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The $\mathrm{Sp}_{k,n}$–local stable homotopy category

Drew Heard

Algebraic & Geometric Topology 23 (2023) 3655–3706

We study the category of (K(k)K(k + 1)K(n))–local spectra, following a suggestion of Hovey and Strickland. 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 n2 + n k at the E2–page for the sphere spectrum. We then study the Picard group of (K(k)K(k + 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.

Morava $E$–theory, Morava $K$–theory, chromatic homotopy
Mathematical Subject Classification
Primary: 55P42, 55P60
Secondary: 55T15
Received: 13 August 2021
Revised: 15 February 2022
Accepted: 23 May 2022
Published: 5 November 2023
Drew Heard
Department of Mathematical Sciences
Norwegian University of Science and Technology

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