INTRODUCTION USING A mod-p version of the lower central series one of us recently [lo] constructed a mod-p version of the spectral sequence of one of the others [3]and showed that for every topological space X there exists a spectral sequence {E'X, d'X) with the following properties:(i) E'X depends only on H,(X; Z,), (ii) for simply connected X the spectral sequence converges in the same sense as the Adams spectral sequence [I].is the Z,-module spectrum freely generated by a single simplex in degree 0, this composition LL + L induces a composition operation z,L,AX x n,L,AS-+ z$+~L~,AX and one readily verifies
The composition operation turns n,LAS into an associative graded algebra with unit and n,LAX into a right module over this algebra.And as n,LIAX = n*AX = H,X part (ii) of Proposition 2.5 follows from 2.3 and 3.3. The right n*LAS map H,X 0 n,LAS-+ n,LAX given by a @ 1 + a for all a, is an isomorphism.
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