Motivic symmetric spectra
type: Item
instance of: scholarly article
title: Motivic symmetric spectra
zbMATH Open document ID: 0969.19004
published in: Documenta Mathematica
publication date: 2000-11-30T00:00:00Z
full work available at URL: https://eudml.org/doc/121235
full work available at URL: 15.html
review text: Let \(T\) denote the quotient of sheaves \({\mathbb A}^1/({\mathbb A}^1-0)\) in motivic homotopy theory, and let \(X\) denote a \(T\)-spectrum. Morel and Voevodsky introduced a motivic stable category, which can be obtained by formally inverting the functor \(X\rightarrow T\wedge X\). This category is fundamental for Voevodsky's proof of the Milnor conjecture. The underlying paper gives a method for importing the stable homotopy theory of symmetric spectra as developed by \textit{M. Hovey, B. Shipley} and \textit{J. Smith} [J. Am. Math. Soc. 13, 149-208 (2000; Zbl 0931.55006)] into Morel's and Voevodsky's motivic stable category. The paper consists of four chapters, two appendices and an index. The first chapter supplies the necessary tools such as motivic homotopy theory, controlled fibrant models, Nisnevich descent and flasque simplicial presheaves. The second chapter deals with motivic stable categories. Topics discussed are level structures, compact objects, stable closed model structures, change of suspension and bounded cofibrations. The third chapter is on fibre and cofibre sequences. It is subdivided into four sections on exact sequences for \(S^1\)-spectra, weighted stable homotopy groups, fibre and cofibre sequences, and \(T\)-suspensions and \(T\)-loops, respectively. The fourth chapter gives the main results. It discusses motivic symmetric spectra. Subjects dealt with are level structures, stable structures, smash product, equivalence of stable categories, and symmetric \(S^1\)-spectra. On the whole the paper is rather technical and probably meant for specialists in the field. The key result states that ``The category \({\mathcal S}pt^{\Sigma}_T(Sm|_S)_ {\text{Nis}}\) of symmetric \(T\)-spectra on the smooth Nisnevich site, and the classes of stable equivalences, stable fibrations and stable cofibrations, together satisfy the axioms for a proper closed simplicial model category''. The same result holds for the category \({\mathcal S}pt^{\Sigma}_{S^1}(Sm|…
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 1527083
zbMATH Keywords: motivic stable category
zbMATH Keywords: motivic symmetric spectra
zbMATH Keywords: motivic homotopy theory
zbMATH Keywords: level structures
zbMATH Keywords: compact objects
zbMATH Keywords: stable closed model structures
zbMATH Keywords: change of suspension
zbMATH Keywords: bounded cofibrations
zbMATH Keywords: cofibre sequence
zbMATH Keywords: fibre sequence
author: J. F. Jardine
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1527083