 5’ = 4 + n consists

of E implies that for e E E, e f 0,

identity and since 111’ is

and so has a

since 111’ is nilpotent

and so finite dimensional.

eat

A functionfE

Hom,(kG, k) i = l,..., m such that

Hom,(kG, k) is called finitary if there are flunctions fi , SI E

for all

g, h E G.

We write %FO(G, k) for the set of all finitary functions from G to k. Then

FO(G, k) is a subalgebra of Hom,(kG, k) under pointwise operations, and

also a right kG-submodule by means of left translation Also cFO(G, k) is the

Page 10

60

IAN M. MUSSON

unique maximal locally finite submodule of Hom,(kG, k) (see [2, Sect. 1.21).

Hence E,,(k) is isomorphic to a submodule of %(G, k).

In addition X0 =&(G, k) carries a Hopf algebra structure [2, Sect. 1.21.

We identify F0 0% with an algebra of functions on G X G. Then the

comultiplication

,u(f) -5 @ 6 satisfies p(f)@, y) =f(xu) for fE X0, x, y E G.

The augmentation

S:.F+.~

In Section 1.2 of [2] Donkin establishes a covariant equivalence between

the category of locally finite right

comodules. In particular the objects of both categories are the same as vector

spaces, and if E is a locally finite right kG-module, then E is injective if and

only if it is injective as an XO(G, k)-comodule. We note that the regular left

comodule .F(G, k) is injective by [.5, (1.5a)]. The Hopf algebra rO(G, k) is

also known as the algebra of all representative functions on G and is studied

by Mostow [ 121 and Magid [lo].

Now suppose that G is a torsion-free nilpotent group of finite rank. Then

it follows from the nilpotent case of Theorem B and the above discussion

that E,,(k) % k[X] a subalgebra and right kG-submodule of.FO(G, k). As we

saw above k[X] = {f~ flG,

k) ]f(g’) = 0 some t}. It follows easily that

k[X] is a subHopf algebra of .&(G, k) ( see the proof of [4, Proposition

51.11).

Following [2], we say that a Hopf algebra A is quasi-affine if the subHopf

algebra generated by each block component of A is a finitely

algebra.

E: 3-t

k is given by a(f) =f(l)

=f(x-‘).

and the antipode

by S(f)(x)

kG-modules and left &(G, k)-

generated

THEOREM A (Nilpotent

finite rank, then .%(G, k) is a quasi-afJine Hopf algebra.

Case). If G is a torsion-free nilpotent group of

ProoJ: We have shown that k[X] is a finitely generated sub-Hopf algebra

of %FO(G, k). We refer to Section 1.2 of [5] for the definition of the coefficient

space cf( V)

of a comodule V. It

cf(k[X]) = k[X]. S ince k[X] is the principal

sub-Hopf algebra generated by every block component of Lc is finitely

generated by Corollary 1.4.4 of [2]. Thus FO(G, k) is a quasi-affine Hopf

algebra.

follows

block component of X0, the

from [S, (l.Zf)] that

We next describe a decomposition

G = H >a A, where A = (y) is infinite cyclic and H is torsion-free soluble of

finite rank. In this case E,,(k) z E&k) @ ELA(k), where EkA(k) is regarded

as a kG-module with trivial H-action and E&k)

module structure. For polycyclic groups this is due to Donkin

of E,,(k)

when G has the form

is given a suitable kG-

[4]. As in

Page 11

THE STRUCTURE OF CERTAIN INJECTI'v'E MODULES

61

osition 3.1.2(i) of 141, the kH-module structure on E,,(k)

Iy finite kG-module structure. We repeat the details since they will be

needed later.

