Subsection 1.4. (0NRH) · ScalingStacks

1.4. Trace methods

One of the major revolutions in the calculational study of algebraic $K$ -theory of rings and schemes in the past two decades has been the development of trace methods, following the ideas of Goodwillie and Bokstedt-Hsiang-Madsen [17]. The cyclotomic trace from $K$ -theory to topological cyclic homology $TC$ and topological Hochschild homology $THH$ (stable homotopy theory generalizations of negative cyclic homology and Hochschild homology) has allowed major calculational advances. The fiber of this map is well understood by work of Goodwillie, McCarthy, and Dundas [58, 26], and the target is relatively computable using the methods of equivariant stable homotopy theory (e.g., see the extensive body of work by Hesselholt and Madsen on the Quillen-Lichtenbaum conjecture [40]). One application of the co-representability of algebraic $K$ -theorem (theorem 1.3) is the complete classification of all natural transformations with source the algebraic $K$ -theory functor.

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Theorem 1.12. (see theorem 10.3) Given an additive invariant

E:\Cat_{\infty}^{\ex}\longrightarrow{\mathcal{S}}_{\infty}

with values in the stable $\infty$ -category of spectra, we have a natural equivalence of spectra

\mathrm{Map}(K,E)\simeq E({\mathcal{S}}_{\infty}^{\omega}),

where $\mathrm{Map}(K,E)$ denotes the spectrum of natural transformations of additive invariants. The analogous result for localizing invariants holds. In the particular case where $E$ is topological Hochschild homology, we obtain an isomorphism

\pi_{0}\mathrm{Map}(K,THH)\simeq\pi_{0}THH({\mathcal{S}}_{\infty}^{\omega})\simeq\pi_{0}THH(\mathbb{S})\simeq{\mathbb{Z}}.

A calculation then provides a canonical construction and conceptual description of the topological Dennis trace map $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}THH$ .

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Corollary 1.13. (see section 10) The set of homotopy classes of natural transformations of additive invariants from connective algebraic $K$ -theory to $THH$ is isomorphic to ${\mathbb{Z}}$ ; furthermore, the topological Dennis trace is characterized up to homotopy as the natural transformation $K\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}THH$ corresponding to the unit $1\in{\mathbb{Z}}$ .

That is, up to scaling, the trace is the only natural transformation of additive invariants between connective algebraic $K$ -theory and $THH$ . This provides a direct proof that all known constructions of the topological Dennis trace map agree up to homotopy.

Working directly with topological cyclic homology ( $TC$ ) is somewhat more complicated; $TC$ does not preserve filtered colimits in general, and is therefore not an additive or localizing functor. However, we deduce an analogous identification of the cyclotomic trace as determined by the unit map; see section 10.

Finally, we note that in the localizing setting our results provide an extension of the cyclotomic trace from non-connective algebraic $K$ -theory to the non-connective versions of $TC$ and $THH$ . This generalizes and extends the non-connective traces constructed in [35] and [10] for rings and schemes.

Andrew J. Blumberg, David Gepner, Goncalo Tabuada

Original source: arXiv:1001.2282v4

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