Invertible spectra of finite type

TL;DR

This paper characterizes the exact numerical conditions under which elements in the Picard group of the $K(2)$-local category at primes $p \\geq 5$ have finitely generated homotopy groups, clarifying finite type properties.

Contribution

It provides a complete numerical criterion for finite type elements in the Picard group of the $K(2)$-local category at primes $p \\geq 5$, advancing understanding of their structure.

Findings

Identifies necessary and sufficient conditions for finite type elements.

Clarifies the structure of the Picard group at prime $p \\geq 5$.

Enhances classification of $K(2)$-local spectra.

Abstract

We describe the necessary and sufficient numerical condition when an element $X$ in the Picard group of $K(2)$-local category at prime $p \geqslant 5$ is of finite type, i.e., $\pi_kX$ is finitely generated as a $\mathbb{Z}_p$-module for all $k \in \mathbb{Z}$.