Localization - Key Properties

Localization preserves and reflects many important algebraic properties, making it a powerful tool for studying rings and modules.

TheoremExactness of Localization

Localization is an exact functor: if $0 \to M' \to M \to M'' \to 0$ is an exact sequence of $R$ -modules, then: $0 \to S^{-1}M' \to S^{-1}M \to S^{-1}M'' \to 0$ is exact as $S^{-1}R$ -modules.

This contrasts with tensor products, which are only right-exact. Exactness makes localization particularly well-behaved for homological computations.

DefinitionSupport of a Module

The support of an $R$ -module $M$ is: $\text{Supp}(M) = \{\mathfrak{p} \in \text{Spec}(R) : M_\mathfrak{p} \neq 0\}$

Geometrically, $\text{Supp}(M)$ is the set of points where the "sheaf" associated to $M$ is non-zero.

ExampleLocalization and Ideals

For an ideal $I \subseteq R$ and multiplicative set $S$ :

$S^{-1}I = \{i/s : i \in I, s \in S\}$ is an ideal of $S^{-1}R$
$(S^{-1}R)/(S^{-1}I) \cong S^{-1}(R/I)$ (localization commutes with quotients)
If $S \cap I \neq \emptyset$ , then $S^{-1}I = S^{-1}R$ (the ideal becomes trivial)

Prime ideals $\mathfrak{p}$ with $\mathfrak{p} \cap S = \emptyset$ correspond bijectively to prime ideals of $S^{-1}R$ .

TheoremLocal Criterion for Properties

Many properties can be checked locally. An $R$ -module $M$ is zero if and only if $M_\mathfrak{m} = 0$ for all maximal ideals $\mathfrak{m}$ .

More generally, a sequence of $R$ -modules is exact if and only if it is exact after localizing at every maximal ideal. This is the algebraic version of checking properties "locally" in geometry.

Remark

The stalk at a prime $\mathfrak{p}$ in algebraic geometry corresponds to the localization $M_\mathfrak{p}$ . The philosophy is that global properties can be understood by studying local properties at all points (prime ideals).

ExampleLocalization Detects Zero Divisors

An element $r \in R$ is a zero divisor if and only if there exists a prime ideal $\mathfrak{p}$ such that $r/1 = 0$ in $R_\mathfrak{p}$ .

Equivalently, $r$ is a zero divisor if and only if $\text{ann}(r) \not\subseteq \mathfrak{p}$ for some prime $\mathfrak{p}$ , where $\text{ann}(r) = \{a \in R : ar = 0\}$ is the annihilator of $r$ .

TheoremLocalization and Tensor Products

Localization commutes with tensor products: $S^{-1}(M \otimes_R N) \cong S^{-1}M \otimes_{S^{-1}R} S^{-1}N$

Combined with exactness, this makes localization compatible with most constructions in homological algebra.

DefinitionFlat Module

An $R$ -module $M$ is flat if the functor $M \otimes_R (-)$ is exact. Localization $S^{-1}R$ is always flat as an $R$ -module, which is another manifestation of the exactness property.

The exactness and compatibility properties of localization make it an indispensable tool for reducing global problems to local ones, embodying the principle "think globally, compute locally" throughout commutative algebra.