Arithmetic Geometry of Non-Commutative Spaces with Large Centres

TL;DR

This paper develops an arithmetic geometry framework for polynomial identity algebras using non-commutative deformation theory, revealing local geometric information hidden in the commutative perspective and establishing a foundation for future research.

Contribution

It introduces a novel approach to arithmetic geometry via non-commutative deformation theory, emphasizing local properties and the non-commutative 'shadow' of classical algebraic geometry.

Findings

Provides a formalism for local geometric analysis in non-commutative settings

Defines a notion of 'infinitesimally close' in non-commutative algebra

Lays groundwork for further exploration of non-commutative arithmetic geometry

Abstract

This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local information that is not visible when looking at the objects from a commutative angle. For instance, it is a precise meaning to be given to two things being ``infinitesimally close'', something being obscured from view when restricting only to a commutative algebraic study. A Platonesque way of looking at this is that the commutative world is a ``shadow'' of a more inclusive non-commutative universe. The present paper aims at laying the foundation for further and deeper study of arithmetic and geometry using non-commutative geometry and non-commutative deformation theory.