The cyclotomic trace and curves on K-theory
The goal of this paper is to reinterpret the topological cyclic homology of \textit{M. Bökstedt, W. C. Hsiang} and \textit{I. Madsen} [Invent. Math. 111, 465--539 (1993; Zbl 0804.55004)] in terms of K-theoretic ingredients, and thus to give a reinterpretation of the cyclotomic trace. More precisely, let \(A\) be a unital and associative ring, \(K(A)\) the K-theory spectrum of \(A\) and \(\widetilde{K}(A[x]/x^n)\) the homotopy fiber of the \(K\)-theory of the projection \(A[x]/x^n\to A\). Let \(\mathcal I\) be the category whose objects are the positive integers, and whose morphisms are generated by maps \(R_s,F_s: rs\to s\) subject to the relations \[ R_1=F_1=\text{id},\quad R_rR_s=R_{rs},\quad F_rF_s=F_{sr}\quad\text{and}\quad R_rF_s=F_sR_r. \] The maps \[ \widetilde{K}(A[x]/x^{mn})\to \widetilde{K}(A[x]/x^{m}) \] given by the projection and by the K-theoretic transfer of the inclusion \(A[x]/x^m\to A[x]/x^m\) sending \(x\) to \(x^n\) define a functor from \(\mathcal I\) to spectra. Likewise for \(n\mapsto \widetilde{\text{TC}}(A[x]/x^n)\). The functoriality of the transfer requires a choice of model, and the authors employ the Barratt-Eccles construction for this purpose, following earlier work by the second order. The authors show that there is a natural equivalence \[ \text{TC}(A)\simeq @<\text{holim}<<\Omega\,\widetilde{\text{TC}}(A[x]/x^n). \] In order to prove this, the authors show that it is enough to do the case when \(A\) is replaced by the sphere spectrum. In this case the fixed points of topological Hochschild homology are more easily describable. \textit{R. McCarthy}'s theorem [Acta Math. 179, 197--222 (1997; Zbl 0913.19001)] asserts that the cylotomic trace \(\widetilde{K}(A[x]/x^n)\to \widetilde{\text{TC}}(A[x]/x^n)\) is an equivalence and so the main result of the authors gives the K-theoretic interpretation of \(\text{TC}\) and the cylotomic trace.
scientific article; zbMATH DE number 2175268