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.
symmetric monoidal (∞,1)-category of spectra
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.
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”.
Let $R$ be a ring and $(S, 1, \cdot)$ be a multiplicative submonoid of $(R, 1, \cdot)$ with monoid monomorphism $i:S \hookrightarrow R$. The localization of $R$ at $S$ is defined as the initial ring $S^{-1} R$ with a ring homomorphism $h:R \to S^{-1} R$ and monoid monomorphism $j:S \hookrightarrow S^{-1} R$ such that for all $s \in S$, $j(s) \cdot h(i(s)) = 1$ and $h(i(s)) \cdot j(s) = 1$: for every other ring $A$ with a ring homomorphism $k:R \to A$ and monoid monomorphism $l:S \hookrightarrow A$ such that for all $s \in S$, $l(s) \cdot k(i(s)) = 1$ and $k(i(s)) \cdot l(s) = 1$, there is a unique ring homomorphism $g:S^{-1} R \to A$.
The localization of a ring at a multiplicative submonoid $S$ which contains $0$ is the trivial ring.
See localization of a commutative ring.
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
Write $r s^{-1}$ for the equivalence class of $(r,s)$. On this set, addition and multiplication is defined by
(e.g. Stacks Project, def. 10.9.1)
(…) By the lifting property of etale morphisms for $E_k$-rings, see here. (…)
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:
cohesion modality | symbol | interpretation |
---|---|---|
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 |
the differential cohomology hexagon/arithmetic fracture squares:
A classical account of localization of commutative rings is in section 1 of
by Andrew Ranicki (pdf)
A constructive account of localization of rings is in chapter 2 section 2 of
Further reviews include
Andreas Gathmann, Localization (pdf)
Joseph NeisendorferA Quick Trip through Localization (pdf)
Discussion of the general concept in noncommutative geometry is in
Last revised on May 26, 2022 at 18:25:46. See the history of this page for a list of all contributions to it.