Dualité de Spanier–Whitehead en géométrie algébrique

Riou, Joël

doi:10.1016/j.crma.2005.02.002

Cited by 47 publications

(28 citation statements)

References 9 publications

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“…D'après la Proposition 3.19, la catégorie DA ét (k, Λ) est compactement engendrée. Étant donné que e * commute aux sommes infinies, on voit que si l'homomorphisme (29) est inversible pour M compact il l'est aussi pour tout M. On peut alors supposer que M est un objet compact et donc fortement dualisable (voir [39] ou encore [6, Lem. 1.3.29]).…”

Section: Le Théorème De Rigidité Relatifunclassified

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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Section: Le Théorème De Rigidité Relatifunclassified

La réalisation étale et les opérations de Grothendieck

Ayoub

2014

Ann. Sci. École Norm. Sup.

159

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“…This is a closed symmetric monoidal category. Riou's statement about Spanier-Whitehead duality says that every object of SH C of the form Σ ∞ T,S 1 X + for X a smooth C-scheme of finite type is (strongly) dualizable in the monoidal category SH C (see [Rio05]).…”

Section: Conjecturesmentioning

confidence: 99%

Topological K-theory of complex noncommutative spaces

Blanc

2015

Compositio Math.

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The purpose of this work is to give a definition of a topological K-theory for dg-categories over C and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cylic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps : taking the topological realization of algebraic K-theory, and inverting the Bott element. The topological realization is the left Kan extension of the functor "space of complex points" to all simplicial presheaves over complex algebraic varieties. Our first main result states that the topological K-theory of the unit dg-category is the spectrum BU. For this we are led to prove a homotopical generalization of Deligne's cohomological proper descent, using Lurie's proper descent. The fact that the Chern character descends to topological K-theory is established by using Kassel's K\"unneth formula for periodic cyclic homology and once again the proper descent result. In the case of a dg-category of perfect complexes on a smooth scheme, we show that we recover the usual topological K-theory. Finally in the case of a finite dimensional associative algebra, we show that the lattice conjecture holds. This gives a formula for the periodic homology groups of a finite dimensional algebra in terms of the stack of projective modules of finite type.Comment: 72 pages. To appear in Compositio Mathematica. Corrected the assumptions in the comparison isomorphism for schemes (now stated for singular schemes, proposition 4.32) as well as some technical assumptions in proposition 3.2

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“…They constitute a triangulated subcategory T pf of T . In the case T = SH(S), we find that SH(S) pf is the pseudo-abelian hull of the triangulated subcategory generated by objects of the form (P 1 ) ∧−n ∧ U + for U ∈ Sm/S (see [35,Proposition 1.2]).…”

Section: Adams Operations On Bgl Qmentioning

confidence: 99%

Algebraic K -theory, A ¹ -homotopy and Riemann-Roch theorems

Riou

2010

Journal of Topology

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In this paper, we show that the combination of the constructions done in SGA 6 and the A 1 -homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.Contents 230 JOËL RIOU Theorem 0.2. Let S be a regular scheme. We let K 0 (−) be the presheaf of sets on Sm/S which maps X to K 0 (X). Then the map induced by Theorem 0.1 is a bijection:where Sm/S opp Sets is the category of presheaves of sets on Sm/S. It follows that the operations defined in [7] at the level of K 0 (for example, λ n , Ψ k ) uniquely lift in H(S). From there, using Theorem 0.1, we can make them act on higher algebraic K-theory. This principle also works for operations involving several operands (for example, products) and in a sense that will be made precise in Section 2, we obtain a machinery that takes as an input the algebraic structures on K 0 and outputs such a structure on Z × Gr inside H(S). Thus, Z × Gr is equipped with a structure of a special λ-ring with duality.Structures of a (special) λ-ring had already been obtained on higher K-theory, with different scales of generality. We may mention constructions of products, λ-operations or Adams' operations by Loday [29], Waldhausen [47], Kratzer [25], Soulé [42], Grayson [18], Lecomte [26] and Levine [27]. We compare the structures on K (X) for X regular obtained by our method to these previous constructions in Section 3. The comparison with Waldhausen's product (see Proposition 3.2.1) may seem surprisingly straightforward, but it is a typical use of Theorem 0.2 and its variants involving several operands (see Theorem 1.1.4).Section 4 relates our results to virtual categories, an insight of Deligne [10]. We show that, after inverting 2, constructions done at the level of K 0 refine to these virtual categories, which embodies both K 0 and K 1 . This theory was used by Dennis Eriksson in his thesis [12] in order to refine Riemann-Roch theorems at the level of these virtual categories.In Section 5, we focus on operations τ :We compute them using the splitting principle. We show that the datum of τ is equivalent to the datum of an element in K 0 (S) [[U ]]. Then we construct, up to a unique isomorphism in the stable homotopy category SH(S), the P 1 -spectrum BGL which represents algebraic K-theory and study its endomorphisms (it is somewhat related but quite different from the methods of [1, Chapter 6; 2, 3]). After tensoring with Q, we show that this spectrum decomposes in SH(S) as the direct sums of 'eigenspaces' for the Adams operations. Alternate interesting descriptions of stable operations on algebraic K-theory (and more general oriented theory) have been obtained by very different methods by Naumann, Østvaer and Spitzweck [32].We prove in Section 6 that these ideas can be used to obtain a homotopical variant of some Riemann-Roch theorems in the case of a smooth and projective morphism f :X → S. Basically, we prove that certain Riemann-Roch formulas are satisfied on zeroth K-groups ...

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Dualité de Spanier–Whitehead en géométrie algébrique

Cited by 47 publications

References 9 publications

La réalisation étale et les opérations de Grothendieck

La réalisation étale et les opérations de Grothendieck

Topological K-theory of complex noncommutative spaces

Algebraic K -theory, A ¹ -homotopy and Riemann-Roch theorems

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Dualité de Spanier–Whitehead en géométrie algébrique

Cited by 47 publications

References 9 publications

La réalisation étale et les opérations de Grothendieck

La réalisation étale et les opérations de Grothendieck

Topological K-theory of complex noncommutative spaces

Algebraic K -theory, A 1 -homotopy and Riemann-Roch theorems

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Product

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Algebraic K -theory, A ¹ -homotopy and Riemann-Roch theorems