Riemann-Roch for $\overline{\text{Spec}\mathbb Z}$

TL;DR

This paper establishes a novel Riemann-Roch theorem for the Arakelov compactification of Spec Z, introducing new cohomological concepts and extending classical ideas to an arithmetic setting.

Contribution

It introduces a new Riemann-Roch theorem for divisors on the Arakelov compactification of Spec Z, utilizing a universal arithmetic theory over the sphere spectrum.

Findings

Proves a Riemann-Roch formula for the arithmetic setting

Defines cohomologies and their dimensions for divisors on Spec Z

Extends classical function field concepts to number fields

Abstract

We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to a divisor), their integer dimension, and Serre duality. These notions directly extend their classical counterparts for function fields. The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2. Both the definitions of the cohomologies and of their dimensions rely on a universal arithmetic theory over the sphere spectrum that we had previously introduced using Segal's Gamma rings. By adopting this new perspective we can parallel Weil's adelic proof of the Riemann-Roch formula for function fields including the use of Pontryagin duality.