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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 subspace YXY \hookrightarrow X on which the elements in SS vanish.

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

Definition

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.

Examples

References

Revised on June 24, 2014 19:25:23 by Urs Schreiber (89.204.130.177)