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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.

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
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