Advanced Studies: Euro-Tbilisi Mathematical Journal

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

R. Kato, Y. Kawamoto, H. Okajima, K. Shimomura

Let 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 L_n as a collection of isomorphism classes of invertible spectra, whose exotic summand Pic⁰(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 L_n if and only if X ⋀ 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 < n ² ≤ 2p+2, we consider the cases for (p, n) = (5, ) and (7, 4). In these cases, we characterize them by the Smith-Toda spectra V(n-1). For this sake, we show that L₃V(2) at the prime five and L₄V(3) at the prime seven are ring spectra.


Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 17(2) (2024), pp. 91-109