16

MARTIN ARKOWITZ AND GREGORY LUPTON

5.6 Proposition Suppose

M

= A(V0

,

V1

;

d)

with both

V0

ffi

V1 and

H* (M)

finite

dimensional. Let n be an integer such that

Hi(M)

= 0 for all i n. Then

£#n(M)

=

£#oo(M).

Proof Suppose that n is as in the hypotheses. Since V0

ffi

F1 and

H*

(M) are both

finite dimensional, then there is at most one generator from F0

ffi

F1 in degrees 2 n,

by [Ha,§2]. If there is no generator of degree n, then we have that

E#n(M)

=

£#oo (M) trivially. So suppose there is a single generator

v

E

F0

ffi

F1 of degree

n, and let

[f]

E

£#n(M). Without loss of generality, by Corollary 3.4, we can

assume that flvo = Llvo. Since v is the only generator of degree 2 n, we have

f(v)

=

.v +X for some )..

E

Q

and decomposable x, and it is sufficient to show

that)..= 1. Since

Hi(M)

=

0 fori n, we have that v

E

V1 and so dv

#

0

E

AF0

.

Then dv

=

f(dv)

=

d(fv)

=

d(.v

+

x)

=

.dv

+

dx and so

d((l-

.)v)

=

dx, for

decomposable X· From our normal form assumption, it follows that ).. = 1. Thus

[f]

E

£#

00

(M),

and the conclusion follows. D

5.7 Proposition Suppose

M

=

A(Fo, F1

;

d)

and

H*(M)

is finite dimensional. Let

n be an integer such that

Hi(M)

=

0 for all i n. If Hom(V1

,

W)

=

0, then

£#n(M)

=

1.

Proof Let

[f]

E

E#n(M)

and assume by Corollary 3.4 that flvo

=

Livo· Let Wk

be a basis element of

v1

of degree :::; n. Then f(wk) - Wk

=

Yk, where Yk is

decomposable. Therefore 0 1(j,L)(wk)

=

[f(wk)- wk]

=

[Yk]· But, by hypothesis,

1101(j,L) = 0: V1-+ W, and hence 11[Yk]

=

0. Thus [Yk]

=

L:;=1Aj[vj],

for some

Aj

E

Q,

i.e.,

r

Yk

=

L

AjVj

+

d(zk)

j=1

for some Zk EM. Since d(zk) is decomposable, Aj

=

0, for all

j.

Thus Yk

=

d(zk)

and we have 0

1

(f, L) ( wk)

=

[Yk]

=

0. But if wk is a basis element of V1 of degree

n, then 0 1(j,L)(wk)

=

0 by hypothesis. Hence

Ol(j,L) =

0, and so f"" L by

Proposition 3.3. D

It

is more interesting to investigate conditions under which the groups

£#n (M)

are infinite. For this it suffices to show £#

00

(M) is infinite.

5.8 Proposition Let

M

=

A(V0

,

F1

;

d) be formal. If Hom-L(Vo,

H*(M))

=

0 and

Hom(V1

,

W)

#

0, then £#

00

(M) is infinite.

MARTIN ARKOWITZ AND GREGORY LUPTON

5.6 Proposition Suppose

M

= A(V0

,

V1

;

d)

with both

V0

ffi

V1 and

H* (M)

finite

dimensional. Let n be an integer such that

Hi(M)

= 0 for all i n. Then

£#n(M)

=

£#oo(M).

Proof Suppose that n is as in the hypotheses. Since V0

ffi

F1 and

H*

(M) are both

finite dimensional, then there is at most one generator from F0

ffi

F1 in degrees 2 n,

by [Ha,§2]. If there is no generator of degree n, then we have that

E#n(M)

=

£#oo (M) trivially. So suppose there is a single generator

v

E

F0

ffi

F1 of degree

n, and let

[f]

E

£#n(M). Without loss of generality, by Corollary 3.4, we can

assume that flvo = Llvo. Since v is the only generator of degree 2 n, we have

f(v)

=

.v +X for some )..

E

Q

and decomposable x, and it is sufficient to show

that)..= 1. Since

Hi(M)

=

0 fori n, we have that v

E

V1 and so dv

#

0

E

AF0

.

Then dv

=

f(dv)

=

d(fv)

=

d(.v

+

x)

=

.dv

+

dx and so

d((l-

.)v)

=

dx, for

decomposable X· From our normal form assumption, it follows that ).. = 1. Thus

[f]

E

£#

00

(M),

and the conclusion follows. D

5.7 Proposition Suppose

M

=

A(Fo, F1

;

d)

and

H*(M)

is finite dimensional. Let

n be an integer such that

Hi(M)

=

0 for all i n. If Hom(V1

,

W)

=

0, then

£#n(M)

=

1.

Proof Let

[f]

E

E#n(M)

and assume by Corollary 3.4 that flvo

=

Livo· Let Wk

be a basis element of

v1

of degree :::; n. Then f(wk) - Wk

=

Yk, where Yk is

decomposable. Therefore 0 1(j,L)(wk)

=

[f(wk)- wk]

=

[Yk]· But, by hypothesis,

1101(j,L) = 0: V1-+ W, and hence 11[Yk]

=

0. Thus [Yk]

=

L:;=1Aj[vj],

for some

Aj

E

Q,

i.e.,

r

Yk

=

L

AjVj

+

d(zk)

j=1

for some Zk EM. Since d(zk) is decomposable, Aj

=

0, for all

j.

Thus Yk

=

d(zk)

and we have 0

1

(f, L) ( wk)

=

[Yk]

=

0. But if wk is a basis element of V1 of degree

n, then 0 1(j,L)(wk)

=

0 by hypothesis. Hence

Ol(j,L) =

0, and so f"" L by

Proposition 3.3. D

It

is more interesting to investigate conditions under which the groups

£#n (M)

are infinite. For this it suffices to show £#

00

(M) is infinite.

5.8 Proposition Let

M

=

A(V0

,

F1

;

d) be formal. If Hom-L(Vo,

H*(M))

=

0 and

Hom(V1

,

W)

#

0, then £#

00

(M) is infinite.