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By Myles Tierney

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Extra info for Categorical Constructions in Stable Homotopy Theory: A Seminar given at the ETH, Zürich, in 1967

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Then an is a of F D - m o d u l e s L e t us d e n o t e b y of F D - m o d u l e s . X e FD ~ operator "finite" is a c o m m u t a t i v e is a s i m p l i c i a l > o boundary from s h o w that the c a t e g o r y version. faces a n d d e g e n e r a c i e s of s i m p l i c i a l and are e q u i v a l e n t "unstable" (NX) n of w As an a p p l i c a t i o n - that F D - s p e c t r a the c a t e g o r y If and w (over K) and the a morphism abelian analogue c a s e of the We r e c a l l FD-module the in the s e n s e of K a n is o b t a i n e d constructions for however, are e q u i v a l e n t .

And then To j u s t i f y this, for FD-modules. definition of the suspension. Again, a shift of one place to the r i g h t w i t h the e x c e p t i o n w h i c h is a d d e d freely. That is, if X ~ FD ~ define SX b y setting (SX) o = O and for n-I (SX)n = Xn-I ~ Sn-i where modulo n-ii (SX) n-i Sn_ is the free k - m o d u l e on symbols (SX) n ) o n-i, n-i Sn_l~ for the r e l a t i o n s n-i n-i n-i Sn_ I (ki~ i + k2~ 2) = k l S n _ l ~ i + k 2 S n _ l a 2 a E (SX) n-i - for of GI,~2 E (SX)n_ 1 (SX)n_l.

FX for j ) o. FX is c l e a r l y a positive functorial FD-spectrum, in X. Lemma T h e r e is a n a t u r a l isomorphism of chain complexes NX ~ NFX i n d u c e d b y the i n j e c t i o n i pr99fs o, i~ is of degree o 9 X and an )FX. FD-morphism, in so far as that m a k e s sense, where - since y u E NX is. 7 that - in each degree - also since i G ~ NFX, o y is. so w e o b t a i n a m o n ~ B o ~ p h i s m o f If chain complexes ) NX NFX . 4, trivially if FX E FD f+ . e. 7. F > (FDf+,s), 9 (FDf~ is c o m p a t i b l e w i t h suspension.

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