# 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 commutative rings

The localization of a commutative 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
$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$

## References

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.

Last revised on May 9, 2021 at 14:15:20. See the history of this page for a list of all contributions to it.