On the coalgebraic ring and Bousfield-Kan spectral sequence for a Landweber exact spectrum

We construct a Bousfield–Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum E with unit and which is related to the homotopy groups of a certain unstable E completion X ∧ E of a space X. For E an S-algebra this completion agrees with that of the first author and Thompson. We also establish in detail the Hopf algebra structure of the unstable cooperations (the coalgebraic module) E∗ (E_∗ ) for an arbitrary Landweber exact spectrum E, extending work of the second author with Hopkins and with Turner and giving basis-free descriptions of the modules of primitives and indecomposables. Taken together, these results enable us to give a simple description of the E 2-page of the E-theory Bousfield–Kan spectral sequence when E is any Landweber exact ring spectrum with unit. This extends work of the first author and others and gives a tractable unstable Adams spectral sequence based on a νn-periodic theory for all n.