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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.

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
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