Invertible objects in Franke's comodule categories
type: Item
instance of: scholarly article
title: Invertible objects in Franke's comodule categories
author: Drew Heard
published in: Mathematica Scandinavica
publication date: 2024-06-25T00:00:00Z
review text: Chromatic homotopy theory studies localizations of the \(p\)-local stable homotopy category with respect to the Morava \(K(n)\)-theories and the Johnson-Wilson spectra \(E(n)\). These categories are known to be more algebraic in nature than the whole stable homotopy category. Unpublished work of \textit{J. Franke} [``Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence'', Unpublished Preprint, available at \url{https://faculty.math.illinois.edu/K-theory/0139/} (1996)] constructs an algebraic model for the \(E(n)\)-local stable homotopy category, which is an equivalence when \(n=1\) and \(p>2\).\N\NFollowing Franke's construction this paper studies quasi-periodic derived categories of comodules over a Hopf algebroid. These algebraic categories are symmetric monoidal and the main results of this paper are on their Picard groups -- the group of objects which are units under the monoidal product.\N\NThe author defines the notion of a descendable Hopf algebroid and uses this to construct an equivalence of stable infinity categories between the quasi-perodic comodule category and a totalisation of a complex of categories of modules. This form of descent is compatible with taking Picard spectra and hence gives a calculation of the Picard group (which is \(\pi_0\) of the Picard spectrum) of the quasi-periodic comodule category.\N\NMore specifically, when \(2p - 2 > n^2 + n\), the Picard group of quasi-periodic \(E_0 E\)-comodules for \(E\) a 2-periodic Landweber exact cohomology theory of height \(n\) (such as Morava \(E\)-theory) is infinite cyclic, generated by the suspension of the unit. Some results along this line have been proven by \textit{D. Barnes} and \textit{C. Roitzheim} [Adv. Math. 228, No. 6, 3223--3248 (2011; Zbl 1246.55009)] and \textit{T. Barthel} et al. [Invent. Math. 220, No. 3, 737--845 (2020; Zbl 1442.55002)]. However, those results rely on calculations from \(E(n)\)-local stable homotopy theory.\N\NFollowing Bar…
reviewed by: David Barnes
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 18G35
Mathematics Subject Classification ID: 18N40
Mathematics Subject Classification ID: 55P43
zbMATH DE Number: 7870739
zbMATH Keywords: chromatic homotopy theory
zbMATH Keywords: stable homotopy theory
zbMATH Keywords: Hopf algebra
zbMATH Keywords: comodules
MaRDI profile type: Publication
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DOI: 10.7146/MATH.SCAND.A-142361
description: scientific article; zbMATH DE number 7870739