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 I ^' 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 s i^ 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 v,q 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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