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# Homology, Homotopy and Applications

## Volume 24 (2022)

### Number 1

### On the Picard group graded homotopy groups of a finite type two $K(2)$-local spectrum at the prime three

Pages: 177 – 203

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a10

#### Authors

#### Abstract

Consider Hopkins’ Picard group of the stable homotopy category of $E(2)$-local spectra at the prime three, consisting of homotopy classes of invertible spectra. Then, it is isomorphic to the direct sum of an infinite cyclic group and two cyclic groups of order three. We consider the Smith–Toda spectrum $V(1)$ and the cofiber $V_2$ of the square $\alpha^2$ of the Adams map, which is a ring spectrum. In this paper, we introduce imaginary elements which make computation clearer, and determine the module structures of the Picard group graded homotopy groups $\pi_\star (V (1))$ and $\pi_\star (V2)$.

#### Keywords

homotopy group, Adams–Novikov spectral sequence, Bousfield–Ravenel localization

#### 2010 Mathematics Subject Classification

55P42, 55Q51, 55Q99, 55T15

Received 21 November 2016

Received revised 23 August 2019

Published 6 April 2022