On localization sequences in the algebraic \(K\)-theory of ring spectra (Q1708556)

Summary: 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 \(\mathrm{K}(\mathrm{BP}\langle n\rangle)\to\mathrm{K}(\mathrm{E}(n))\) and of \(\mathrm{K}(\mathrm{BP}\langle n-1\rangle)\) in rational topological Hochschild homology.