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Topological cyclic homology via the norm. (English) Zbl 1417.55015

M. Bökstedt developed coherence machinery that enabled the definition of topological Hochschild homology \(THH\) [“Topological Hochschild Homology”, Preprint (1990)]. However, it seems that the original construction makes it difficult to understand the equivariance of various additional algebraic structures on \(THH\), for example, the Adams operations and relevance to topological cyclic homology \(TC\). In the paper under review, the authors have developed a new approach to the construction of the cyclotomic structure on \(THH\) of a ring spectrum using an interpretation of \(THH\) in terms of the Hill-Hopkins-Ravenel multiplicative norm [M. A. Hill et al., in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. II: Invited lectures. Seoul: KM Kyung Moon Sa. 1219–1243 (2014; Zbl 1373.55023)].
Let \(G\) be a Lie group and \(U\) a fixed universe of \(G\)-representations. We denote by \({\mathcal S}^G_U\) the category of orthogonal \(G\)-spectra indexed on \(U\). In the paper, after reviewing necessary details on the category of orthogonal \(G\)-spectra and the geometric fixed point and norm functors, the new approach to \(THH\) is introduced. It turns out that the construction enables us to consider the cyclotomic structure on \(THH\). Here a cyclotomic spectrum is an \(S^1\)-spectrum equipped with compatible equivalences of \(S^1\)-spectra \(t_n : \rho_n^*L\Phi^{C_n}X \to X\) for which \(\rho_n : S^1\cong S^1/C_n\) is the \(n\)th root isomorphism and \(L\Phi^{C_n}\) is the left derived functor of the geometric fixed point functor \(\Phi^{C_n} : {\mathcal S}_U^{S^1} \to {\mathcal S}_{U^{C_n}}^{S^1/C_n}\).
For a ring orthogonal spectrum \(R\), the cyclic object \(N^{\text{cyc}}_\wedge R\) in orthogonal spectra is defined by \(k\)-simplices \([k] \mapsto R^{\wedge (k+1)}\). Then we define the functor \(N_e^{S^1} : {\mathcal Ass} \to {\mathcal S}_U^{S^1}\) from the category of associative ring orthogonal spectra to be the composite functor \(R \mapsto {\mathcal I}_{{\mathbb R}^\infty}^U|N^{\text{cyc}}_\wedge R|\) with \( {\mathcal I}_{{\mathbb R}^\infty}^U\) the change of universe functor [M. A. Mandell and J. P. May, Equivariant orthogonal spectra and S-modules. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1025.55002)]. The first main theorem (Theorem 1.5) asserts that if \(R\) is a cofibrant associative or cofibrant commutative ring orthogonal spectrum, then \(N_e^{S^1}R\) has a natural structure of a cyclotomic spectrum.
The homotopy limit construction with the Frobenius maps and the equivalences \(t_n\) mentioned above defines \(TR\)-theory and topological cyclic homology \(TC(R)\). Moreover, a relative version \(_AN_e^{S^1}R\) of the norm above for an algebra \(R\) over a commutative ring orthogonal spectrum \(A\) is introduced. The underlying non-equivariant spectrum of \(_AN_e^{S^1}R\) is denoted by \({}_ATHH(R)\). With also a relative version of \(TC\), the authors deduce the result (Theorem 1.11) on the existence of an \(A\)-relative cyclotomic trace map \(K(R) \to TC(R) \to {}_ATC(R)\) whose second map is a lift of the natural map \(THH(R) \to {}_ATHH(R)\) in the stable category.
Furthermore, as announced in the Introduction, the paper deals with and studies the Adams operations on \({}_AN_e^{S^1}R\) and spectral sequences for \(TR\)-theory whose \(E_2\)-terms are described by Mackey functors.

MSC:

55P91 Equivariant homotopy theory in algebraic topology
19D55 \(K\)-theory and homology; cyclic homology and cohomology
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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