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
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 $R$ is a Cohen-Macaulay ring. Then, the polynomial ring $R[x]$ is also a Cohen-Macaulay ring.
strongly local property of commutative unital rings	Yes	Cohen-Macaulay is strongly local	The following are eqiuvalent for a commutative unital ring $R$: (i) $R$ is Cohen-Macaulay, (ii) the localization at any prime ideal of $R$ is Cohen-Macaulay, (iii) the localization at any maximal ideal of $R$ is Cohen-Macaulay.
quotient-closed property of commutative unital rings	No	Cohen-Macaulay is not quotient-closed	It is possible to have a Cohen-Macaulay ring $R$ and an ideal $I$ in $R$ such that $R/I$ is not Cohen-Macaulay. However, if $I$ 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.
finite direct product-closed property of commutative unital rings	Yes	Cohen-Macaulay is finite direct product-closed	Suppose $R_1,R_2,\dots,R_n$ are all Cohen-Macaulay rings. Then, the external direct product $R_1 \times R_2 \times \dots \times R_n$ is also a Cohen-Macaulay ring.

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Regular local ring		regular local implies Cohen-Macaulay	Cohen-Macaulay not implies regular local	click here
regular ring	its localization at any prime ideal is a regular local ring	Regular implies Cohen-Macaulay	Cohen-Macaulay not implies regular	click here
polynomial ring over a field	has the form $K[x]$ where $K$ is a field			click here
multivariate polynomial ring over a field	has the form $K[x_1,x_2,\dots,x_n]$ where $K$ is a field			click here
principal ideal domain	every ideal is a principal ideal, i.e., it is generated by one element			click here
Dedekind domain	one-dimensional Noetherian normal domain			click here
Finite-dimensional algebra over a field				click here
Artinian ring	every descending chain of ideals stabilizes at a finite length	Artinian implies Cohen-Macaulay		click here
one-dimensional Noetherian domain	one-dimensional Noetherian integral domain	One-dimensional Noetherian domain implies Cohen-Macaulay		click here

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 $k[x,y,z]/(xz,yz)$.

However:

• The spectrum need not be irreducible: The ring $k[x,y]/(xy)$ 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,