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
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
= (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).
1.1.13. If $M = 0, then the G-structure of ft M is given by
(a) B(ftxM) = 0
i = 0 mod p
(b) P^ft (x) = \
^i^p Px)
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
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