Motivic symmetric spectra

By John F. Jardine. Documenta Mathematica, 2000

303 citations recorded by the source corpus.

This paper demonstrates the existence of a theory of symmetric spectra for the motivic stable category. The main results together provide a categorical model for the motivic stable category which has an internal symmetric monoidal smash product. The de- tails of the basic construction of the Morel-Voevodsky proper closed simplicial model structure underlying the motivic stable category are required to handle the symmetric case, and are displayed in the rst three sections of this paper.

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Citations in the corpus

On the motivic spectra representing algebraic cobordism and algebraic $K$-theory (2007, methodology): “Jardine’s motivic symmetric spectra [7]; in particular, this is a much stronger than the assertion that algebraic K-theory defines a presheaf of (ordinary) E∞ spectra on an appropriate site.”
Galois equivariance and stable motivic homotopy theory (2014, background): “As shown in [43] there is a monoidal Quillen equivalence between SptP1(S) and Jardine’s model category of motivic symmetric P(1)-spectra [31].”
CATEGORICAL HOMOTOPY THEORY (2006, background): “The development of these theories and their applications was initiated in the 1980s [7], [12], [19] and continues to the present [8], [16].”
HZ -algebra spectra are differential graded algebras (2002, background): “On the other hand, although this corollary applies to the rational motivic stable homotopy theory of schemes from [ Jar98 , Voe98] no such explicit algebraic model is known.”
Aspects of Enumerative Geometry with Quadratic Forms (2017, methodology): “We use [21, 23, 36] as references for these constructions and their basic properties.”
Stable model categories are categories of modules (2003, background): “In [30], Jardine provides the details of the construction of model categories of T spectra over the spectrum of a field k.”
Théories homotopiques dans les topos (2002, background): “Le r&esultat qui suit est une l&eg= ere am&elioration du pr&ec&edent, et sa d&emonstration s’inspire d’un cas particulier trait&e dans un appendice de [ 10 ], lequel apparaˆ Tt en corollaire peu apr= es (4.11).”
La réalisation étale et les opérations de Grothendieck (2014, background): “Il induit alors une équivalence en tre la catégorieSH(S) de [29] et celle de [5].”
On the Motivic π0 of the Sphere Spectrum (2004, methodology): “In fact following [10, 7] one can put a symmetric monoidal structure on SH 1 s (k) whose product, denoted by ∧, is indeed induced by the smashproduct of pointed simplicial sheaves of sets on VNis.”
Motivic cell structures (2003, background): “(1) It is flasque, in the sense of [ Ja , Section 1.4];”
Morita theory in abelian, derived and stable model categories (2003, methodology): “he ‘Tate-sphere’) is an important tool in Voevodsky’s proof of the Milnor conjecture [Voe]. There are several stable model categories underlying this motivic stable homotopy category, see for example [Jar00b], [Hov01b], [Hu03] or [DØR]. 4.2. Symmetric ring and module spectra. In this section we give”
The motivic adams spectral sequence (2009, background): “The categories of symmetric spectra (see Hovey, Shipley and Smith [10]) and of motivic symmetric spectra (see Jardine [13]) will work fine.”
Topological hypercovers and 1-realizations (2004, methodology): “... and Voevodsky [MV] produced a model category Spc(k) which captures the ‘motivic homotopy theory’ of smooth schemes over k. Here Spc(k) stands for ‘spaces over k’. There are several possible choices for what model category to choose, all Quillen equivalent in the end, but we choose our reference ”
A UNIVERSALITY THEOREM FOR VOEVODSKY'S ALGEBRAIC COBORDISM SPECTRUM (2007, methodology): “when equipped with model structures as described in [1] (see [3, A.”