nLab projective dimension



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

0P nP n1P 1P 0M0 0\to P_n\to P_{n-1}\to\cdots\to P_1\to P_0\to M\to 0

where P n,,P 1,P 0P_n,\ldots,P_1,P_0 are projective objects in CC.

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 MM.

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

gl.dim(C):=sup{i0|M,NC,Ext i(M,N)=0} 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 RR is the global dimension of the category of left (right) RR-modules.

See also global dimension theorem.

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.