Segal's Gamma rings and universal arithmetic

Alain Connes; Caterina Consani

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arXiv · 2004.08879

Segal's Gamma rings and universal arithmetic

Alain Connes·Caterina Consani

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Abstract

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 of S-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D, we show how to associate to D a Gamma-space that encodes, in homotopical terms, the Riemann-Roch problem for D.

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Alain Connes, Caterina Consani. 2020-04-19. Segal's Gamma rings and universal arithmetic. https://arxiv.org/abs/2004.08879

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