The Spk,n–local stable homotopy category

Heard, Drew

doi:10.2140/agt.2023.23.3655

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

Drew Heard

Algebraic & Geometric Topology 23 (2023) 3655–3706

DOI: 10.2140/agt.2023.23.3655

Abstract

We study the category of $(K (k) \lor K (k + 1) \lor \dots \lor K (n))$ –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 $n^{2} + n - k$ at the $E_{2}$ –page for the sphere spectrum. We then study the Picard group of $(K (k) \lor K (k + 1) \lor \dots \lor K (n))$ –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.

Keywords

Morava $E$–theory, Morava $K$–theory, chromatic homotopy

Mathematical Subject Classification

Primary: 55P42, 55P60

Secondary: 55T15

References

Publication

Received: 13 August 2021

Revised: 15 February 2022

Accepted: 23 May 2022

Published: 5 November 2023

Authors

Drew Heard
Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim Norway

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Volume 23, issue 8 (2023)