For a kH-module V affording the representation p and g E

representation pg: hi + End,(V) by pg(h) = p(ghg-‘)

by Vy the k-space V regarded as a kiTmodule

Ivow it is easily checked that E,,(k)Yis

an essential extension of k. Hence there is a

“k”: E,,(k)” + E&k). We define

extends to a

define a

e denote

for h E

via

an inje

module

ule isomorphism

which is

e . y = Y(e)‘)

for e E E,,,(k),

e.y-’

= y-‘(&),

where

module structure on E,,(k)

Thus we can extend the kITmodule

structure (cf, 14, Proposition 3.1.21). We say that E,,(k)

Y, and write E,,(k), for E&k)

-’ is regarded as a map EkN(k)Y-’ -Ekl,(k).

such that e . yP ’ . h y = e . h?‘ for all h E li.

structure on E,,(k)

This defines 2 kA-

lo a kG-module

is a kG-module via

considered as a kG-module in this way.

PROPOSITION

2.2.

With the above notation

E,,(k) z E,,(k), 0 E,,(k)

as kG-modules.

ProojI

E = E&k),+, @ EkA(k) is the one-dimensional trivial module. By Lemma 2.i

E&l ” E,,(E) = 1 is locally

finite dimensional. We can now adapt the

proof of Proposition 3.1.1(i) of [4] to show that E is injective so E FZ EkJk).

PJow let r be a finite extension of a torsion-free soluble group of finite

rank

G such that G/N is free abelian for a normal nilpotent subgroup N of

G. Then G has a series

As in Proposition 3.1.2(ii) of [4] we can show that the socle of

G=G,IG,... xG,~+~=N~

where G,, I a G for 1 < i < s with Gi/Gi+ i infinite cyclic.

(2.1)

THEOREM A.

of finite rank. Then <FO(T, k) is a quasi-affine

PT-OOJ We know that FO(N, k) is a quasi-affine

&potent case. The proof proceeds as in Theorem 1.65 of j2] by induction

on tke length of the series (2.1) using the results of 12, Sect. 1.5] for the

finite quotient T/G.

From this result we can deduce the following

the corollaries G is a finite extension of a torsion-free soluble group of finite

rank, and k is a field of characteristic zero.

Let r be a $nite extension of a torsion free soluble group

opf algebra.

opf algebra by the

corollaries as in [2]. En all

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62

IAN M. MUSSON

COROLLARY Al.

module is artinian.

Any essential extension of a $nite-dimensional kG-

COROLLARY A2.

endomorphism ring End&E(V))

Noetherian.

If V is a finite-dimensional kG-module, then the

of the injective hull E(V) of V is

COROLLAR~_Y A3.

completion kG, of kG with respect to I is Noetherian.

If I is an ideal of finite codimension in kG, then the

WeAemark that for torsion-free nilpotent groups of finite rank, the fact

that kG, is Noetherian follows from Theorem 3.4 of [ 1 I]. Using this result

and the methods of [14], one can obtain proofs of Corollaries Al and A2 in

this special case.

To prove Theorem B we shall need a slight change of notation

Section 1. Relative to a given NO-sequence of N = Fit(G) choose elements xi,

1 < i < n, as in Section 1 and relabel xi as xi+ s. Now choose elements xi,

1 < i < s, such that the image of xi in G/Gi+ I generates Gi/Gi+ ]. Then every

element of G has a unique representation in the form

from

x = X(b, )...) b,) = xfr . . . XpxfI

with bi E Q,

(2.2)

where we have written r = n + s.

Let Xi E Hom,(kG, k) be defined by X,(x) = b, when x is written as in

(2.2).

For 1 < i < s let k[Xli be the polynomial ring in X,, X,_ , ,..., X,, 1 ,..., Xi.

Thus we relabel the object k[X] constructed in Section 1, using the given NO-

sequence of N as k[X],+ 1. Also k[X], denotes the polynomial algebra in all

the indeterminates X,, X, ,..., X,.. From now on we refer to this algebra as

WA.

We have shown that N = G,, r acts as a group of algebra automorphisms

of k[X],+,

and Eke,+,(k) 2 k[X],+,

induction that Gi, I acts as a group of algebra automorphisms of k[Xli+,

and EkCi+, g k[Xli+ 1 for some integer i with 1 < i < s. Write y for the

generator of G,/G,+, and as before choose an isomorphism Yi: (k[Xli+ ,)” --f

k[Xli+ r and regard k[X],+, as a kGi-module via Yi. Then Proposition 2.2

gives k[X], = k[X]i+ r ,,,i @ k[Xi] g E,ai(k) as right kGi-modules. We have

proved the first statement of Theorem B.

as right kG,+,-modules. Suppose by

THEOREM B.

where k[X] is a jinitely generated polynomial algebra. Moreover for a

suitable choice of the module isomorphisms Yi, G acts as a group of algebra

automorphisms on k[X].

