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 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.
Previous Page Next Page