Morava K-theories and localisation
by Mark Hovey and Neil P. Strickland
We study the structure of the categories of $K(n)$-local and
$E(n)$-local spectra, using the axiomatic framework developed in
earlier work of the authors with John Palmieri. We classify
localising and colocalising subcategories, and give
characterisations of small, dualisable, and $K(n)$-nilpotent
spectra. We give a number of useful extensions to the theory of
$v_n$ self maps of finite spectra, and to the theory of Landweber
exactness. We show that certain rings of cohomology operations are
left Noetherian, and deduce some powerful finiteness results. We
study the Picard group of invertible $K(n)$-local spectra, and the
problem of grading homotopy groups over it. We prove (as announced
by Hopkins and Gross) that the Brown-Comenetz dual of $M_nS$ lies in
the Picard group. We give a detailed analysis of some examples when
$n=1$ or $2$, and a list of open problems.