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

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.