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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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