Projective module

The projective dimension is a homological notion of commutative algebra. It measures how far away from a module is to be projective. A projective module has projective dimension zero.

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.

  • 4.1 Examples
  • 4.2 Characterization of regular rings

Definition

Projective dimension of a module over a ring is the minimum number, so that there is an exact sequence

With projective modules (ie a projective resolution) all, if there ever is such a number, otherwise infinite.

The projective dimension of a module over a ring is ( inter alia ) with

Noted.

Three sets of the projective dimension

There are the following levels:

First Set

Is a module over a ring, are equivalent:

  • .
  • All modules and all Extn (M, N ) = 0

Second sentence

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

Here, the depth of the module.

Third set

Is

An exact sequence of -modules, a module has finite projective dimension one if and only if the other two modules have finite projective dimension.

In this case:

Example

Is a regular local ring with residue class field, then

In particular, there are so examples of modules of any projective dimension.

Global dimension

If a module, then under the global dimension (also: cohomological dimension), the " number" understood with:

Examples

  • The global dimension of a body is zero.
  • The global dimension of a Dedekind ring is 1, if it is not a body.

Characterization of regular rings

A Noetherian local ring is regular if its global dimension is finite. In this case, its global dimension is equal to its Krull dimension.

It follows in particular the statement that the localization of regular local rings is regular again.

Injective dimension

In analogy to the projective dimension of the injective dimension is defined as the smallest length of a injective resolution.

Projective object Krulldimension Robin Hartshorne
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