Localization of Rings

Localization generalizes the construction of fractions, allowing formal inversion of elements in a ring. It is the algebraic counterpart of "zooming in" on a point in geometry.

Construction

Definition5.3Localization

Let $R$ be a commutative ring and $S \subseteq R$ a multiplicative set ( $1 \in S$ , $S$ closed under multiplication). The localization $S^{-1}R$ consists of equivalence classes of pairs $(r, s)$ with $r \in R$ , $s \in S$ , where $(r,s) \sim (r', s')$ iff $t(rs' - r's) = 0$ for some $t \in S$ . We write the class as $r/s$ .

Operations: $r/s + r'/s' = (rs' + r's)/(ss')$ , $(r/s)(r'/s') = (rr')/(ss')$ .

ExampleKey localization examples

Field of fractions: $S = R \setminus \{0\}$ for an integral domain $R$ . Then $S^{-1}R = \mathrm{Frac}(R)$ . For $R = \mathbb{Z}$ : $\mathrm{Frac}(\mathbb{Z}) = \mathbb{Q}$ .
Localization at a prime: For a prime ideal $\mathfrak{p}$ , take $S = R \setminus \mathfrak{p}$ . Then $R_\mathfrak{p} = S^{-1}R$ is a local ring with unique maximal ideal $\mathfrak{p}R_\mathfrak{p}$ .
Inverting a single element: $S = \{1, f, f^2, \ldots\}$ . Then $R_f = S^{-1}R \cong R[x]/(xf-1)$ .

Properties

Theorem5.3Properties of localization

Let $\iota: R \to S^{-1}R$ be the natural map $r \mapsto r/1$ .

Universal property: For any ring homomorphism $\varphi: R \to T$ with $\varphi(s)$ invertible for all $s \in S$ , there exists a unique $\bar{\varphi}: S^{-1}R \to T$ with $\bar{\varphi} \circ \iota = \varphi$ .
Ideals of $S^{-1}R$ correspond to ideals of $R$ that are $S$ -saturated: $I = \{r : r/1 \in S^{-1}I\}$ .
Prime ideals of $S^{-1}R$ correspond bijectively to prime ideals of $R$ disjoint from $S$ : $\mathrm{Spec}(S^{-1}R) \cong \{\mathfrak{p} \in \mathrm{Spec}(R) : \mathfrak{p} \cap S = \emptyset\}$ .
Localization is exact: it preserves short exact sequences of modules.

RemarkLocal-global principle

Many properties of rings and modules can be checked locally: a module $M$ is zero iff $M_\mathfrak{p} = 0$ for all prime ideals $\mathfrak{p}$ ; a homomorphism $f$ is injective (resp. surjective) iff $f_\mathfrak{p}$ is injective (resp. surjective) for all $\mathfrak{p}$ . This "local-global principle" is one of the most powerful tools in commutative algebra and algebraic geometry.