  Let K(n) be the nth Morava K--theory at a prime p. This paper is a thorough
study of questions like the following: to what extent does the
K(n)--localization, or the K(n)--homology, of a spectrum X determine the
K(n)--homology of its 0th space X_0?
  Our methods combine techniques from modern homotopical algebra with chromatic
homotopy. In particular, we use the telescopic functors of Bousfield and the
author (dependent on the Nilpotence Theorem of Devanitz, Hopkins, and Smith),
as wel as Topological Andre--Quillen Homology and Goodwillie calculus in
nonconnective settings.
