# nLab localization of a commutative ring

Contents

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

## 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 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.

### For $E_\infty$-rings

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

## Examples

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