Open Access
March 2015 Projective geometry in characteristic one and the epicyclic category
Alain Connes, Caterina Consani
Nagoya Math. J. 217: 95-132 (March 2015). DOI: 10.1215/00277630-2887960


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 integers Zmax. 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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Alain Connes. Caterina Consani. "Projective geometry in characteristic one and the epicyclic category." Nagoya Math. J. 217 95 - 132, March 2015.


Published: March 2015
First available in Project Euclid: 6 May 2015

zbMATH: 1338.19004
MathSciNet: MR3343840
Digital Object Identifier: 10.1215/00277630-2887960

Primary: 19D55
Secondary: 12K10 , 20N20 , 51E26

Keywords: Characteristic one , cyclic category , Hyperstructures , projective geometry , Semifields

Rights: Copyright © 2015 Editorial Board, Nagoya Mathematical Journal

Vol.217 • March 2015
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