Among all the projective resolutions of , 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 such that for all ). If such a subset is non-empty, then the projective dimension of is defined to be the smallest number such that
We denote this by . If this subset is empty, then we define .
In an abelian category having enough projectives, the projective dimension of an object always exists (whether it is finite or not).
Then for all .
iff is a projective object.
In the (abelian) category of left (right) -modules, the projective dimension of a left (right) -module is denoted by .
Likewise, given an abelian category and a object having at least one injective resolution. Then the injective dimension, denoted by is the minimum number such that
if such an injective resolution exists. Otherwise, set . This is the dual notion of projective dimension.
|Date of creation||2013-03-22 14:50:56|
|Last modified on||2013-03-22 14:50:56|
|Last modified by||CWoo (3771)|