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

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$⟨n⟩$) $\to$ K(E$(n)$) and of K(BP$⟨n-1⟩$) in rational topological Hochschild homology.