Banerjee, Abhishek 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 Ann. Math. Blaise Pascal 21, No. 1, 1-23 (2014). 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 PDFBibTeX XMLCite \textit{A. Banerjee}, Ann. Math. Blaise Pascal 21, No. 1, 1--23 (2014; Zbl 1327.58008) Full Text: DOI References: [1] Connes, A., Cohomologie cyclique et foncteurs \({E}xt^n\), C. R. Acad. Sci. Paris Sér. I. Math., 296(23), 953-958 (1983) · Zbl 0534.18009 [2] Connes, A., Géométrie non commutative (1990) · Zbl 0745.46067 [3] Connes, A.; Consani, C.; Marcolli, M., Noncommutative geometry and motives : the thermodynamics of endomotives, Adv. Math., 214(2), 761-831 (2007) · Zbl 1125.14001 [4] Deligne, P., The Grothendieck Festschrift Vol II, Progr. Math. Vol. 87, 111-195 (1990) · Zbl 0727.14010 [5] Hakim, M., Topos annelés et schémas relatifs, Ergebnisse der Mathe-matik und ihrer Grenzgebiete Band 64 (1972) · Zbl 0246.14004 [6] Jones, J. D. S.; Kassel, C., Bivariant cyclic theory, \(K\)-Theory, 3(4), 339-365 (1989) · Zbl 0755.18008 [7] Joyal, A.; Street, R., Braided tensor categories, Adv. Math., 102(1), 20-78 (1993) · Zbl 0817.18007 [8] Lenstra, H., Galois Theory for Schemes (1985) · Zbl 0571.10005 [9] Loday, J. L., Cyclic homology, Appendix E by Maria O. Ronco (1992) · Zbl 0780.18009 [10] Mazza, C.; Voevodsky, V.; Weibel, C., Lecture Notes on motivic cohomology (2006) · Zbl 1115.14010 [11] Toen, B.; Vaquié, M., Au-dessous de Spec(Z), J. K-Theory, 3(3), 437-500 (2009) · Zbl 1177.14022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.