Modules over motivic cohomology

Röndigs, Oliver; Østvær, Paul Arne

doi:10.1016/j.aim.2008.05.013

Cited by 94 publications

(111 citation statements)

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“…Nous travaillons avec des T -spectres de préfaisceaux à valeurs dans les caté-gories de modèles stables de S 1 -spectres ou de Λ[1]-spectres. Dans [42], les auteurs suivent de plus près la recette de Jardine [28] consistant à prendre des T -spectres à valeurs dans des catégo-ries de modèles instables. Bien entendu, les deux recettes aboutissent à des catégories de modèles Quillen-équivalentes : le point est que T est A 1 -équivalent à une suspension.…”

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La réalisation étale et les opérations de Grothendieck

Ayoub

2014

Ann. Sci. École Norm. Sup.

159

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In this article, we construct étale realization functors defined on the categories DAét(X, Λ) of étale motives (without transfers) over a scheme X. Our construction is natural and relies on a relative rigidity theorem à la Suslin-Voevodsky that we will establish first. Then, we show that these realization functors are compatible with Grothendieck operations and the "nearby cycles" functors. Along the way, we prove a number of properties concerning étale motives. LA RÉALISATION ÉTALE ET LES OPÉRATIONS DE GROTHENDIECK par Joseph AyoubRésumé. -Dans cet article, nous construisons des foncteurs de réalisation étale définis sur les caté-gories DA ét (X, Λ) des motifs étales (sans transferts) au-dessus d'un schéma X. Notre construction est naturelle et repose sur un théorème de rigidité relatif à la Suslin-Voevodsky que nous devons établir au préalable. Nous montrons ensuite que ces foncteurs sont compatibles aux opérations de Grothendieck et aux foncteurs « cycles proches ». Au passage, nous démontrons un certain nombre de propriétés concernant les motifs étales.Abstract. -In this article, we construct étale realization functors defined on the categories DA ét (X, Λ) of étale motives (without transfers) over a scheme X. Our construction is natural and relies on a relative rigidity theorem à la Suslin-Voevodsky that we will establish first. Then, we show that these realization functors are compatible with Grothendieck's operations and the "nearby cycles" functors. Along the way, we prove a number of properties concerning étale motives.

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La réalisation étale et les opérations de Grothendieck

Ayoub

2014

Ann. Sci. École Norm. Sup.

159

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“…Let MZ denote motivic cohomology, considered as an object of MSS (see the next section). In [10] we compare MZ-mod with MSS tr . Since the monoid axiom in [11] holds for motivic symmetric spectra [6], the module category over motivic cohomology acquires a model structure.…”

Section: The Stable Theorymentioning

confidence: 99%

“…Our goal is to show it is a Quillen equivalence. Equivalently, the unit map MZ ∧ U + → Ψ Φ(MZ ∧ U + ) is a weak equivalence of motivic symmetric spectra for every smooth quasi-projective S-scheme U [10]. We shall analyze the unit map using the category MF of motivic functors fM → M introduced in [2].…”

Section: Outline Of Proofsmentioning

confidence: 99%

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Motives and modules over motivic cohomology

Röndigs

Østvær

2006

Comptes Rendus. Mathématique

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“…Thanks to the work of Bloch [6], reformulated by Riou in [35, §6], we have KGL R ≃ i∈Z HZ R (i)[2i]. On the other hand, thanks to the work of Rödings-Østvaer [37], DM(k; R) identifies with the homotopy category Ho(Mod(HZ R )). As a consequence, base-change along HZ R → KGL R gives rise to an R-linear symmetric monoidal triangulated basechange functor − ∧ HZ R KGL R making the following diagram commute: The base-change functor − ∧ HZ R KGL R preserves arbitrary direct sums.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

Tabuada

2015

Sel. Math. New Ser.

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Abstract. This article is the sequel to [27]. We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub's weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute the associated Hopf dg algebra using solely the classical category of mixed Artin-Tate motives. Finally, we establish a short exact sequence relating the Hopf algebra of continuous functions on the absolute Galois group with the motivic Hopf dg algebras of the base field k and of its algebraic closure. Along the way, we describe the behavior of Ayoub's weak Tannakian formalism with respect to orbit categories and relate the category of NC mixed motives with Voevodsky's category of mixed motives.

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Modules over motivic cohomology

Cited by 94 publications

References 23 publications

La réalisation étale et les opérations de Grothendieck

La réalisation étale et les opérations de Grothendieck

Motives and modules over motivic cohomology

Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

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