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Projective geometry in characteristic one and the epicyclic category

Published online by Cambridge University Press:  11 January 2016

Alain Connes
Affiliation:
Collège de France, Paris F-75005, France, alain@connes.org
Caterina Consani
Affiliation:
Department of Mathematics, Johns Hopkins University Baltimore, Maryland 21218, USA, kc@math.jhu.edu
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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 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.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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