If $C$ is an abelian category (usually assumed with enough projectives), and $M$ an object in $C$, then its projective dimension$pd(M)$ is the minimal integer $n$ (if it exists) such that there is a resolution of $M$ by projective modules of length $n$, that is the exact sequence of the form

An important identity involving the projective dimension of a module of finite projective dimension over a commutative Noetherian local ring is the Auslander-Buchsbaum formula.

Dually, we define the injective dimension of $M$.

Id $R$ is a ring then both the category of all right $R$-modules and the category of all left $R$-modules have enough projectives. The left (right) global dimension of $R$ is the supremum of projective dimensions of all left (right) $R$-modules. More generally, a global dimension of an abelian category$C$ may be defined as