Localization - Applications

Localization provides powerful techniques for detecting and verifying module-theoretic properties locally.

TheoremLocal-Global Principle

Let $M$ be an $R$ -module. Then $M = 0$ if and only if $M_\mathfrak{m} = 0$ for all maximal ideals $\mathfrak{m}$ of $R$ .

More generally, a homomorphism $f: M \to N$ is:

Injective $\Leftrightarrow$ $f_\mathfrak{m}: M_\mathfrak{m} \to N_\mathfrak{m}$ is injective for all maximal $\mathfrak{m}$
Surjective $\Leftrightarrow$ $f_\mathfrak{m}$ is surjective for all maximal $\mathfrak{m}$
An isomorphism $\Leftrightarrow$ $f_\mathfrak{m}$ is an isomorphism for all maximal $\mathfrak{m}$

This principle reduces global questions to local verification, embodying the "think globally, check locally" philosophy.

TheoremSupport and Associated Primes

For a finitely generated $R$ -module $M$ : $\text{Supp}(M) = \{\mathfrak{p} \in \text{Spec}(R) : M_\mathfrak{p} \neq 0\} = V(\text{ann}(M))$

where $\text{ann}(M) = \{r \in R : rM = 0\}$ is the annihilator. The support is the vanishing locus of the annihilator, providing a geometric interpretation.

ExampleComputing Support

For $M = \mathbb{Z}/6\mathbb{Z}$ as a $\mathbb{Z}$ -module:

$\text{ann}(M) = 6\mathbb{Z}$
$\text{Supp}(M) = V(6\mathbb{Z}) = \{(2), (3)\}$
Indeed, $M_{(p)} \neq 0$ if and only if $p \in \{2, 3\}$

The support captures which primes "see" the module.

TheoremLocalization and Finite Generation

If $M$ is a finitely generated $R$ -module, then for any multiplicative set $S$ , the localization $S^{-1}M$ is a finitely generated $S^{-1}R$ -module.

Moreover, if $m_1, \ldots, m_n$ generate $M$ over $R$ , then $m_1/1, \ldots, m_n/1$ generate $S^{-1}M$ over $S^{-1}R$ .

Remark

The converse holds for finitely presented modules: if $M_\mathfrak{m}$ is finitely generated over $R_\mathfrak{m}$ for all maximal ideals $\mathfrak{m}$ , and $M$ is finitely presented, then $M$ is finitely generated over $R$ .

TheoremSaturation and Localization

For ideals $I, J \subseteq R$ , the saturation is $(I : J^\infty) = \{r \in R : rJ^n \subseteq I \text{ for some } n\}$ .

If $S$ is the multiplicative set generated by $J$ , then: $(I : J^\infty) = \phi^{-1}(S^{-1}I)$ where $\phi: R \to S^{-1}R$ is the natural map. Saturation can be computed via localization.

ExamplePrimary Decomposition and Localization

If $I = Q_1 \cap \cdots \cap Q_n$ is a primary decomposition with associated primes $\mathfrak{p}_1, \ldots, \mathfrak{p}_n$ , then localizing at $\mathfrak{p}_i$ isolates the $\mathfrak{p}_i$ -primary component: $I R_{\mathfrak{p}_i} = Q_i R_{\mathfrak{p}_i}$

This shows how localization "separates" primary components.

TheoremDiscrete Valuation Rings

A local integral domain $(R, \mathfrak{m})$ is a discrete valuation ring (DVR) if and only if $\mathfrak{m}$ is principal, say $\mathfrak{m} = (\pi)$ .

DVRs arise as localizations of Dedekind domains at prime ideals, providing the local perspective in algebraic number theory and algebraic geometry.

These applications demonstrate how localization transforms global module-theoretic problems into tractable local questions, making it indispensable for computational and theoretical work.