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

Abstract

Following a suggestion of Hovey and Strickland, we study the category of $K(k)\vee K(k+1)\vee \cdots \vee K(n)$-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 $p\gg n$ 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 $K(k)\vee K(k+1)\vee \cdots \vee K(n)$-local spectra, showing that this group is algebraic, in a suitable sense, when $p\gg n$. We also consider a version of Gross--Hopkins duality in this category. A key concept throughout is the use of descent.

arXiv comment34 pages, comments welcome. v2: version accepted for publication in AG&T

References in corpus (9)

• Nilpotence and descent in equivariant stable homotopy theory
• The Picard group of topological modular forms via descent theory
• On Hopkins' Picard groups for the prime 3 and chromatic level 2
• Local duality in algebra and topology
• Separable extensions in tensor-triangular geometry and generalized Quillen stratification
• K-theory, reality, and duality
• The $E_2$-term of the $K(n)$-local $E_n$-Adams spectral sequence
• Centralizers in good groups are good
• Variations of the telescope conjecture and Bousfield lattices for localized categories of spectra

Cited by in corpus (2)

• Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality
• The inverse limit topology and profinite descent on Picard groups in $K(n)$-local homotopy theory