By Jan-Erik Roos

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**Extra info for Algebra, Algebraic Topology and their Interactions: Proceedings of a Conference held in Stockholm, Aug. 3 - 13, 1983, and later developments **

**Sample text**

On the other hand R = S = Zp R (and S) concentrated in degree zero then also has a free model ~

The following assertions are then equivalent: (i) The Serre spectral sequence collapses at (ii) The morphism (iii) H*(E;~) Proof: H*(E;~) ~ H*(F;~) is a free H~(B;~) (resp. module In the topological case denote H*(E) ~ H~(F) is identified with E2 . H(S) ~ TorR(s;~)) (resp. H(S) ApL(B) ~ ApL(E) is surjective. is a free also by H(R)-module). R ~ S. Then H(S) ~ TorR(s;k). In either case, this is one edge homomorphism for the spectral sequence; the other is H(R) ~ H(S). Now clearly (i) (ii) and (iii).

The composite map which in a sense does not change the (~ ~3 was used by Jacobsson [Ja] to disprove a conjecture by Lemaire. Properties of As mentioned, we are concerned in this paper with describing the set ~7~ . It seems unlikely that there is any easy analytic way to characterize the elements of ~ , for we shall see in theorem 5 that J rather complicated operations. is closed under some Of special interest, however, are the rates of " 34 growth of the sequences analytic properties exponentially, {rank(An)}n20 of the series , and these rates are reflected in the A(z) .