View source for Local property

''in commutative algebra''

A property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601801.png" /> of a [[Commutative ring|commutative ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601802.png" /> or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601803.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601804.png" /> that is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601805.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601806.png" />) if and only if a similar property holds for the localizations (cf. [[Local ring|Local ring]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601807.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601808.png" />) with respect to all prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l0601809.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018010.png" />. This terminology becomes clear if one associates to the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018011.png" /> the topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018012.png" /> (the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018013.png" />) consisting of all prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018014.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018015.png" /> is integrally closed in its field of fractions if and only if the localizations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018016.png" /> are integrally closed for all maximal ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018018.png" />. A homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018019.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018020.png" /> is an isomorphism (monomorphism, epimorphism, null morphism) if and only if the mapping of localized modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018021.png" /> is an isomorphism (monomorphism, epimorphism, null morphism) for all maximal ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018022.png" />.

However, the property of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018023.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060180/l06018024.png" /> of being free is not local.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>



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