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Localization for \(THH(ku)\) and the topological Hochschild and cyclic homology of Waldhausen categories. (English) Zbl 1441.19001

Memoirs of the American Mathematical Society 1286. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4178-4/pbk; 978-1-4704-6140-9/ebook). v, 113 p. (2020).
Trace maps to topological Hochschild homology (\(\mathrm{THH}\)) and topological cyclic homology (\(\mathrm{TC}\)) are among the most powerful tools for the notoriously difficult task of computing the higher algebraic \(K\)-theory of a ring or a structured ring spectrum. While one general input for algebraic \(K\)-theory constructions are Waldhausen categories, the usual definitions of topological Hochschild homology apply to rings, structured ring spectra or more generally to spectral categories, that is, categories enriched in a suitable symmetric monoidal category of spectra.
In the paper under review, the authors give a construction of \(\mathrm{THH}\) and \(\mathrm{TC}\) that applies to Waldhausen categories with a compatible spectral enrichment. For one choice of input data (called the non-connective variant), this construction gives rise to the relative \(\mathrm{THH}\)- and \(\mathrm{TC}\)-terms that appear in the localization sequence studied by the authors in [Geom. Topol. 16, No. 2, 1053–1120 (2012; Zbl 1282.19004)]. This localization sequence matches the localization sequence in algebraic \(K\)-theory established by R. W. Thomason and T. Trobaugh [Prog. Math. 88, 247–435 (1990; Zbl 0731.14001)]. For a different choice of input (called the connective variant), they obtain \(\mathrm{THH}\)- and \(\mathrm{TC}\)-terms that participate in another type of localization sequence. The construction of these localization sequences is the main result of this paper. They generalize the localization sequence associated with a discrete valuation ring that was studied by L. Hesselholt and I. Madsen [Ann. Math. (2) 158, No. 1, 1–113 (2003; Zbl 1033.19002)]. One key application of the new localization sequences are the \(\mathrm{THH}\)- and \(\mathrm{TC}\)-localization sequences associated to the transfer map built from the Eilenberg–Mac Lane spectrum \(H\mathbb Z\) and the connective complex topological \(K\)-theory spectrum \(ku\). They match the corresponding localization sequence in algebraic \(K\)-theory established by the authors in [Acta Math. 200, No. 2, 155–179 (2008; Zbl 1149.18008)].
The paper is organized as follows. In the first chapter, the authors review the construction and the basic properties of \(\mathrm{THH}\) and \(\mathrm{TC}\) for spectral categories. In the second chapter, they introduce a notion of simplicially enriched Waldhausen categories that extends that of Waldhausen’s categories with cofibrations and weak equivalences. They also supply a relative version of the former, called enhanced simplicially enriched Waldhausen categories. Suitable subcategories of cofibrant objects in simplicial model categories where all objects are fibrant provide examples of simplicially enriched Waldhausen categories. The authors then go on to show that their variants of Waldhausen categories give rise to spectral categories so that one can form \(\mathrm{THH}\) or \(\mathrm{TC}\) and the desired trace maps from the algebraic \(K\)-theory of the underlying Waldhausen categories. In the third chapter, the authors show that some of the key theorems about Waldhausen \(K\)-theory have analogues for their variants of \(\mathrm{THH}\) and \(\mathrm{TC}\), among them the additivity theorem, the fibration theorem and the sphere theorem established by F. Waldhausen [Lect. Notes Math. 1126, 318–419 (1985; Zbl 0579.18006)].
In chapter 4, the authors apply their general theory to produce localization sequences for the \(\mathrm{THH}\) or \(\mathrm{TC}\) of rings and structured ring spectra. As a first application, they obtain localization sequences relating the \(\mathrm{THH}\) and \(\mathrm{TC}\) of a discrete valuation ring, its quotient field and its field of fractions. These localization sequences were first studied by L. Hesselholt and I. Madsen [Ann. Math. (2) 158, No. 1, 1–113 (2003; Zbl 1033.19002)]. As a second application, they obtain localization sequences relating the \(\mathrm{THH}\) and \(\mathrm{TC}\) of the connective complex topological \(K\)-theory spectrum \(ku\) and of the integers. The third term in these localization sequences should be viewed as a replacement of the \(\mathrm{THH}\) or \(\mathrm{TC}\) of the periodic complex topological \(K\)-theory spectrum \(KU\), which would not participate in such a localization sequence. There is also a version of the localization sequence for the \(p\)-local and the \(p\)-completed Adams summand, and it is shown that the localization sequences are compatible with the corresponding localization sequences in algebraic \(K\)-theory. In the last chapter, they introduce a notion of weakly exact functors and a construction of simplicially enriched Waldhausen categories from ordinary Waldhausen categories satisfying a certain mild assumption.
Due to the long time between the submission of the manuscript and its publication, several newer developments in the field are unfortunately not reflected in this memoir, for example the new approach to topological cyclic homology established by T. Nikolaus and P. Scholze [Acta Math. 221, No. 2, 203–409 (2018; Zbl 1457.19007)] and the subsequent work it has generated.

MSC:

19-02 Research exposition (monographs, survey articles) pertaining to \(K\)-theory
19D55 \(K\)-theory and homology; cyclic homology and cohomology
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
19L41 Connective \(K\)-theory, cobordism
19D10 Algebraic \(K\)-theory of spaces
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References:

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