Localization - Core Definitions

Localization is the algebraic analogue of restricting attention to an open subset in geometry, allowing us to "invert" elements and study local properties of rings.

DefinitionMultiplicative Set

A subset $S \subseteq R$ is multiplicative if:

$1 \in S$
If $s, t \in S$ , then $st \in S$

Multiplicative sets are closed under multiplication and contain the identity, but need not contain $0$ (though they may).

ExampleStandard Multiplicative Sets

Powers of an element: $S = \{1, f, f^2, f^3, \ldots\}$ for any $f \in R$
Complement of prime ideal: $S = R \setminus \mathfrak{p}$ for a prime ideal $\mathfrak{p}$
Units: $S = R^\times$ (the group of units)
Non-zero-divisors: $S = R \setminus \bigcup \mathfrak{p}$ where the union is over associated primes

DefinitionLocalization of a Ring

Given a multiplicative set $S \subseteq R$ , the localization $S^{-1}R$ is constructed as follows. Consider pairs $(r, s)$ with $r \in R$ and $s \in S$ , and define an equivalence relation: $(r, s) \sim (r', s') \iff \exists t \in S : t(rs' - r's) = 0$

The localization $S^{-1}R$ consists of equivalence classes $r/s = [r,s]$ , with operations: $\frac{r}{s} + \frac{r'}{s'} = \frac{rs' + r's}{ss'}, \qquad \frac{r}{s} \cdot \frac{r'}{s'} = \frac{rr'}{ss'}$

DefinitionLocal Ring

A ring $R$ is local if it has a unique maximal ideal. Local rings capture the notion of "functions near a point" in algebraic geometry.

For a prime ideal $\mathfrak{p} \subseteq R$ , the localization $R_\mathfrak{p} = (R \setminus \mathfrak{p})^{-1}R$ is a local ring with maximal ideal $\mathfrak{p}R_\mathfrak{p}$ .

ExampleKey Localizations

Field of fractions: $S = R \setminus \{0\}$ $S = R ∖ {0}$ for integral domain $R$ $R$ gives $\text{Frac}(R)$ $Frac (R)$
- Example: $\mathbb{Q} = \text{Frac}(\mathbb{Z})$
Localization at prime: $R_\mathfrak{p} = (R \setminus \mathfrak{p})^{-1}R$ $R_{p} = (R ∖ p)^{- 1} R$
- Example: $\mathbb{Z}_{(p)}$ consists of fractions $a/b$ where $p \nmid b$
Localization at element: $R_f = \{1, f, f^2, \ldots\}^{-1}R$ $R_{f} = {1, f, f^{2}, \dots}^{- 1} R$
- Example: $k[x]_x = k[x, x^{-1}]$ (Laurent polynomials)

Remark

The natural map $\phi: R \to S^{-1}R$ given by $\phi(r) = r/1$ is a ring homomorphism. Elements of $S$ become units in $S^{-1}R$ , which is the universal property of localization: any ring homomorphism $R \to T$ that sends $S$ to units factors uniquely through $S^{-1}R$ .

DefinitionLocalization of Modules

For an $R$ -module $M$ and multiplicative set $S$ , the localization $S^{-1}M$ consists of fractions $m/s$ with $m \in M, s \in S$ , where: $\frac{m}{s} = \frac{m'}{s'} \iff \exists t \in S : t(sm' - s'm) = 0$

The module $S^{-1}M$ is an $S^{-1}R$ -module via $(r/s) \cdot (m/t) = (rm)/(st)$ .

Localization allows us to focus on local behavior near prime ideals, providing a bridge between global ring-theoretic properties and local geometric properties.