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