The Category of H-modules over a Spectrum 5
defined. For each (R,q), there is a space F{R,q) all of whose homotopy
groups vanish and a map e: F(R,q) - Map (M A A ,B ) . There are maps
U R q q+LR
c, a', and W similar to those above. The category of H-modules over
r, s
M has the H-modules over M as objects and equivalence classes of words
[F I •••IF,]: A - B as morphisms, where F.: C. , - C. is an H-map, C^ = A,
n1 ' 1 1 l-l
^' 0
and C = B. We prove
2.12. Proposition. There is a functor from the category of H-modules
over M to the category of TT^(M)-modules and TT^(M)-homomorphisms which
assigns TT^(A) to A and f ^ : TT^(A) - TT^(B) to f: A - B.
In section 3 we construct a mapping cylinder and a mapping cone for a
morphism f: A - B. The constructions are far from unique. We define an
equivalence relation, H-equivalence, and prove that the mapping cylinder and
mapping cone are unique up to H-equivalence.
Let x e T T (M) be represented by f: S - M . For an H-module N,
p v
p+v p+v
there are an H-module S A N , an H-map F: S A N - N, and hence a
mapping cone which we denote by N(f). In particular, N(f) is the
mapping cone of the composition
p+v fAl v,q
A N y
M A N '-^ * N
q v q q+v
where N e N(v,q). This kind of mapping cone is especially interesting
since the process can be repeated. Furthermore, if N is a convergent
H-module, TT^(N) and ir^(N(f)) are related. N is a convergent H-module if
N(R,q) contains only one point whenever q _ 0 or r. _ 0 for some i,
and if both M and N are convergent spectra. We prove that if N is a
convergent H-module over M, and if p _ 0 and v _ 0, then N(f) is a
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