If is an abelian category (usually assumed with enough projectives), and an object in , then its projective dimension is the minimal integer (if it exists) such that there is a resolution of by projective modules of length , that is the exact sequence of the form
where are projective objects in .
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 .
Id is a ring then both the category of all right -modules and the category of all left -modules have enough projectives. The left (right) global dimension of is the supremum of projective dimensions of all left (right) -modules. More generally, a global dimension of an abelian category may be defined as
Thus the left (right) global dimension of a ring is the global dimension of the category of left (right) -modules.
See also global dimension theorem.
Last revised on October 6, 2023 at 08:24:10. See the history of this page for a list of all contributions to it.