Regular sequence

Regular sequences play a role in commutative algebra and algebraic geometry. You will be required to define the depth of a module and Cohen- Macaulay rings and to make statements about complete intersections.

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-

If a Noetherian 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

The modifier "- " is omitted when the context is clear which module is meant.

The special case, when a local ring, and the module itself is most important. In this case, all terms of the sequence lie in the maximal ideal.

Regular system parameters

Is local and the maximal ideal, then a minimal generating system is called by a regular system of parameters.

Properties

• A maximum - regular sequence is finite and all maximal - regular sequences have the same length.
• Is a finite module over a Noetherian local ring and is a regular sequence, then:

(The dimension of. )

• For a regular ring local with maximal ideal and is equivalent to:

In particular, a minimal generating system of a regular sequence.

• Conversely, if a Noetherian local ring with maximal ideal which is generated by a regular sequence of length, as is regular and.

• General: If a Noetherian local ring and a regular sequence, then every permutation of the train is regular. ( This does not apply to any Noetherian rings. )

Examples

• In the polynomial ring over a field every sequence of variables is a - regular sequence.
• The local ring body corresponds to the geometric intersection of two affine surfaces in four-dimensional space. The ring is two-dimensional, but regular sequences have length 1, since the ring contains only zero divisors and units modulo a non- zero divisor, which is not a unit. In particular, this ring is not Cohen- Macaulay ring.