Depth (ring theory)

The depth of a module, in particular of an ideal is studied in commutative algebra. It is an important invariant, which plays a role in various definitions and sentences.

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.

Definition

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

The notation for the depth of a module is not uniform in the literature: in addition to and is also and find.

Properties

If a local noetherian ring with maximal ideal and is a finite module (which is not trivial, ie equal to 0) is on, then:

• If the residue field, then:

• The following applies:

Modules ( or rings ) in which equality holds are called Cohen- Macaulay modules ( or Cohen- Macaulay rings).

( Is the set of prime ideals associated to of. )

• Has a finite projective dimension, then:

It is particularly

Examples

• Is a vector space over a field of the vector space dimension, so its depth as - module is the same.
• The depth of a regular local ring is its Krull dimension.