The algebraic K-theory of the K(1)-local sphere via TC

TL;DR

This paper investigates the algebraic K-theory of the K(1)-local sphere and type 2 finite spectra at prime 2, linking it to K-theory of discrete rings and topological cyclic homology, revealing new torsion classes.

Contribution

It provides a novel description of the algebraic K-theory of the K(1)-local sphere and type 2 spectra using topological cyclic homology, identifying new torsion classes.

Findings

Identifies an infinite family of 2-torsion classes in K_0 of type 2 spectra at prime 2

Provides constructions for representatives of these K_0 classes

Relates algebraic K-theory to topological cyclic homology and discrete rings

Abstract

We describe the algebraic K-theory of the $K(1)$-local sphere and the category of type 2 finite spectra in terms of K-theory of discrete rings and topological cyclic homology. We find an infinite family of 2-torsion classes in the $K_0$ of type 2 spectra at the prime 2, and explain how to construct representatives of these $K_0$ classes.