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)