The Thomified Eilenberg-Moore spectral sequence

where Xs+1 is the fiber of gs. We get an exact couple of homotopy groups and a spectral sequence with E 1 = πt−s(Ks) and dr : E s,t r → Es+r,t+r−1 r . This spectral sequence converges to π∗(X) (where X = X0) if the homotopy inverse limit lim←Xs is contractible and certain lim 1 groups vanish. When X is connective, it is a first quadrant spectral sequence. For more background, see [Rav86]. In the case of the classical Adams spectral sequence, we have some additional conditions on on (1.1), namely • Each spectrum Ks is a generalized mod p Eilenberg-Mac Lane spectrum, and • each map gs induces a monomorphism in mod p homology These conditions enable us to identify the E2-term as an Ext group over the Steenrod algebra, and to prove convergence when X is connective and p-adically complete.