Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Noncommutative localization in noncommutative geometry   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 one-sided modules.
  We present the basics of Ore localization of rings and modules in much
detail. Common practical techniques are studied as well. We also describe a
counterexample for a folklore test principle. 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 localization of
differential calculi.
  To aid the geometrical viewpoint, localization is studied with emphasis on
descent formalism, flatness, abelian categories of quasicoherent 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, 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. Cohn
localization is here related to the quasideterminants of Gelfand and Retakh;
and this may help understanding both subjects.

14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
3
The categories of this paper are: physics
18
The categories of this paper are: econ
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga