Antieau, Benjamin; Barthel, Tobias; Gepner, David On localization sequences in the algebraic \(K\)-theory of ring spectra. (English) Zbl 06852551 J. Eur. Math. Soc. (JEMS) 20, No. 2, 459-487 (2018). 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. Cited in 1 Document MSC: 19D55 \(K\)-theory and homology; cyclic homology and cohomology 55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 18E30 Derived categories, triangulated categories (MSC2010) 19D10 Algebraic \(K\)-theory of spaces Keywords:algebraic \(K\)-theory; structured ring spectra; trace methods PDF BibTeX XML Cite \textit{B. Antieau} et al., J. Eur. Math. Soc. (JEMS) 20, No. 2, 459--487 (2018; Zbl 06852551) Full Text: DOI arXiv