# 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 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`

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

polynomial-closed property of commutative unital rings | Yes | Cohen-Macaulay is polynomial-closed | Suppose is a Cohen-Macaulay ring. Then, the polynomial ring 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 : (i) is Cohen-Macaulay, (ii) the localization at any prime ideal of is Cohen-Macaulay, (iii) the localization at any maximal ideal of 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 and an ideal in such that is not Cohen-Macaulay. However, if 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 are all Cohen-Macaulay rings. Then, the external direct product 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

### 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,