Journal of the European Mathematical Society

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Volume 20, Issue 2, 2018, pp. 489–527
DOI: 10.4171/JEMS/772

Published online: 2018-01-31

Logarithmic topological Hochschild homology of topological $K$-theory spectra

John Rognes[1], Steffen Sagave[2] and Christian Schlichtkrull[3]

(1) University of Oslo, Norway
(2) Radboud Universiteit Nijmegen, The Netherlands
(3) University of Bergen, Norway

In this paper we continue our study of logarithmic topological Hochschild homology. We show that the inclusion of the connective Adams summand$\ell$ into the $p$-local complex connective $K$-theory spectrum $ku_{(p)}$, equipped with suitable log structures, is a formally log THH-├ętale map, and compute the $V(1)$-homotopy of their logarithmic topological Hochschild homology spectra. As an application, we recover Ausoni's computation of the $V(1)$-homotopy of the ordinary THH of $ku$.

Keywords: Logarithmic ring spectrum, Adams summand, complex topological $K$-theory spectrum

Rognes John, Sagave Steffen, Schlichtkrull Christian: Logarithmic topological Hochschild homology of topological $K$-theory spectra. J. Eur. Math. Soc. 20 (2018), 489-527. doi: 10.4171/JEMS/772