Cohen-Macaulay ring
From Commalg
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 outsideView 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
Definition
Symbol-free definition
A commutative unital ring is termed Cohen-Macaulay if it is Noetherian and 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
Equivalence of definitions
Further information: Equivalence of definitions of Cohen-Macaulay
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
polynomial-closed property of commutative unital rings | Yes | Cohen-Macaulay is polynomial-closed | Suppose ![]() ![]() |
strongly local property of commutative unital rings | Yes | Cohen-Macaulay is strongly local | The following are eqiuvalent for a commutative unital ring ![]() ![]() ![]() ![]() |
quotient-closed property of commutative unital rings | No | Cohen-Macaulay is not quotient-closed | It is possible to have a Cohen-Macaulay ring ![]() ![]() ![]() ![]() ![]() |
finite direct product-closed property of commutative unital rings | Yes | Cohen-Macaulay is finite direct product-closed | Suppose ![]() ![]() |
Relation with other properties
Conjunction with other properties
Conjunction | Other component of conjunction | Comments |
---|---|---|
Cohen-Macaulay domain | integral domain | |
local Cohen-Macaulay ring | local ring |
Stronger properties
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Noetherian ring | every ideal is finitely generated | click here | ||
universally catenary ring | Cohen-Macaulay implies universally catenary | universally catenary not implies Cohen-Macaulay | click here | |
catenary ring | Cohen-Macaulay implies catenary | catenary not implies Cohen-Macaulay | click here |
Spectrum
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 .
However:
- 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,