Cohen–Macaulay ring

In the mathematical subfield of commutative algebra is meant by a Cohen- Macaulay ring a Noetherian ring which is not necessarily regular, but whose depth is equal to its Krull dimension. A Cohen- Macaulay singularity is a singularity whose local ring is a Cohen- Macaulay ring. The rings were named after Irvin Cohen and Francis Macaulay.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

Definitions

Regular follow-

When a module over a ring is, as an element is called regular if for a always follows.

A sequence of elements of is called - regular sequence if the following conditions are met:

  • For the image of not a zero divisor in

Depth of a module

When a module over a ring, so is from the depth of the cardinality of a maximum - regular sequence of elements.

Dimension of a module

The dimension of a module over a ring is defined as the dimension of Krull. ( Is the annihilator of M. )

If a finitely generated module over a ring noetherschem, then:

(For the notation: denotes the set of prime ideals associated to the carrier of the module. )

Even true for a finitely generated module over a Noetherian local ring:

Cohen- Macaulay

A finitely generated module over a ring is called Cohen- Macaulay noetherschem module if for all maximal ideals of:

Is called Cohen- Macaulay ring if the module is a Cohen- Macaulay module.

Cohen- Macaulay rings

  • Any localization of a Cohen- Macaulay ring is a Cohen- Macaulay ring.
  • Every 0 -dimensional Noetherian ring is a Cohen- Macaulay ring.
  • Each one-dimensional reduced Noetherian ring is a Cohen- Macaulay ring.
  • Every regular noetherian ring is a Cohen- Macaulay ring.
  • Every Gorenstein ring is a Cohen- Macaulay ring.
  • Every Cohen- Macalay ring is a chain ring.

Examples

  • A body, then the variety, which consists of the X-axis and the Y axis is described by the coordinates of the ring.
  • A more complicated singularity consists in the ring
Krulldimension Francis Sowerby Macaulay Associated prime Reduced ring Catenary ring Robin Hartshorne
196422
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