# 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 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 ringsVIEW 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`

## 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.

### Stronger properties

### Weaker properties

- 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*

## 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 isView 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

View other strongly local properties of commutative unital rings

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

*This property of commutative unital rings is not quotient-closed: in other words, a quotient of a commutative unital ring with this property, need not have this property*

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.

### Direct products

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*

## 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,

- Standard non-basic definitions in commutative algebra
- Standard terminology
- Properties of commutative unital rings
- Polynomial-closed properties of commutative unital rings
- Strongly local properties of commutative unital rings
- Local properties of commutative unital rings
- Localization-closed properties of commutative unital rings
- Finite direct product-closed properties of commutative unital rings