Local property

in commutative algebra

A property $P$ of a commutative ring $A$ or an $A$-module $M$ that is true for $A$ (or $M$) if and only if a similar property holds for the localizations (cf. Local ring) of $A$ (or $M$) with respect to all prime ideals of $A$, that is, a property that holds globally if and only if it holds locally everywhere. Often, instead of the set of all prime ideals one can restrict oneself to the set of maximal ideals of $A$. This terminology becomes clear if one associates to the ring $A$ the topological space $\text{Spec}\,A$ (the spectrum of $A$) consisting of all prime ideals of $A$. Then the assertion "$P$ is true for $A$" is equivalent to the assertion "$P$ holds on the whole space $\text{Spec}\,A$" , and the assertion "$P$ is true for all $A_{\mathfrak{P}}$" is equivalent to the assertion "every point $\mathfrak{P}$ of $\text{Spec}\,A$ has a neighbourhood in which $P$ holds" .

Examples of local properties. An integral domain $A$ is integrally closed in its field of fractions if and only if the localizations $A_{\mathfrak{m}}$ are integrally closed for all maximal ideals $\mathfrak{m}$ of $A$. A homomorphism of $A$-modules $f : M \rightarrow N$ is an isomorphism (monomorphism, epimorphism, null morphism) if and only if the mapping of localized modules $f_{\mathfrak{m}}: M_{\mathfrak{m}} \rightarrow N_{\mathfrak{m}}$ is an isomorphism (monomorphism, epimorphism, null morphism) for all maximal ideals $\mathfrak{m}$ of $A$.

However, the property of an $A$-module $M$ of being free is not local.

#### References[edit]

[1] |  N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)

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#### Comments[edit]

For the term "local property" in algebraic systems (such as groups) as well as in topology see Local and residual properties.