  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.
