The Category of H-modules over a Spectrum 21
A spectrum (M,a) is a sequence *•', M , M ., ••• of pointed
spaces indexed by 4Z, each having the homotopy type of a CW complex, and
4
a map a, : S A M„ - M„ for each 4n e 4Z. See [9] . (M,a) is a
4n 4n 4n+4
convergent spectrum if M is (4n-l) -connected for n _ 1, and if
M = * for n _ 0. Convergent spectra have been defined more generally,
but this definition is suitable for our purposes. We will often omit the
subscripts on the map a , and we will often denote (M,a) by M only.
We assume henceforth that if M is a space of a spectrum, then n £ 4Z.
We define T T (M) = lim T T (M ) , where the homomorphisms of the
k k+n n
n
-+oo
direct system are the compositions
c4 4 a*
T T (M
)
- TT,

(S A M )

TT,
A
(M
J .
k+n n k+n+4 n k+n+4 n+4
If M is a convergent spectrum, then T T (M) = 0 for k 0.
A map of spectra f: (M,a) (N,B) is a sequence of maps
•••,f,f „,•••, f:M -•Nn, such that
n n+4' n n
4 a
S
A M
* M
n n+4
1
A
f I i f ,
n n+4
4 6
S AN N
A
n n+4
is homotopy commutative for all n £ 4Z. A map of spectra f: M -*N induces
a homomorphism f^: TT^(M) - TT^(N) . Given maps of spectra f: M - N and
g: N - P, the composition g°f:M-P is a map of spectra, and
(gof), - g* « f«: irt(M) •* TT,(P).
A subspectrum (N,3) of (M,a) is a spectrum such that N. is a
closed subspace of M.; (M.,N.) has the homotopy type of a CW pair;
i
M./N. is Hausdorff;
a.(S4
A N.) C N. .; and 3. = a.l 4 . Then there
1 1 1 1 1+4
I I
!S
A N .
l
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