With the above notation E,,(k) g k[X] as kG-modules,

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THE STRUCTURE OF CERTAIN INJECTIVE MODULES

63

We prove the second statement by induction on the length of the series

(2.1). To do this we need some information on the socle series of k[%] which

is proved in Section 3. To apply induction we suppose that G is a subgroup

of some larger group r such that r/IV is free abelian and T/G infinite cyclic,

generated by the image of y E I’. Suppose we have shown EkG(k) z k[X],

where G acts on k[X] as a group of algebra automorphisms and consider the

kS-module k[X], 0 k[ Y] = k[X, Y], where y acts on k[Xj via a suitable kG-

module isomorphism Yy: (k[X])y + k[X] and G acts trivially

need to show that F acts as a group of algebra automorphisms on k[X, Y].

By the definition of the tensor product of modules ] 17, pa 271, it suffices to

show that r acts as a group of algebra automorphisms on each of the factors

k[Y]. Since G acts trivially

infinite cyclic group T/G. Thus r acts in the desired fashion on k[ Y] by the

nilpotent case (see also, the example at the end of Section I.3 of [Z]).

It remains only to show that r acts on k[X]y

automorphisms. We already know that G acts in this manner. Since the k-

!inear map k[X] --) (k[X])”

sending a monomial m to m’ is an algebra map,

it is enough to show that the module isomorphism rY, (kj Y])” --f kjX] can be

chosen to be an algebra map. This is proved in Lemma 3.5. We use only the

assumption that G acts as a group of algebra automor~hisms on k[X] to

obtain the information on the socle series of k[X] which is needed for the

roof of this lemma.

In certain situations the isomorphisms !Pi can be chosen to be the identity

map, and then G acts on the algebra k[X] by left translation.

group G

its over its Fitting subgroup, G = A7 M A w-ith A torsion-free

abelian.

can choose the elements x1,..., x, as generators of A. Then

N= G,,, acts on k[X],+,

= E&k) by left translation.

We regard k[X]i+l as a subalgebra of XO(Gill,

by conjugation on k[Xli+ r . That is for all Qi E k[X] i+ r, the function $Xi7

defined by #“‘(g) = #(xi gxi ‘) for g E Gj+ r, lies in k[Xli+ r . Clearly Xj”i = xi

for 1 < i <j < s since the group elements x1 ,-‘> x, commute. It remains to

show that (k[X],+,)“i = k[X],+,

= E&k).

regard EkN(k) as a union of submodules (kN/n’)*

acts on each kN/n’ by conjugation and hence on E,,(k).

Therefore in this case we can choose each Yj to be the identity map. In

summary, we see that N acts on k[X],+l

on k[YJ.

on k[ Y], k[Y] is a module for the

as a group of algebra

Suppose the

k) and show that xi acts

The easiest way to do this is to

as in Section 1. Then A

by left translation and also,

if l<j<s

and

j < i,

then Xj. xi=Xjy

(2.3)

for

1 <i<S, Xi.Xj=Xi+ 1.

(2.5)

481/85/l-5

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64

IAN M. MUSSON

By checking each case separately we see that

(Xi * Xj)(X) = Xi(XjX)

for all

i, j,

and xE G.

In other words G acts on k[X] by left translation

THEOREM Bl. If G splits over N = Fit(G),

generated subalgebra k[X] of Hom,(kG,

lation such that k[X] g E,,(k).

then there is a finitely

K) on which G acts by left trans-

We now present some examples using perhaps some of the simplest groups

under consideration. The first group is free nilpotent

generators. Example 2 is a polycyclic

while Example 3 is a finitely presented

is not polycyclic-by-finite.

of class two on two

group which is not nilpotent-by-finite

metabelian group of finite rank which

EXAMPLE 1. LetG=(x,y,z](x,y)=z,(x,z)=(y,z)=1).LetG,=G,

G, = (x, z), G, = (z).

