Generalized Moore spectra and Hopkins' Picard groups for a smaller chromatic level

2021/11/05 by Kato, Ryo, Kawamoto, You-na, Okajima, Hiroki +1
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2111.03291

Abstract

Let \mathcal L_n for a positive integer n denote the stable homotopy category of v_n^-1BP-local spectra at a prime number p. Then, M.~Hopkins defines the Picard group of \mathcal L_n as a collection of isomorphism classes of invertible spectra, whose exotic summand Pic⁰(\mathcal L_n) is studied by several authors. In this paper, we study the summand for n with n²≤ 2p+2. For n²≤ 2p-2, it consists of invertible spectra whose K(n)-localization is the K(n)-local sphere. In particular, X is an exotic invertible spectrum of \cL_n if and only if X\wedge MJ is isomorphic to a v_n^-1BP-localization of the generalized Moore spectrum MJ for an invarinat regular ideal J of length n. For n with 2p-2

Related

• On Hopkins' Picard group
• Picard groups of quotient ring spectra
• Bounding the K(p-1)-local exotic Picard group at p>3
• Chromatic Picard groups at large primes
• The Picard group in equivariant homotopy theory via stable module categories