# Segal’s Gamma rings and universal arithmetic

@article{Connes2020SegalsGR, title={Segal’s Gamma rings and universal arithmetic}, author={Alain Connes and Caterina Consani}, journal={arXiv: Algebraic Geometry}, year={2020} }

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 functor-ring of Almkvist for the simplest Gamma-ring S. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between Lambda-rings and the Arithmetic site. Then, we concentrate on the Arakelov compactification of Z which acquires a structure sheaf…

## 5 Citations

Generalizations of Loday's assembly maps for Lawvere's algebraic theories

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In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and…

## References

SHOWING 1-10 OF 45 REFERENCES

Au-dessous de SpecZ .

- Mathematics
- 2009

In this article we use the theories of relative algebraic geometry and of homotopical algebraic geometry (cf. [HAGII]) to construct some categories of schemes defined under Spec ℤ. We define the…

Projective geometry in characteristic one and the epicyclic category

- MathematicsNagoya Mathematical Journal
- 2015

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…

The cyclic and epicyclic sites

- Mathematics
- 2014

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…

On the arithmetic of the BC-system

- Mathematics
- 2011

For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-system…