# 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. 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 $\operatorname{pd}(M)=d$ 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. 3.

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

4. 4.

In the (abelian) category of left (right) $R$-modules, the projective dimension of a left (right) $R$-module $M$ is denoted by $\operatorname{pd}_{R}(M)$.

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.

Title projective dimension ProjectiveDimension 2013-03-22 14:50:56 2013-03-22 14:50:56 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 16E10 msc 13D05 msc 18G20 injective dimension