nLab localization of a ring

Contents

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.

Context

Higher algebra

higher algebra

universal algebra

Contents

Idea

General

Given a (possibly noncommutative) unital ring $R$ there are many situations when certain elements or matrices can be inverted in a universal way obtaining a new “localized” ring $S^{-1}R$ equipped with a localization homomorphism $R\to S^{-1}R$ under which all elements in $S$ 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 $S\subset R$ 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 $S$ a set of primes, “localize at $S$” means “invert what is not divisible by $S$”; so for $p$ prime, localizing “at $p$” means considering only $p$-torsion.

Adjoining inverses $[S^{-1}]$ is pronounced “localized away from $S$”. Inverting a prime $p$ is localizing away from $p$, which means ignoring $p$-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 $R$ “at $p$” (or its Ab-enriched version, if saying that is necessary) really means the localization-of-the-ring R “away from p”.

Definition

For noncommutative rings

Definition

The localization of a ring $R$ at a multiplicative subset $S$ is the commutative ring whose underlying set is the set of equivalence classes on $R \times S$ under the equivalence relation

$(r_1, s_1) \sim (r_2, s_2) \;\;\Leftrightarrow\;\; \exists u \in S \; (r_1 s_2- r_2 s_1) u = 0 \;\in R \,.$

Write $r s^{-1}$ for the equivalence class of $(r,s)$. On this set, addition and multiplication is defined by

$r_1 s_1^{-1} + r_2 s_2^{-1} \coloneqq (r_1 s_2 + r_2 s_1) (s_1 s_2)^{-1}$
$(r_1 s_1^{-1})(r_2 s_2^{-1}) \coloneqq r_1 r_2 (s_1 s_2)^{-1} \,.$

For $E_k$-rings

(…) By the lifting property of etale morphisms for $E_k$-rings, see here. (…)

Properties

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 modalitysymbolinterpretation
flat modality$\flat$formal completion at
shape modality$ʃ$torsion approximation
dR-shape modality$ʃ_{dR}$localization away
dR-flat modality$\flat_{dR}$adic residual
$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$

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

Further reviews include

Discussion of the general concept in noncommutative geometry is in

• Zoran ?koda?, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (pdf), ed. A. Ranicki; pp. 220–313, math.QA/0403276.