nLab projective dimension

Definitions

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 objects of length $n$ , that is the exact sequence of the form

0\to P_n\to P_{n-1}\to\cdots\to P_1\to P_0\to M\to 0

where $P_n,\ldots,P_1,P_0$ are projective objects in $C$ .

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

gl.dim(C) := sup\{ i\geq 0\,|\, \exists M,N\in C,\, Ext^i(M,N) = 0\}

Thus the left (right) global dimension of a ring $R$ is the global dimension of the category of left (right) $R$ -modules.