Cyclotomic complexes (Q2866960)

This article is a continuation of a previous article of the author [Mosc. Math. J. 11, No. 4, 723--803 (2011; Zbl 1273.18025)], and depends heavily on that. The goal of both this and the previous is to understand the connection between the purely topological notions in the noncommutative Hodge-to-de Rham degeneration and the topological cyclic homology of the cyclotomic trace and algebraic \(K\)-theory of spaces. This project involves constructing the appropriate homological counterparts of most of the notions in stable homotopy theory.NEWLINENEWLINEThe derived Mackey functors was treated in the first article. These are the homological analogues of the genuine \(G\)-equivariant spectra.The abelian category of Mackey functors is sufficiently known in stable homotopy theory and plays an important role. This is not the case for its naive derived generalization, and so a slightly different derived version of the Mackey functors was constructed and studied in the first article.NEWLINENEWLINEThe present article tries to do a similar job for the cyclotomic spectra studied by Bökstedt and Hsiang in the \(K\)-theoretic view, and this needs additional theory and methods. Cyclotomic spectra are equivariant with respect to the group \(S^1\) while the Mackey functors only deal with finite groups. It is not possible to construct good homological analogues of all \(S^1\)-spectra, but a category \(\mathcal D\mathcal M\Lambda(\mathbb Z)\) of \textit{cyclic Mackey functors} can be constructed which captures the parts of the equivariant stable category relevant for topological cyclic homology. Given this, the author introduces the triangulated category \(\mathcal D\Lambda R(\mathbb Z)\) of \textit{cyclotomic complexes}. The relation between cyclotomic complexes and and cyclotomic spectra should be given by correspondences \(\mathcal D\Lambda R(\mathbb Z)\rightarrow\text{Cycl}\), \(\text{Cycl}\rightarrow\text{StHom}\), \(\mathcal D\Lambda R(\mathbb Z)\rightarrow\mathcal D(\mathbb Z)\), \(\mathcal D(\mathbb Z)\rightarrow\text{StHom}\) of understandable new schemes (which are e.g. derived categories of several kinds): \(\text{StHom}\) is the stable homotopy category, \(\mathcal D(\mathbb Z)\) is the derived category of abelian groups and \(\text{Cycl}\) is the category of cyclotomic spectra. As this diagram, and in particular the category of cyclotomic spectra, are hard to construct, at least such that the correspondence diagram is almost cartesian, the following are the possible partial steps at the time being:NEWLINENEWLINE(i) The first correspondence is approached by a construction of an equivariant homology cyclotomic complex \(C_\bullet(T)\) for each cyclotomic spectrum \(T\).NEWLINENEWLINE(ii) A topological cyclic homology functor \(\text{TC}\) is constructed on the category \(\mathcal D\Lambda R(\mathbb Z)\) such that for every cyclotomic spectrum \(T\) the complex \(\text{TC}(C_\bullet(T))\) is naturally identified with the homology of the spectrum \(\text{TC}(T)\).NEWLINENEWLINEFor the category \(\mathcal D\Lambda R(\mathbb Z)\) it can be proved that cyclotomic complexes are essentially equivalent to the filtered Diudonné modules which are linear-algebraic objects which supply a \(p\)-adic counterpart of Deligne's notion of a mixed Hodge structure. This gives a motivic view on the theory.NEWLINENEWLINEThe author focuses on pure linear algebra and leave the geometric applications for future research. However, it is proved that for profinitely complete cyclotomic complexes, the topological cyclic homology \(\text{TC}\) coincides with the syntomic cohomology.NEWLINENEWLINEThe article gives a model for \(S^1\)-equivariant spaces and their cohomology. a combinatorial approach is chosen, using Connes' category \(\Lambda\). A construction of the cyclic Mackey functors is given, a study of the cyclotomic complexes then follows. The author constructs equivariant homology functors from \(S^1\)-spectra to \(\mathcal D\mathcal M\Lambda(\mathbb Z)\), and from cyclotomic spectra to \(\mathcal D\Lambda R(\mathbb Z)\). A theorem comparing cyclotomic spectra and filtered Dieudonné modules is proved, and finally, topological cyclic homology is studied, and comparison theorems for \(\text{TC}\) are given.NEWLINENEWLINEThe article is technical and involved. It should also be mentioned that it depends on a deep understanding of \(A_\infty\)-algebras and functors of such. It digs deep in the theory to achieve the final goal of understanding the connection given initially.