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
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
- 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
Equivalence of definitions
Further information: Equivalence of definitions of Cohen-Macaulay
Relation with other properties
Conjunction with other properties
- Cohen-Macaulay domain: A Cohen-Macaulay ring that is also an integral domain.
- Local Cohen-Macaulay ring: A Cohen-Macaulay ring that is also a local ring.
- Noetherian ring
- Universally catenary ring: For proof of the implication, refer Cohen-Macaulay implies universally catenary and for proof of its strictness (i.e. the reverse implication being false) refer universally catenary not implies Cohen-Macaulay
- Catenary ring: For proof of the implication, refer Cohen-Macaulay implies catenary and for proof of its strictness (i.e. the reverse implication being false) refer Catenary not implies Cohen-Macaulay
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
A polynomial ring over a Cohen-Macaulay ring is Cohen-Macaulay. For full proof, refer: Cohen-Macaulay is polynomial-closed
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. For full proof, refer: Cohen-Macaulay is strongly local
Closure under taking quotients
A quotient of a Cohen-Macaulay ring by an ideal need not be Cohen-Macaulay. However, if the ideal is generated by a regular sequence, then the quotient is a Cohen-Macaulay ring. Thus, for instance, the quotient by a principal ideal generated by an element which is not a zero divisor, is again a Cohen-Macaulay ring.
This property of commutative unital rings is finite direct product-closed: a finite direct product of rings with this property, also has this property
View other finite direct product-closed properties of commutative unital rings
A finite direct product of Cohen-Macaulay rings is Cohen Macaulay. For full proof, refer: Cohen-Macaulay is finite direct product-closed
The spectrum of a Cohen-Macaulay ring has the following important geometric property: If two irreducible components intersect, they must have the same dimension. Here, by the dimension of an irreducible component, we mean the dimension of the corresponding minimal prime ideal, or more explicitly, the Krull dimension of the quotient ring by that minimal prime ideal.
This rules out, for instance, rings like .
- The spectrum need not be irreducible: The ring is Cohen-Macaulay, although its spectrum has two irreducible components.
- All components of the spectrum need not have the same dimension: For instance, we could have a ring that is a disjoint union of irreducible subsets of different dimensions
- Not every irreducible space is the spectrum of a Cohen-Macaulay ring. In other words, not every integral domain, or even every affine domain, is Cohen-Macaulay,