nLab localization of a commutative ring

Idea

The localization of a commutative ring $R$ at a set $S$ of its elements is a new ring $R[S^{-1}]$ in which the elements of $S$ 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 $X$ (its spectrum), then localizing it at $S$ corresponds to restricting to the complement of the subspace $Y \hookrightarrow X$ on which the elements in $S$ 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 $S$ 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

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

The following gives the universal property of the localization.

Definition

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

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

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

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

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

Remark

The formal duals $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ of the localization maps $R \longrightarrow R[S^{-1}]$ (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 $R$ 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

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

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

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

Remark

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

For $E_\infty$-rings

(…) By the lifting property of etale morphisms for $E_\infty$-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 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:

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

Examples

• field of fractions

Related concepts

• completion of a ring

• localization,

• localization of a monoid

• localization of an abelian group

• localization of a module

• localization of a ring, localization of a category

• Cohn localization, Ore localization.

• Bousfield localization of spectra

  • localization of a space (and of a spectrum)

  • p-localization, p-completion

• quotient ring

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:

• Henri Lombardi, Claude Quitté chapter 2 section 2 in: Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) [doi:10.1007/978-94-017-9944-7, pdf]

Further review:

• Andreas Gathmann, Localization (pdf)

• Joseph NeisendorferA Quick Trip through Localization (pdf)

Other accounts of the basics include

• Andreas Gathmann, Localization (pdf)

• The Stacks Project, 10.9. Localization

• Yoshifumi Tsuchimoto, Review of commutative (usual) affine schemes. Localization of a commutative ring