Abstract

The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of “spaces” locally described by noncommutative rings and their categories of modules.

We present the basics of the Ore localization of rings and modules in great detail. Common practical techniques are studied as well. We also describe a counterexample to a folklore test principle for Ore sets. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of the localization of differential calculi.

To aid the geometrical viewpoint, localization is studied with emphasis on descent formalism, flatness, the abelian categories of quasi-coherent sheaves and generalizations, and natural pairs of adjoint functors for sheaf and module categories. The key motivational theorems from the seminal works of Gabriel on localization, abelian categories and schemes are quoted without proof, as well as the related statements of Popescu, Eilenberg-watts, Deligne and Rosenberg.

The Cohn universal localization does not have good flatness properties, but it is determined by the localization map already at the ring level, like the perfect localizations are. Cohn localization is here related to the quasideterminants of Gelfand and Retakh; and this may help the understanding of both subjects.

Introduction

Objectives and scope. This is an introduction to Ore localizations and generalizations, with geometric applications in mind.