ABSTRACT. We study the structure of the categories of Zf(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 subcat-
egories, and give characterisations of small, dualisable, and if(n)-nilpotent
spectra. We give a number of useful extensions to the theory of vn 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(ri)-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 MnS lies
in the Picard group. We give a detailed analysis of some examples when n = 1
or 2, and a list of open problems.
1991 Mathematics Subject Classification. 55P42, 55P60, 55N22,55T15.
Key words and phrases. Morava KT-theory, stable homotopy category, Bousfield localisation,
Picard group, phantom maps, Landweber exact homology theories, Adams spectral sequence,
spectrum, Brown-Comenetz duality, thick subcategory.
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