nLab localization of a commutative ring

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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 Y↪XY \hookrightarrow X on which the elements in SS vanish.

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

Remark

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”.

Definition

For commutative rings

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

The following gives the universal property of the localization.

Definition

The localization L S:R→R[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 s∈S↪Rs \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 R→R˜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.

Remark

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

Explicitly:

Definition

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)⇔∃u∈S(r 1s 2−r 2s 1)u=0∈R. (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)

Remark

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 } \,,

Examples

References

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.

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