Some spectral sequences in Morava E-theory
by Mark Hovey
mhovey@wesleyan.edu
The Morava E-theory of X is the homotopy of the K(n)-localization of E
smash X, where E is the completed and extended version of E(n) on which
the Morava stabilizer group acts. Because K(n)-localization is not
smashing, Morava E-theory is not a homology theory; it is exact, but
does not preserve coproducts. Nevertheless, it is the most important
theory to use in understanding the K(n)-local stable homotopy category;
for example, X is small in the K(n)-local stable homotopy category if
and only if the Morava E-theory of X is degreewise finite.
In the paper at hand, we show how the usual spectral sequences used with
homology theories work for Morava E-theory. The most interesting such
spectral sequence is a spectral sequence that converges to the Morava
E-theory of an infinite coproduct. The E_2-term involves the derived
functors of direct sum in the category of "L-complete" E_*-modules.
There are (n-1) such derived functors (n if we try to compute filtered
homotopy colimits). Thus, Morava E-theory is "n derived functors away
from being a homology theory". In particular, when n=1, we see that
p-completed K-theory actually commutes with coproducts, in the category
of Ext-p-complete abelian groups. It follows that K(1)-local homotopy also
commutes with coproducts as a functor to Ext-p-complete abelian groups.