ANDREW R. KUSTIN
J?* h a
A? Lemma 1.11. Adopt Data 1.2. Let ap 6 /\iP
p
F*, bq
f\q
F, ar f\ F\ and M
and w be integers.
(a) / / / + 1 + w p + r, then £ (-!)*("-*) fi(ap) A [y/(6,)](ar) = 0.
|J|= t
(b) Ifw + p+1 q, then £
( - I ^ P + D ^ ) / ,
L
p
A M&,)](a
r
) = 0
i=Z
Proof. We first prove (a). For each pair of integers (M,w), let
hM,w: f\9F*®/\F®/\*F*-®/\*F*
be the homomorphism which is given by
i 2
|J| = t
It is clear that HM,W is the zero homomorphism whenever w 0. Lemma 1.9.d
shows that
hM${ap®bq®ar) = (-l)pqbq(ap A ar);
thus, the conclusion holds whenever w 0. The proof proceeds by induction on w.
Observe that
*i
[^M,W
(aP ® &g ® ar)]
for all 61 /\ F. Indeed, the left side of (1.12) is equal to A + B, where
A = Y, (-l)ip(Ww"% [//(o,)] A M M K ) and
iez
| / | = t
B = X] ( - l y ^ r j U K ) A 6 1 (b/(6,)](ar)) -
i Z
|/ | = i
Use Proposition 1.1.a to write
h (k/(6,)](o
P
)) = [61 A^/(fc,)](ar) = ([MPi)](&«)) (a
r
) + (-1)' (Pi(bi Abqj) (ar).
Apply Lemma 1.9.e to see that B = B\ + Bi for
Bi= £ (-l) £ p + ^( M r)/H&iK)]Ab/(6,)](a
r
) and
i Z
|i |
=
i - l
B*=Y
(-
1
)
i P + P
(
M
u
7
£
)//(
t t
p)
A
^ 1 A6,)](a
r
) =
(-l)phM,w(ap
® fc A 6, ® a
r
) .
iGZ
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