projective dimension

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

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 $i\ge n$). If such a subset is non-empty, then the projective dimension of $M$ is defined to be the smallest number $d$ such that

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

   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

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