The $\mathop{Sp}_{k,n}$-local stable homotopy category

Abstract

Following a suggestion of Hovey and Strickland, we study the category of K(k) ∨ K(k +1)∨• • •∨K(n)-local spectra. When k = 0, this is equivalent to the category of E(n)-local spectra, while for k = 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 + n − k at the E 2 -page for the sphere spectrum. We then study the Picard group ofshowing 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. Contents1. Introduction 1 2. The category of Sp k,n -local spectra 4 3. Thick subcategories and (co)localizing subcategories 8 4. Descent theory and the E(n, J k )-local Adams spectral sequence 17 5. Dualizable objects in Sp k,n 22 6. The Picard group of the Sp k,n -local category 26 7. E(n, J k )-local Brown-Comenetz duality 30 References 32 2. The category of Sp k,n -local spectra2A. Chromatic spectra. We begin by introducing some of the main spectra that we will be interested in.2.1. Definition. Let BP denote the Brown-Peterson ring spectrum with coefficient ring2.2. Remark. The generators v i are not well-defined; for example they could be the Hazewinkel generators or the Araki generators. However, the ideals I n = (p, v 1 , . . . , v n−1 ) for 0 ≤ n ≤ ∞ are well-defined.By taking quotients and localizations of BP (for example, using the theory of structured ring spectra [EKMM97, Chapter V]), we can form new ring spectra. In particular, let J k denote a fixed invariant regular sequence p i0 , v i1 1 , . . . , v i k−1 k−1 of length k. Then we can form the associative ring spectrum BP J k with (BP J k ) * ∼ = BP * /J k .