Local property

in commutative algebra

A property of a commutative ring or an -module that is true for (or ) if and only if a similar property holds for the localizations (cf. Local ring) of (or ) with respect to all prime ideals of , 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 . This terminology becomes clear if one associates to the ring the topological space (the spectrum of ) consisting of all prime ideals of . Then the assertion "P is true for A" is equivalent to the assertion "P holds on the whole space SpecA" , and the assertion "P is true for all AP" is equivalent to the assertion "every point P of SpecA has a neighbourhood in which P holds" .

Examples of local properties. An integral domain is integrally closed in its field of fractions if and only if the localizations are integrally closed for all maximal ideals of . A homomorphism of -modules is an isomorphism (monomorphism, epimorphism, null morphism) if and only if the mapping of localized modules is an isomorphism (monomorphism, epimorphism, null morphism) for all maximal ideals of .

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

References

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

Comments

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