projective dimension

Let 𝒜 be an abelian categoryMathworldPlanetmathPlanetmathPlanetmath and MOb(𝒜) such that a projective resolution of M exists:


Among all the projective resolutions of M, consider the subset consisting of those projective resolutions that contain only a finite number of non-zero projective objects (there exists a non-negative integer n such that Pi=0 for all in). If such a subset is non-empty, then the projective dimension of M is defined to be the smallest number d such that


We denote this by pd(M)=d. If this subset is empty, then we define pd(M)=.


  1. 1.

    In an abelian category having enough projectives, the projective dimension of an object always exists (whether it is finite or not).

  2. 2.

    If pd(M)=d and


    Then Pi0 for all 0id.

  3. 3.

    pd(M)=0 iff M is a projective object.

  4. 4.

    In the (abelian) categoryMathworldPlanetmath of left (right) R-modules, the projective dimension of a left (right) R-module M is denoted by pdR(M).

Likewise, given an abelian category and a object N having at least one injective resolution. Then the injective dimension, denoted by id(N) is the minimum number d such that


if such an injective resolution exists. Otherwise, set id(N)=. This is the dual notion of projective dimension.

Title projective dimension
Canonical name ProjectiveDimension
Date of creation 2013-03-22 14:50:56
Last modified on 2013-03-22 14:50:56
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 16E10
Classification msc 13D05
Classification msc 18G20
Defines injective dimension