On Relations between Adams Spectral Sequences, with an Application to the Stable Homotopy of a Moore Space

abutting to the stable homotopy of X. It has long been recognized that a map A +B of ring-spectra gives rise to information about the differentials in this spectral sequence. The main purpose of this paper is to prove a systematic theorem in this direction, and give some applications. To fix ideas, let p be a prime number, and take B to be the modp EilenbergMacLane spectrum H and A to be the Brown-Peterson spectrum BP at p. For p odd, and X torsion-free (or for example X a Moore-space V= So Up e’), the classical Adams E2-term E2(X;H) may be trigraded; and as such it is E2 of a spectral sequence (which we call the May spectral sequence) converging to the AdamsNovikov Ez-term E2(X; BP). One may say that the classical Adams spectral sequence has been broken in half, with all the “BP-primary” differentials evaluated first. There is in fact a precise relationship between the May spectral sequence and the H-Adams spectral sequence. In a certain sense, the May differentials are the Adams differentials modulo higher BP-filtration. One may say the same for p=2, but in a more attenuated sense. In this paper we restrict attention to dz, although I believe that the machinery developed here sheds light on the higher differentials as well. Assertions similar to these, in case X is torsion-free, have been made by Novikov [24], who however provided only the barest hint of a proof. I have attempted to provide in Section 1 a convenient account of part of the abstract theory of spectral sequences of Adams type, and in Sections 3, 5, and 6, I construct the May spectral sequence and prove the theorem outlined above. The constructions here are