The Bockstein and the Adams spectral sequences

We show that, above the appropriate "vanishing line", the Adams spectral sequence of a connective spectrum can be read off from its Bockstein spectral sequence. In this short note, we prove a basic folkdore theorem which relates the modp homology Bockstein spectral sequence of X to the Adams spectral sequence {ErX) converging from E2X = ExtA(H*X, Z4) to 7r*X where X is a bounded below spectrum with integral homology of finite type. As usual, we grade { EX) so that E2'tX = Exte'(H*X, Z.), with 4: Ers,X~r Es+r,t+r-lX, the total degree being t s. We have a natural homomorphism E?2* X -> H X which factors the mod p Hurewicz homomorphism, and we shall sometimes identify elements of H*X with their inverse images in EOj* X. We have a pairing of spectral sequences ErS ? ErX -> ErX, where S is the sphere spectrum. Finally, we have an infinite cycle ao E E2 "S such that if x E EsJX and if y E FsT,-sX projects to x, then py E Fs+ 'iT,-sX and py projects to aox. Our main theorem will be a consequence of the following vanishing theorem, which is due to Adams [1] when p = 2 and to Liulevicius [4] when p > 2. Let Ao = E { / ) c A and recall that an A0-module M is free if and only if H(M; ) = 0. THEOREM 1. Let M be an (m + 1)-connected AO-free A-module. Then Ext'`(M, Z) = 0for s> 1 and t-s 2 and, if p =2, f(4k) = 8k + 1, f(4k + 1) = 8k + 2, f(4k + 2) = 8k + 3, and f(4k + 3) = 8k + 5. DEFINITION 2. Let M be an A-module. We say that x E ExtA(M, ZP) generates a spike if x is not of the form aox' and if aox # 0 for all i. The set of spikes in ExtA(M, ZP) has its evident meaning. The same language will be applied to each ErX. Let K(R, n) denote the nth Eilenberg-Mac Lane spectrum of R and abbreviate HR = K(R, 0). Lety denote the canonical generator of Ho(HZpr) and let /3r denote the rth modp Bockstein (in homology or cohomology according to context); let Y = f3rZ. Received by the editors September 23, 1980. 1980 Mqthernatics Subject Classification. Primary 55T15; Secondary 55P42. i American Mathematical Society 0002-9939/81 /0000-0429/$0 1.75