Chromatic Picard groups at large primes

Pstrągowski, Piotr

doi:10.48550/arxiv.1811.05415

Cited by 5 publications

(6 citation statements)

References 8 publications

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“…In particular, Theorem 1.1 yields a complete classification of homotopy types of E-local spectra, and of homotopy classes of maps between them. It also implies that any E * E-comodule has a canonical realization as a homology of an E-local spectrum; as we show in a companion paper, this alone already forces algebraicity of the Picard group of the corresponding K(n)-local category [Pst18a].…”

Section: Statement Of Resultsmentioning

confidence: 86%

Chromatic homotopy is algebraic when $p > n^{2}+n+1$

Pstrągowski

2018

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We show that if E is a p-local Landweber exact homology theory of height n and p > n 2 + n + 1, then there exists an equivalence hSp E ≃ hD(E * E) between homotopy categories of E-local spectra and differential E * E-comodules, generalizing Bousfield's and Franke's results to heights n > 1. 1.4. Acknowledgements 5 2. Realization of comodules 5 3. Derived ∞-category of differential comodules 9 4. Goerss-Hopkins theory 10 5. The Bousfield adjunction 14 6. An algebraic model for the E-local homotopy category 16 Appendix A. Connectivity results for undercategories 20 Appendix B. The homotopy category of a stable ∞-category 21 References 23

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Section: Statement Of Resultsmentioning

confidence: 86%

Chromatic homotopy is algebraic when $p > n^{2}+n+1$

Pstrągowski

2018

Preprint

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“…The special case of this result for invertible Morava modules is the main theorem of [Pst18b], and our proof is closely modelled on Pstrągowski's argument. In fact, motivated by the algebraicity results of [BSS20, Pst18a, BSS19], we suspect that this theorem can be promoted to an equivalence of categories, but we will not pursue this question at this point.…”

Section: Then Statement (2) Implies Statement (1) and The Converse Is...mentioning

confidence: 84%

“…Assume throughout this subsection that 2p − 2 > n 2 + n. Our goal is to prove that all finitely generated Morava modules can be realized by K(n)-locally dualizable spectra. The special case of invertible comodules is the main result of [Pst18b]. We recall that we write E = E n and K = K(n) for brevity.…”

Section: 3mentioning

confidence: 99%

On conjectures of Hovey--Strickland and Chai

Barthel¹,

Heard²,

Naumann³

2020

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We prove the height two case of a conjecture of Hovey and Strickland that provides a K(n)-local analogue of the Hopkins-Smith thick subcategory theorem. Our approach first reduces the general conjecture to a problem in arithmetic geometry posed by Chai. We then use the Gross-Hopkins period map to verify Chai's Hope at height two and all primes. Along the way, we show that the graded commutative ring of completed cooperations for Morava E-theory is coherent, and that every finitely generated Morava module can be realized by a K(n)-local spectrum as long as 2p − 2 > n 2 + n. Finally, we deduce consequences of our results for descent of Balmer spectra.

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“…Using Theorem 1.2 we deduce the following result (Theorem 6.8). In the case k = n, this is a theorem of Pstrągowski [Pst18].…”

Section: Introductionmentioning

confidence: 88%

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

Heard¹

2021

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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 .

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Chromatic Picard groups at large primes

Cited by 5 publications

References 8 publications

Chromatic homotopy is algebraic when $p > n^{2}+n+1$

Chromatic homotopy is algebraic when $p > n^{2}+n+1$

On conjectures of Hovey--Strickland and Chai

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

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