Projective dimension

The projective dimension is a homological term from commutative algebra . It measures how far a module is from being projective . A projective module has the projective dimension zero.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

definition

The projective dimension of a module over a ring is the smallest number , making it an exact sequence ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle r}$

${\displaystyle 0\longrightarrow P_{r}\longrightarrow P_{r-1}\longrightarrow \dots \longrightarrow P_{0}\longrightarrow M\longrightarrow 0}$

with projective modules (i.e. a projective resolution ), if there is such a number at all, otherwise infinite. ${\displaystyle P_{i}}$

The projective dimension of a module over a ring is (among other things) with ${\displaystyle M}$ ${\displaystyle A}$

${\displaystyle r=\mathrm {proj.dim} _{A}(M)}$

written down.

Three sentences on the projective dimension

The following rates apply:

First sentence

If a module is over a ring , the following are equivalent: ${\displaystyle M}$ ${\displaystyle A}$

• ${\displaystyle r=\mathrm {proj.dim} _{A}(M)}$.
• For all modules and all , Ext ⁿ (M, N ) = 0. ${\displaystyle A}$ ${\displaystyle N}$ ${\displaystyle n>r}$

Second sentence

If a finitely generated module is over a Noetherian local ring , then is ${\displaystyle M}$ ${\displaystyle A}$

${\displaystyle \mathrm {proj.dim} _{A}(M)+\mathrm {tf} (M)=\mathrm {tf} (R)}$

Where is the depth of the module. ${\displaystyle \mathrm {tf} (M)}$

Third sentence

Is

${\displaystyle 0\longrightarrow M_{1}\longrightarrow M_{2}\longrightarrow M_{3}\longrightarrow 0}$

an exact sequence of modules, a module has a finite projective dimension if and only if the other two modules have a finite projective dimension. ${\displaystyle A}$ ${\displaystyle M_{i}}$

In this case:

${\displaystyle \mathrm {proj.dim} _{A}(M_{2})\leq \mathrm {max} (\mathrm {proj.dim} _{A}(M_{1}),\mathrm {proj.dim} _{A}(M_{3}))}$

example

If is a regular local ring with remainder class field , then is ${\displaystyle A}$ ${\displaystyle k}$

${\displaystyle \mathrm {proj.dim} _{A}(k)=\mathrm {dim} (A)}$

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

Global dimension

If there is a module, the global dimension (also: cohomological dimension) is understood to mean the "number" with: ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle r\in \mathbb {N} \cup \{\infty \}}$

${\displaystyle r=\mathrm {sup} \{\mathrm {proj.dim} _{A}M|M{\text{ ist A-Modul}}\}}$

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 and only if its global dimension is finite. In this case, its global dimension is equal to its Krull dimension .

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

Injective dimension

Analogous to the projective dimension, the injective dimension is defined as the smallest length of an injective resolution .

literature

• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Atiyah, Macdonald: Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
• Robin Hartshorne : Algebraic Geometry . Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9