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Difference between revisions of "Local property"

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''in commutative algebra''
 
''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" .
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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 ring|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 <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" />.
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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 <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.
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However, the property of an $A$-module $M$ of being [[Free module|free]] is not local.
  
 
====References====
 
====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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====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]].
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For the term  "local property"  in algebraic systems (such as groups) as well as in topology see [[Local and residual properties]].
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Latest revision as of 08:07, 1 January 2017

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

[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.

How to Cite This Entry:
Local property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_property&oldid=40135
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article