Segal's Gamma rings and universal arithmetic
Abstract
Segal's Gammarings provide a natural framework for absolute algebraic geometry. We use Almkvist's global Witt construction to explore the relation with J. Borger F1geometry and compute the Witt functorring of Almkvist for the simplest Gammaring S. We prove that it is isomorphic to the Galois invariant part of the BCsystem, and exhibit the close relation between Lambdarings and the Arithmetic site. Then, we concentrate on the Arakelov compactification of Z which acquires a structure sheaf of Salgebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D, we show how to associate to D a Gammaspace that encodes, in homotopical terms, the RiemannRoch problem for D.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.08879
 Bibcode:
 2020arXiv200408879C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 19D55;
 13F35;
 19D55;
 13F35;
 14G40;
 18G55;
 18G30;
 19L20
 EPrint:
 25 pages