Model theory and geometry of representations of rings of integers

TL;DR

This paper explores a geometric perspective on the ring of integers by linking representations of integral extensions to model theory, revealing a new structure connected to pseudo-finite fields and their theories.

Contribution

It introduces a novel category of representations of integral extensions of al and connects it with the model theory of pseudo-finite fields, providing a new geometric framework.

Findings

The category of representations is superstable with trivial pregeometry.

The structure encodes complex mathematics of pseudo-finite fields.

A basic approach linking algebraic geometry and model theory for al.

Abstract

The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what geometry this object carries. A.Connes and C.Consani published recently an important paper which introduces a much more complex structure called {\em the arithmetic site} which includes $\mathrm{Spec}(\mathbb{Z}).$ Our approach is based on the generalisation of constructions applied by the first author for similar purposes in non-commutative (and commutative) algebraic geometry. The current version is quite basic. We describe a category of certain representations of integral extensions of $\Z$ and establish its tight connection with the space of elementary theories of pseudo-finite fields. From model-theoretic point of view the category of representations is a multisorted structure which we prove to be superstable with pregeometry of trivial type. It comes as some surprise that a structure like this can code a rich mathematics of pseudo-finite fields.