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Tate motives and the periodicity operators of Connes. (Les motifs de Tate et les opérateurs de périodicité de Connes.) (French. English summary) Zbl 1327.58008

Summary: We define a category \(\widetilde{\text{Mot}}_{\mathbb{C}}\) of motives over a symmetric monoidal category \((\mathbb{C},\otimes,1)\) satisfying certain conditions. The role of spaces over \((\mathbb{C},\otimes,1)\) is played by monoid objects (not necessarily commutative) in \(\mathbb{C}\). To define morphisms in the category \(\widetilde{\text{Mot}}_{\mathbb{C}}\), we use classes in bivariant cyclic homology groups. The aim is to show that the Connes periodicity operators induce morphisms \(M\otimes \mathbb{T}^{\otimes 2}\to M\) in \(\widetilde{\text{Mot}}_{\mathbb{C}}\), where \(\mathbb{T}\) is the Tate motive in \(\widetilde{\text{Mot}}_{\mathbb{C}}\).

MSC:

58B34 Noncommutative geometry (à la Connes)
14F42 Motivic cohomology; motivic homotopy theory
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
46L80 \(K\)-theory and operator algebras (including cyclic theory)
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
14G20 Local ground fields in algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14L05 Formal groups, \(p\)-divisible groups
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References:

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