k

John R. Harper

F'(n) = F(n)/F(.n)3

where F(n)3 is the left ideal generated "by 3i .

(a) ftF'(n) = F'(n-l) for n 1

(b) For n 1, F'(XL) is A-projective.

1.1.11. EXACTNESS PROPERTIES OF ft [6], [20]. From the definition 1.1.6. it

follows that ft is right exact. We denote "by ft% the left derived

functors offt.

(a) ft = 0 for r 2.

Proof. Let P# - - M "bea projective resolution of M. By

definition

ft^M = H^CftP*) .

Since projectives are A-projective, the sequence below is

exact (maps not preserving dimension)

0 -•P# -^ P# - fiP# -*0 ,

which gives (a)by taking homology.

(b) As Z/pZ-modules,

(^M)21*-1 = (ker X) 2 n

(ftlM)2np+1

= (ker X)

2 n + 1

(ftxM)k = 0 k i ± 1 mod2p

Proof. From the definition, ft M can be identified with

ker X in the sequence

0-ftM-M-^M + ftM+0

with regrading as listed.

1.1.12. If x e ker X we denote its image in ft,M byft(x).

even

1.1.13. If $M = 0, then the G-structure of ft M is given by

(a) B(ftxM) = 0

ft

(p^'Px)

i = 0 mod p

(b) P^ft (x) = \

^i^p Px)

j

E

1 mod p, dim x odd

0 otherwise

Proof. Routine use of l.l.U.

1.1.Ik. GENERALIZED EILENBERG-MACLANE SPACES [2?]. Via the identification