Every element

Z, X, YE Hom,(kG,

the polynomial

kG-modules

of G has the form g = za$‘yC for integers

k) by Z(g) = a, X(g) = b and Y(g) = c and let k[X] be

algebra in these indeterminates.

where the action is given by left translation,

a, b, c. Define

Then E,,(k) z k[X] as right

for example,

(2 . y)(z”x”y’) = Z(yz”xbyC) = Z(Za-bxbyc+ ‘)

=a-b.

Therefore

Y~j=Y+l,Xz=Xy=Xand

As an alternative

the dual space of kG/ g3 which has a basis consisting

Zy = Z - X. Similarly Zx = Z and Zz = Z + 1. Also Xx = X + 1,

Yx=Yz=Y.

to these calculations we may consider the action of G on

of elements

B*(a,b,c)= [(z - 1)" (X - I)" (y - l)']"

with 2a + b + c < 3, and then use the formula (1.4). For example, y(x - 1) =

(x-l)(y-1)+(x-1)-(z-l)modg3

These and similar calculations show that (z - 1)” y = (z - l)* - (x - l)*

and formula (1.4) gives Zy = Z - X.

The comultiplication map ,u: k[X] j k[X] @ k[X] is determined

X~~+~@X,~(Y)=Y~~+~@Y~~~,U(Z)=Z@~+~@Z-Y@X.

Tne antipode S satisfies 5’(X) = -X, S(Y) = -Y and S(Z) = -Z - XY.

and y(z-l)=(z-l)modg3.

by p(X) =

EXAMPLE 2. Let G = (a, b, x / (a, b) = 1, ax = b, b” = ab). Any element

of G can be written uniquely as

g = aPbqxr.

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THE STRUCTURE OF CERTAININJECTIVEMODULES

65

Define A,B,X

polynomial

translation.

It is easily checked that

by A(g) =p, B(g) = q, X(g) = r. Let k[X] be the

algebra in A, B, 1. Then k[X] z E,,(k), where G acts by left

h=A+l,

Ab=A, Bb=B+ 1,

BL2=3,

Ax=B-A,

Bx=A, Xx=X+ 1,

Xa=Xb=x

EXAMPLE 3. Let G=(x,yIx’=x*).

Then every element of G may be expressed uniquely in the form g = xi?;’

where s E ZI, Y E Z [il. Define functions X, Y: G --f k by

Kg> = 7,

Y(g) = s.

Let kjX\ be the polynomial algebra in X, Y then k[Xj z E,,(k) where G acts

by left translation. Easy calculations show that

Xx=X+ 1,

xy = +x,

Yy= Y+ 1,

Yx = Y.

3. THE SOCLE SERIES OF E&k)

Let G be a torsion-free soluble group of finite rank such that G/N is free

abelian, where N is the Fitting subgroup

E, = sot,(E) and o”t = C,(E,) for t > 1. If G is torsion-free abelian it is easily

seen that E, is faithful for G. Hence in general, if I is the exponent of the

torsion subgroup of G/G’ we have o”f s G’ for t > 2.

The subgroups fit play an important role in the study of the socle series of

E,,(k) and we show first that they are independent of the field k. To this end

we set E, = E,,(k), E,,, = soc,(E,)

extension of k we write VF for the module V @ Jo

of G. rite E = E,,(k).

and fik,, = C,V(-Ek,f). If F is a field

LEMMA 3.3.

(ii)

(iii)

(i) E,gEL,

Ek,l z Ek,, for all t > 1,

5”k,t = BQ,* for all t > 1.

Brooj

E&k)

inductive method of Proposition 3.2.lfii)

(ii) As a kG-module Ek,,, has secular height at most t, and by Lemma

5.1 of [14], this module is an essential extension of k. Hence EL,( is

isomorphic to a submodule of E,,,.

(i) The description

z &,&Q)“.

of E&k) given in Theorem B shows that

and we may use the

of [4].

In general G/N is polycyclic-by-finite