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Projective geometry in characteristic one and the epicyclic category

  • Alain Connes (a1) and Caterina Consani (a2)

Abstract

We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield of max-plus integersmax. Finite-dimensional vector spaces are replaced by modules defined by restriction of scalars from the one-dimensional free module, using the Frobenius endomorphisms of ℤmax. The associated projective spaces are finite and provide a mathematically consistent interpretation of Tits's original idea of a geometry over the absolute point. The self-duality of the cyclic category and the cyclic descent number of permutations both acquire a geometric meaning.

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[1] Bourbaki, N., Éléments de mathématiques: Algèbre, chapitre II, Actualités Sci. Indust. 1032, Hermann, Paris, 1947. MR 0023803.
[2] Burghelea, D., Fiedorowicz, Z., and Gajda, W., Power maps and epicyclic spaces, J. Pure Appl. Algebra 96 (1994), 114. MR 1297435. DOI 10.1016/0022-4049(94)90081-7.
[3] Connes, A., Cohomologie cyclique et foncteurs Ext n , C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 953958. MR 0777584.
[4] Connes, A. and Consani, C., Schemes over 𝔽1 and zeta functions , Compos. Math. 146 (2010), 13831415. MR 2735370. DOI 10.1112/S0010437X09004692.
[5] Connes, A. and Consani, C., “Characteristic 1, entropy and the absolute point” in Noncommutative Geometry, Arithmetic, and Related Topics, Johns Hopkins University Press, Baltimore, 2011, 75139. MR 2907005.
[6] Connes, A. and Consani, C., The hyperring of adèle classes, J. Number Theory 131 (2011), 159194. MR 2736850. DOI 10.1016/j.jnt.2010.09.001.
[7] Connes, A. and Consani, C., Cyclic structures and the topos of simplicial sets, J. Pure Appl. Algebra 219 (2015), 12111235. MR 3282133. DOI 10.1016/j.jpaa.2014.06.002.
[8] Connes, A. and Consani, C., Cyclic homology, Serre's local factors and the ƛ-operations, preprint, arXiv:1211.4239v1 [math.AG].
[9] Connes, A. and Consani, C., The universal thickening of the field of real numbers, preprint, arXiv:1202.4377v2 [math.NT].
[10] Dundas, B. I., Goodwillie, T. G., and McCarthy, R., The Local Structure of Algebraic K-theory, Algebr. Appl. 18, Springer, London, 2013. MR 3013261.
[11] Faure, C.-A. and Frölicher, A., Morphisms of projective geometries and semilinear maps, Geom. Dedicata 53 (1994), 237262. MR 1311317. DOI 10.1007/BF01263998.
[12] Golan, J. S., Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999. MR 1746739. DOI 10.1007/978-94-015-9333-5.
[13] Gondran, M. and Minoux, M., Graphs, Dioids and Semirings: New Models and Algorithms, Oper. Res./Comput. Sci. Interfaces Ser. 41, Springer, New York, 2008. MR 2389137.
[14] Goodwillie, T. G., Cyclic homology, derivations, and the free loop space, Topology 24 (1985), 187215. MR 0793184. DOI 10.1016/0040-9383(85)90055-2.
[15] Goodwillie, T. G., personal communication to F. Waldhausen, August 10, 1987.
[16] Henry, S., Symmetrization of monoids as hypergroups, preprint, arXiv: 1309.1963v1 [math.AG].
[17] Loday, J.-L., Cyclic Homology, Grundlehren Math. Wiss. 301, Springer, Berlin, 1998. MR 1600246. DOI 10.1007/978-3-662-11389-9.
[18] Mac Lane, S. and Moerdijk, I., Sheaves in Geometry and Logic: A First Introduction to Topos Theory, corrected reprint, Universitext, Springer, New York, 1994. MR 1300636.
[19] Tits, J., Sur les analogues algébriques des groupes semi-simples complexes, Centre Belge de Recherches Mathématiques Établissements Ceuterick, Louvain; Librairie Gauthier-Villars, Paris, 1957, 261289. MR 0108765.
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Projective geometry in characteristic one and the epicyclic category

  • Alain Connes (a1) and Caterina Consani (a2)

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