There is no "Weil-"cohomology theory with real coefficients for arithmetic curves

TL;DR

This paper explains why a Weil cohomology theory with real coefficients cannot exist for arithmetic schemes over Spec Z, extending Serre's argument from varieties over finite fields to arithmetic schemes.

Contribution

It provides a conceptual explanation for the non-existence of Weil cohomology theories with real coefficients for arithmetic schemes over Spec Z.

Findings

No Weil cohomology theory with real coefficients exists for arithmetic schemes over Spec Z.

Extends Serre's argument from varieties over finite fields to arithmetic schemes.

Clarifies limitations of cohomology theories in arithmetic geometry.

Abstract

A well known argument by Serre shows that there is no Weil cohomology theory with real coefficients for smooth projective varieties over $\bar{\mathbb{F}}_p$. In this note we explain why no "Weil-"cohomology theory with real coefficients can exist for arithmetic schemes over spec $\mathbb{Z}$, even for spectra of number rings.