projective dimension
Let $\mathcal{A}$ be an abelian category^{} and $M\in \mathrm{Ob}(\mathcal{A})$ such that a projective resolution of $M$ exists:
$$\text{xymatrix}\mathrm{\dots}\text{ar}[r]\mathrm{\&}{P}_{n}\text{ar}[r]\mathrm{\&}\mathrm{\dots}\text{ar}[r]\mathrm{\&}{P}_{1}\text{ar}[r]\mathrm{\&}{P}_{0}\text{ar}[r]\mathrm{\&}M\text{ar}[r]\mathrm{\&}0.$$ 
Among all the projective resolutions of $M$, consider the subset consisting of those projective resolutions that contain only a finite number of nonzero projective objects (there exists a nonnegative integer $n$ such that ${P}_{i}=0$ for all $i\ge n$). If such a subset is nonempty, then the projective dimension of $M$ is defined to be the smallest number $d$ such that
$$\text{xymatrix}0\text{ar}[r]\mathrm{\&}{P}_{d}\text{ar}[r]\mathrm{\&}\mathrm{\dots}\text{ar}[r]\mathrm{\&}{P}_{1}\text{ar}[r]\mathrm{\&}{P}_{0}\text{ar}[r]\mathrm{\&}M\text{ar}[r]\mathrm{\&}0.$$ 
We denote this by $\mathrm{pd}(M)=d$. If this subset is empty, then we define $\mathrm{pd}(M)=\mathrm{\infty}$.
Remarks.

1.
In an abelian category having enough projectives, the projective dimension of an object always exists (whether it is finite or not).

2.
If $\mathrm{pd}(M)=d$ and
$$\text{xymatrix}0\text{ar}[r]\mathrm{\&}{P}_{d}\text{ar}[r]\mathrm{\&}\mathrm{\dots}\text{ar}[r]\mathrm{\&}{P}_{1}\text{ar}[r]\mathrm{\&}{P}_{0}\text{ar}[r]\mathrm{\&}M\text{ar}[r]\mathrm{\&}0.$$ Then ${P}_{i}\ne 0$ for all $0\le i\le d$.

3.
$\mathrm{pd}(M)=0$ iff $M$ is a projective object.

4.
In the (abelian) category^{} of left (right) $R$modules, the projective dimension of a left (right) $R$module $M$ is denoted by ${\mathrm{pd}}_{R}(M)$.
Likewise, given an abelian category and a object $N$ having at least one injective resolution. Then the injective dimension, denoted by $\mathrm{id}(N)$ is the minimum number $d$ such that
$$\text{xymatrix}0\text{ar}[r]\mathrm{\&}N\text{ar}[r]\mathrm{\&}{I}_{0}\text{ar}[r]\mathrm{\&}{I}_{1}\text{ar}[r]\mathrm{\&}\mathrm{\dots}\text{ar}[r]\mathrm{\&}{I}_{d}\text{ar}[r]\mathrm{\&}0,$$ 
if such an injective resolution exists. Otherwise, set $\mathrm{id}(N)=\mathrm{\infty}$. This is the dual notion of projective dimension.
Title  projective dimension 

Canonical name  ProjectiveDimension 
Date of creation  20130322 14:50:56 
Last modified on  20130322 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 