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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).

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

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

2. 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 following gives an explicit description of the localization

Definition

For $R$ a commutative ring and $s \in R$ an element, the localization of $R$ at $s$ is

$$A[s^{-1}] = A[X](X s - 1) \,.$$

(e.g. 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.

For $E_\infty$-rings

(…) By the lifting property of etale morphisms for $E_\infty$-rings, see here. (…)

Examples

• Cohn localization

Related concepts

• localization of an abelian group

• localization of a module

• localization of a ring, localization of a category

• completion of a ring

References

A classical set of lecture notes is

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