The Category of H-modules over a Spectrum 23
- TT ( N ) TT (M) TT ( M / N )
Y-
IT
n
( N ) +
q q q q- 1
( f |
N
) * ^ f* J 1 F , I ( f |
N
) ,
. . . - ^ (
N
' ) y ^ (
M
' ) ^ ^ ( M ' / N ' ) TT , ( N ' ) - '
q q q q- 1
Let (A,a), (B,3), and (D,y) be spectra. A pairing f: (A,B) - D
is a double sequence of maps {f }, f : A A B - ^ D , such that
p,q p,q p q p+q
A . A B
p+4 q
a A 1.
1 A f
4 p.a 4
S A A A B —• •'•" S AD
p q p+q
4 1 A
3
, ^
L]?'q+4
A A S A B =—* A A B
A
p q p q+4
is a homotopy commutative diagram for all p, q. A pairing f: (A,B) - D
induces a homomorphism f ^: TT^(A) ® TT^(B) + TT^(D). We denote f ^ (x®y) by
x
A
y.
The sphere spectrum S is defined by setting S = S for n _ 4
4
and S = * for n 0. We let a : S AS - S
A
be the identity map
n n n n+4
for n _ 4, and we let a be the base point map for n _ 0. For any
spectrum (M,a) there are natural pairings h:(S,M) - M and h': (M,S) M.
For p 4 , h :S A M -•M is the composition
- Prq p q p+q
1
A
a . 1
A
a o i
w S AM S A M ••• M
p q p-4 q+4 p+q
For p 4, h' : M A S - M is the composition
^ - qP q P P+q
T l A a l A a a
M A S S
A M
» S
A M
M
q p P q
P-4.
q+4 q+p
For p 0, h and h' are the base point maps.
^ - P,q q*P
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