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).

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.

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
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