Cohen-Macaulay ring: Difference between revisions
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* For any [[prime ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]] | * For any [[prime ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]] | ||
* For any [[ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]] | * For any [[ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]] | ||
==Relation with other properties== | |||
==Metaproperties== | |||
{{poly-closed curing property}} | |||
{{local curing property}} | |||
The property of being a Cohen-Macaulay ring is ''local'' in the sense that a commutative unital ring is Cohen-Macaulay if and only if its localizations at all maximal ideals are Cohen-Macaulay, if and only if its localizations at all prime ideals are Cohen-Macaulay. | |||
Revision as of 11:20, 7 August 2007
This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
Definition
Symbol-free definition
A commutative unital ring is termed Cohen-Macaulay if it satisfies the following equivalent conditions:
- For any maximal ideal, the depth equals the codimension
- For any prime ideal, the depth equals the codimension
- For any ideal, the depth equals the codimension
Relation with other properties
Metaproperties
Closure under taking the polynomial ring
This property of commutative unital rings is polynomial-closed: it is closed under the operation of taking the polynomial ring. In other words, if
is a commutative unital ring satisfying the property, so is
View other polynomial-closed properties of commutative unital rings
Local nature
This property of commutative unital rings is local in the following sense: a commutative unital ring satisfies the property iff its localizations at all prime ideals satisfy the property
The property of being a Cohen-Macaulay ring is local in the sense that a commutative unital ring is Cohen-Macaulay if and only if its localizations at all maximal ideals are Cohen-Macaulay, if and only if its localizations at all prime ideals are Cohen-Macaulay.