This entry is about the general notion of localization of a possible noncommutative ring. For the more restrictive but more traditional notion of localization of a commutative ring see there.
Algebras and modules
Model category presentations
Geometry on formal duals of algebras
Given a (possibly noncommutative) unital ring there are many situations when certain elements or matrices can be inverted in a universal way obtaining a new “localized” ring equipped with a localization homomorphism under which all elements in are mapped to multiplicatively invertible elements (units). The latter property must be modified for Cohn localization at multiplicative set of matrices.
We can typically invert elements in a left or right Ore subset or much more generally some multiplicative set or matrices (Cohn localization) etc. There are also some specific localizations like Martindale localizations in ring theory.
Localization “at” and “away from”
The common terminology in algebra is as follows.
For a set of primes, “localize at ” means “invert what is not divisible by ”; so for prime, localizing “at ” means considering only -torsion.
Adjoining inverses is pronounced “localized away from ”. Inverting a prime is localizing away from , which means ignoring -torsion.
See also lecture notes such as (Gathmann) and see at localization of a space for more discussion of this.
Evidently, this conflicts with more-categorial uses of “localized”; “inverting weak equivalences” is called localization, by obvious analogy, and is written as “localizing at weak equivalences”. This is confusing! It’s also weird: since a ring is a one-object Ab-enriched category with morphisms “multiply-by”, the localization-of-the-category “at ” (or its Ab-enriched version, if saying that is necessary) really means the localization-of-the-ring R “away from p”.
For noncommutative rings
The localization of a ring at a multiplicative subset is the commutative ring whose underlying set is the set of equivalence classes on under the equivalence relation
Write for the equivalence class of . On this set, addition and multiplication is defined by
(e.g. Stacks Project, def. 10.9.1)
For commutative rings
See localization of a commutative ring.
(…) By the lifting property of etale morphisms for -rings, see here. (…)
As a modality in arithmetic cohesion
Localization away from a suitably tame ideal may be understood as the dR-shape modality in the cohesion of E-infinity arithmetic geometry:
cohesion in E-∞ arithmetic geometry:
the differential cohomology hexagon/arithmetic fracture squares:
A classical account of localization of commutative rings is in section 1 of
- Dennis Sullivan, Geometric topology: localization, periodicity and Galois symmetry, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface by Andrew Ranicki (pdf)
Further reviews include
- Andreas Gathmann, Localization (pdf)
Discussion of the general concept in noncommutative geometry is in
- Zoran Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (pdf), ed. A. Ranicki; pp. 220–313, math.QA/0403276.