Morava K-theories and localisation

We study the structure of the categories of K(n)-local and E(n)local spectr a, using the axiomatic framework developed in earlier work of the authors w ith John Palmieri. We classify localising and colocalising subcategories, a nd give characterisations of small, dualisable, and K(n)-nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of fin ite spectra, and to the theory of Landweber erectness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerf ul finiteness results. We study the Picard group of invertible K(n)-local s pectra, and the problem of grading homotopy groups over it. We Drove las an nounced by Hopkins and Gross) that the Brown-Comenetz dual of MnS lies in t he Picard group. We give a detailed analysis of some examples when n = 1 or 2, and a list of open problems.