On localization sequences in the algebraic K-theory of ring spectra

TL;DR

This paper explores the algebraic K-theory of ring spectra, identifying fibers of localizations via endomorphism spectra, and provides a negative answer to a question about the relationship between K-theories of certain spectra.

Contribution

It introduces a new description of K-theoretic fibers of localizations in ring spectra using endomorphism spectra, and addresses an open question by comparing traces in topological Hochschild homology.

Findings

Identifies K-theoretic fibers using endomorphism spectra.

Provides a negative answer to Rognes' question for n>1.

Compares traces in rational topological Hochschild homology.

Abstract

We identify the $K$-theoretic fiber of a localization of ring spectra in terms of the $K$-theory of the endomorphism algebra spectrum of a Koszul-type complex. Using this identification, we provide a negative answer to a question of Rognes for $n>1$ by comparing the traces of the fiber of the map $K(BP(n))\rightarrow K(E(n))$ and of $K(BP(n-1))$ in rational topological Hochschild homology.