Hovey, Mark; Strickland, Neil P. Morava \(K\)-theories and localisation. (English) Zbl 0929.55010 Mem. Am. Math. Soc. 666, 100 p. (1999). Among generalized homology theories the Morava K-theories, \(K(n)^*(\cdot)\), and the theories \(E(n)^*(\cdot)\) originally defined by D. Johnson and W.S. Wilson have been of particular interest in recent years. This paper studies the categories \({\mathcal K}\) and \({\mathcal L}\) of \(K(n)\)-local and \(E(n)\)-local spectra. In a previous paper [Axiomatic stable homotopy theory, Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)] the authors, together with J. H. Palmieri, developed a general theory of stable homotopy categories, i.e., triangulated categories with a compatible closed symmetric monoidal structure. Here, they show that \(\mathcal L\) and \(\mathcal K\) are both examples of these. They establish a nilpotence theorem for \(\mathcal L\), and classify thick subcategories, the localizing subcategories and the co-localizing subcategories of \(\mathcal L\). The category \(\mathcal K\) is shown to have no nontrivial localizing or co-localizing subcategories, and so to be, in a sense, irreducible. Small, and dualizable spectra in \(\mathcal K\) are characterized, a representability theorem for homology and cohomology theories in \(\mathcal K\) is established, and a version of Brown-Comenetz duality for \(\mathcal K\) is described. A nilpotence theorem and an analog of the Krull-Schmidt theorem are proved for dualizable spectra in \(\mathcal K\) and \(K\)-nilpotent spectra are characterized. The paper concludes with a description of some examples in the cases \(n=1\) and \(n=2\), and a list of open problems. Reviewer: Keith Johnson (Halifax) Cited in 6 ReviewsCited in 84 Documents MSC: 55P42 Stable homotopy theory, spectra 55P60 Localization and completion in homotopy theory 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 55T15 Adams spectral sequences Keywords:Morava \(K\)-theory; stable homotopy theory; localization Citations:Zbl 0881.55001 PDFBibTeX XMLCite \textit{M. Hovey} and \textit{N. P. Strickland}, Morava \(K\)-theories and localisation. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0929.55010) Full Text: DOI