A relation among Hopkins’ Picard groups of the localized categories with respect to finite wedges of the Morava 𝐾-theories

Abstract

We work in the stable homotopy category of p p -local spectra for a fixed prime number p p . Let E E be a spectrum and L E \mathcal {L}_E denote the stable homotopy category of localized spectra with respect to E E in the sense of Bousfield. Then, M. Hopkins introduced a Picard group P i c ( L E ) Pic(\mathcal {L}_E) of the category L E \mathcal {L}_E . If the spectra E E and F F satisfy the relation ⟨ E ⟩ ≥ ⟨ F ⟩ \left \langle {E}\right \rangle \ge \left \langle F\right \rangle of the Bousfield classes, then we have a homomorphism ℓ : P i c ( L E ) → P i c ( L F ) \ell \colon Pic(\mathcal {L}_E)\to Pic(\mathcal {L}_F) . We consider the spectra K m n = E ( n ) ∧ M J m K_m^n=E(n)\wedge MJ_m for the n n -th Johnson-Wilson spectrum E ( n ) E(n) and a type m m generalized Moore spectrum M J m MJ_m for 0 ≤ m ≤ n 0\le m\le n . For E = K m n E=K_m^n , we have a subgroup P i c 0 ( L E ) Pic^0({\mathcal {L}_E}) of P i c ( L E ) Pic(\mathcal {L}_E) consisting of exotic elements. In this paper, we study the homomorphism ℓ : P i c 0 ( L E ( n ) ) → P i c 0 ( L K m n ) \ell \colon Pic^0({\mathcal {L}_{E(n)}})\to Pic^0({\mathcal {L}_{K_m^n}}) , and give conditions under which it is an isomorphism. This is a generalization of the result P i c 0 ( L 2 ) ≅ κ 2 Pic^0(\mathcal {L}_2)\cong \kappa _2 (P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk [J. Topol. 8 (2015), 267–294, Remark. 6.5]) for ( p , n , m ) = ( 3 , 2 , 2 ) (p,n,m)=(3,2,2) .