Cyclotomic complexes
type: Item
instance of: scholarly article
title: Cyclotomic complexes
zbMATH Open document ID: 1301.19004
published in: Izvestiya: Mathematics
publication date: 2013-12-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.2810
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 6236931
zbMATH Keywords: cyclotomic spectrum
zbMATH Keywords: cyclotomic complex
zbMATH Keywords: cyclotomic spectra
zbMATH Keywords: \(p\)-adic Hodge structure
zbMATH Keywords: filtered Dieudonné module
zbMATH Keywords: Mackey functors
zbMATH Keywords: \(A_\infty\)-algebras
zbMATH Keywords: \(A_\infty\)-functors
zbMATH Keywords: Connes'cyclic category
zbMATH Keywords: \(S^1\) equivariant spaces
zbMATH Keywords: derived cyclic categories
zbMATH Keywords: admissible groups
zbMATH Keywords: admissible functors
author: Dmitry Kaledin
MaRDI profile type: Publication
arXiv ID: 1003.2810
cites work: Derived Mackey functors
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cites work: Q3769025
cites work: Q4367927
cites work: Q3774051
cites work: Relative algebraic K-theory and cyclic homology
cites work: Q4660482
cites work: Constructions of factorization systems in categories
OpenAlex ID: W3037033473
DOI: 10.1070/IM2013V077N05ABEH002663
review text: 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 \(\mathcal D\Lambda R(\mathbb Z)\rightarrow\text{Cycl}\), \(\text{Cycl}\rightarrow\text{StHom}\), \(\mathcal D\Lambda R(\mathbb Z)\rightarrow\mathcal D(\mathbb Z)\),…
reviewed by: Arvid Siqveland
description: scientific article; zbMATH DE number 6236931