# nLab projective dimension

Definitions

## 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 modules 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.

category: algebra

Last revised on October 6, 2023 at 08:24:10. See the history of this page for a list of all contributions to it.