Segal’s Gamma rings and universal arithmetic

Abstract

Abstract Segal’s Γ-rings provide a natural framework for absolute algebraic geometry. We use G. Almkvist’s global Witt construction to explore the relation with J. Borger ${\mathbb F}_1$-geometry and compute the Witt functor-ring ${\mathbb W}_0({\mathbb S})$ of the simplest Γ-ring ${\mathbb S}$. We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between λ-rings and the Arithmetic Site. Then, we concentrate on the Arakelov compactification ${\overline{{\rm Spec\,}{\mathbb Z}}}$ which acquires a structure sheaf of ${\mathbb S}$-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor D on ${\overline{{\rm Spec\,}{\mathbb Z}}}$, we show how to associate to D a Γ-space that encodes, in homotopical terms, the Riemann–Roch problem for D.