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localization of a commutative ring

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Idea

The localization of a commutative ring RR at a set SS of its elements is a new ring R[S] 1R[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).

Definition

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.

Definition

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

  1. 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);

  2. 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 following gives an explicit description of the localization

Definition

For RR a commutative ring and sRs \in R an element, the localization of RR at ss is

A[s 1]=A[X](Xs1). A[s^{-1}] = A[X](X s - 1) \,.

(e.g. 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 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.

For E E_\infty-rings

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

Examples

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

Revised on August 18, 2014 21:44:12 by Urs Schreiber (89.204.153.66)