## You are here

Homeprojective dimension

## Primary tabs

# projective dimension

Let $\mathcal{A}$ be an abelian category and $M\in\operatorname{Ob}(\mathcal{A})$ such that a projective resolution of $M$ exists:

$\xymatrix{{\ldots}\ar[r]&P_{n}\ar[r]&{\ldots}\ar[r]&P_{1}\ar[r]&P_{0}\ar[r]&M% \ar[r]&0}.$ |

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 $P_{i}=0$ for all $i\geq n$). If such a subset is non-empty, then the *projective dimension* of $M$ is defined to be the smallest number $d$ such that

$\xymatrix{0\ar[r]&P_{d}\ar[r]&{\ldots}\ar[r]&P_{1}\ar[r]&P_{0}\ar[r]&M\ar[r]&0}.$ |

We denote this by $\operatorname{pd}(M)=d$. If this subset is empty, then we define $\operatorname{pd}(M)=\infty$.

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 $\operatorname{pd}(M)=d$ and

$\xymatrix{0\ar[r]&P_{d}\ar[r]&{\ldots}\ar[r]&P_{1}\ar[r]&P_{0}\ar[r]&M\ar[r]&0}.$ Then $P_{i}\neq 0$ for all $0\leq i\leq d$.

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

4.

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

$\xymatrix{0\ar[r]&N\ar[r]&I_{0}\ar[r]&I_{1}\ar[r]&{\ldots}\ar[r]&I_{d}\ar[r]&0},$ |

if such an injective resolution exists. Otherwise, set $\operatorname{id}(N)=\infty$. This is the dual notion of projective dimension.

## Mathematics Subject Classification

16E10*no label found*13D05

*no label found*18G20

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections