Arithmetic geometry of non-commutative spaces with large centres

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 algebraic number theory and geometry using non-commutative geometry and non-commutative deformation theory. As such we prove no deep theorems. Instead we strive to illustrate, for instance by supplying detailed examples, how non-commutative algebraic geometry can be used to view number-theoretic objects from a different perspective.