Projective geometry in characteristic one and the epicyclic category

@article{Connes2015ProjectiveGI,
  title={Projective geometry in characteristic one and the epicyclic category},
  author={Alain Connes and Caterina Consani},
  journal={Nagoya Mathematical Journal},
  year={2015},
  volume={217},
  pages={95 - 132}
}
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 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… 
View on Cambridge Press
cambridge.org
12 Citations
Background Citations
4

12 Citations

The cyclic and epicyclic sites
We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic…
Semiring Congruences and Tropical Geometry
One of the main motivations and inspirations for this thesis is the still open question of the definition of geometry in characteristic one. This is geometry over a structure, called an idempotent…
Algebraic Geometry Over Hyperrings
On the dimension of polynomial semirings
Segal’s Gamma rings and universal arithmetic
Segal's Gamma-rings provide a natural framework for absolute algebraic geometry. We use Almkvist's global Witt construction to explore the relation with J. Borger F1-geometry and compute the Witt…
Hyperstructures and Idempotent Semistructures
Much of this thesis concerns hypergroups, multirings, and hyperfields. These are analogous to abelian groups, rings, and fields, but have a multivalued addition operation. M. Krasner introduced the…
Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials
A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative…
Cyclic Theories
We describe a geometric theory classified by Connes-Consani’s epicylic topos and two related theories respectively classified by the cyclic topos and by the topos [ℕ∗,Set]$[{\mathbb N}^{\ast },…
FINITE EXTENSIONS OF Zmax
We classify the semifields and division semirings containing the max-plus semifield Zmax, which are finitely generated as Zmax-semimodules.
Valuations of semirings
  • Jaiung Jun
  • Mathematics
    Journal of Pure and Applied Algebra
  • 2018
...
...

References

SHOWING 1-10 OF 39 REFERENCES
Power maps and epicyclic spaces
Morphisms of projective geometries and semilinear maps
Morphisms between projective geometries are introduced; they are partially defined maps satisfying natural geometric conditions. It is shown that in the arguesian case the morphisms are exactly those…
Characteristic 1 , entropy and the absolute point
We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion…
Characteristic one, entropy and the absolute point
We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion…
Cyclic homology, Serre's local factors and the -operations
We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ringA∞ = Q ν|∞ Kν, provides the right theory to obtain, using the λ-operations,…
Cyclic structures and the topos of simplicial sets
The hyperring of adèle classes
The universal thickening of the eld of real numbers
We dene the universal 1-adic thickening of the eld of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion…
Schemes over 𝔽1 and zeta functions
Abstract We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ $C=\overline {\mathrm {Spec}\,\Z }$ over 𝔽1, whose corresponding zeta function is the complete Riemann…
Universal Thickening of the Field of Real Numbers
We define the universal thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process.…
...
...