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Projective geometry in characteristic one and the epicyclic category

Part of: Other generalizations of groups Finite geometry and special incidence structures Higher algebraic $K$-theory Generalizations of fields

Published online by Cambridge University Press:  11 January 2016

Alain Connes  and
Caterina Consani
Alain Connes
Affiliation:
Collège de France, Paris F-75005, France, [email protected]
Caterina Consani
Affiliation:
Department of Mathematics, Johns Hopkins University Baltimore, Maryland 21218, USA, [email protected]
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Abstract

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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.

MSC classification

Secondary: 19D55: $K$-theory and homology; cyclic homology and cohomology 12K10: Semifields 51E26: Other finite linear geometries 20N20: Hypergroups
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 217 , March 2015 , pp. 95 - 132
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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