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**Motivic symmetric spectra.**
*(English)*
Zbl 0969.19004

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 M. Hovey, B. Shipley and 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 \({\mathcal S}pt^{\Sigma}_{S^1}(Sm|_S)_{\text{Nis}}\) of symmetric \(S^1\)-spectra. The first appendix is on properness, and the second deals with motivic homotopy theory of presheaves.

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 \({\mathcal S}pt^{\Sigma}_{S^1}(Sm|_S)_{\text{Nis}}\) of symmetric \(S^1\)-spectra. The first appendix is on properness, and the second deals with motivic homotopy theory of presheaves.

Reviewer: W.W.J.Hulsbergen (Haarlem)

### MSC:

19E15 | Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) |

14F42 | Motivic cohomology; motivic homotopy theory |

55P42 | Stable homotopy theory, spectra |

18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |