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On localization sequences in the algebraic \(K\)-theory of ring spectra. (English) Zbl 06852551
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.

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
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