On the coalgebraic ring and Bousfield-Kan spectral sequence
for a Landweber exact spectrum
Martin Bendersky and John R. Hunton
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 $\xe$ of a space $X$. For $E$
an S-Algebra this completion agrees with that of the first author
and R. Thompson. We also establish in detail the Hopf algebra
structure of the unstable cooperations (the coalgebraic module)
$E_*(\EE_*)$ for an arbitrary Landweber exact spectrum $E$,
extending work of the second author and M. Hopkins\cite 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 $E_2$-term 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 $v_n$-periodic theory for all~$n$.