The Spk,n–local stable homotopy category

Abstract

We study the category of .K.k/_K.k C 1/_ _K.n//-local spectra, following a suggestion of Hovey and Strickland. When k D 0, this is equivalent to the category of E.n/-local spectra, while for k D n, this is the category of K.n/-local spectra, both of which have been studied in detail by Hovey and Strickland. Based on their ideas, we classify the localizing and colocalizing subcategories, and give characterizations of compact and dualizable objects. We construct an Adams-type spectral sequence and show that when p n it collapses with a horizontal vanishing line above filtration degree n 2 C n k at the E 2 -page for the sphere spectrum. We then study the Picard group of .K.k/_K.k C 1/_ _K.n//-local spectra, showing that this group is algebraic, in a suitable sense, when p n. We also consider a version of Gross-Hopkins duality in this category. A key concept throughout is the use of descent. 55P42, 55P60; 55T152 The category of Sp k;n -local spectra 2A Chromatic spectraWe begin by introducing some of the main spectra that we will be interested in.Definition 2.1 Let BP denote the Brown-Peterson homotopy ring spectrum with coefficient ring BP Š Z .p/ OEv 1 ; v 2 ; : : :Remark 2.2 The classes v i are not intrinsically defined, and so the definition of BP depends on a choice of sequence of generators; for example, they could be the Hazewinkel generators or the Araki generators. However, the ideals I n D .p; v 1 ; : : : ; v n 1 / for 0 Ä n Ä 1 do not depend on this choice.