nLab localization of a commutative ring




The localization of a commutative ring RR at a set SS of its elements is a new ring R[S 1]R[S^{-1}] in which the elements of SS become invertible (units) and which is universal with this property.

When interpreting a ring under Isbell duality as the ring of functions on some space XX (its spectrum), then localizing it at SS corresponds to restricting to the complement of the subspace YXY \hookrightarrow X on which the elements in SS vanish.

See also commutative localization and localization of a ring (noncommutative).


Localization “at” and “away from”

The common terminology in algebra is as follows.

For SS a set of primes, “localize at SS” means “invert what is not divisible by SS”; so for pp prime, localizing “at pp” means considering only pp-torsion.

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


For commutative rings

Let RR be a commutative ring. Let SU(R)S \hookrightarrow U(R) be a multiplicative subset of the underlying set.

The following gives the universal property of the localization.


The localization L S:RR[S 1]L_S \colon R \to R[S^{-1}] is a homomorphism to another commutative ring R[S 1]R[S^{-1}] such that

  • for all elements sSRs \in S \hookrightarrow R the image L S(s)R[S 1]L_S(s) \in R[S^{-1}] is invertible (is a unit);

  • for every other homomorphism RR˜R \to \tilde R with this property, there is a unique homomorphism R[S 1]R˜R[S^{-1}] \to \tilde R such that we have a commuting diagram

    R L S R[S 1] R˜. \array{ R &\stackrel{L_S}{\to}& R[S^{-1}] \\ & \searrow & \downarrow \\ && \tilde R } \,.

The special case of inverting an element rr of RR, in which SS is the set {r,r 2,r 3,}\{ r, r^{2}, r^{3}, \ldots \}, is discussed at localisation of a commutative ring at an element. See also for example Sullivan 70, first pages.


The formal duals Spec(R[S 1])Spec(R)Spec(R[S^{-1}]) \longrightarrow Spec(R) of the localization maps RR[S 1]R \longrightarrow R[S^{-1}] (under forming spectra) serve as the standard open immersions that define the Zariski topology on algebraic varieties.



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

(r 1,s 1)(r 2,s 2)uS(r 1s 2r 2s 1)u=0R. (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 rs 1r s^{-1} for the equivalence class of (r,s)(r,s). On this set, addition and multiplication is defined by

r 1s 1 1+r 2s 2 1(r 1s 2+r 2s 1)(s 1s 2) 1 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 1s 1 1)(r 2s 2 1)r 1r 2(s 1s 2) 1. (r_1 s_1^{-1})(r_2 s_2^{-1}) \coloneqq r_1 r_2 (s_1 s_2)^{-1} \,.

(e.g. Stacks Project, def. 10.9.1)


The above definitions also work for non-commutative rings RR as well, so long as the multiplicative subset SS is a submonoid of the center Z(R)Z(R) of the multiplicative monoid of RR.

For E E_\infty-rings

(…) By the lifting property of etale morphisms for E E_\infty-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 modalitysymbolinterpretation
flat modality\flatformal completion at
shape modalityʃʃtorsion approximation
dR-shape modalityʃ dRʃ_{dR}localization away
dR-flat modality dR\flat_{dR}adic residual

the differential cohomology hexagon/arithmetic fracture squares:

localizationawayfrom𝔞 𝔞adicresidual Π 𝔞dR 𝔞X X Π 𝔞 𝔞dRX formalcompletionat𝔞 𝔞torsionapproximation, \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 set of lecture notes:

  • Dennis Sullivan, Localization, Periodicity and Galois Symmetry (The 1970 MIT notes) edited by Andrew Ranicki, K-Monographs in Mathematics, Dordrecht: Springer (pdf)

Concretely on localization of commutative rings:

  • Dennis Sullivan, Section 1 of: 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)

Discussion in constructive mathematics:

Further review:

Other accounts of the basics include

Last revised on July 29, 2023 at 19:11:03. See the history of this page for a list of all contributions to it.