# nLab localization of a commutative ring

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 subspace $Y \hookrightarrow X$ on which the elements in $S$ vanish.

## Definition

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

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

## References

Revised on April 6, 2014 02:00:51 by Urs Schreiber (185.37.147.12)