Cohen-Macaulay ring

This article is about a definition in group theory that is standard among the commutative algebra community (or sub-community that dabbles in such things) but is not very basic or common for people outside

View a list of other standard non-basic definitions

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
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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

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

$R$

is a commutative unital ring satisfying the property, so is

$R[x]$

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Strong local nature

This property of commutative unital rings is strongly local in the following sense: a commutative unital ring has the property iff its localization at each prime ideal has the property, iff its localization at each maximal ideal has the property
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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.