In commutative algebra, the AuslanderâBuchsbaum formula, introduced by Auslander and Buchsbaum ( 1957, theorem 3.7), states that if R is a commutative Noetherian local ring and M is a non-zero finitely generated R-module of finite projective dimension, then:
Here pd stands for the projective dimension of a module, and depth for the depth of a module.
The AuslanderâBuchsbaum theorem implies that a Noetherian local ring is regular if, and only if, it has finite global dimension. In turn this implies that the localization of a regular local ring is regular.
If A is a local finitely generated R-algebra (over a regular local ring R), then the AuslanderâBuchsbaum formula implies that A is CohenâMacaulay if, and only if, pdRA = codimRA.
In commutative algebra, the AuslanderâBuchsbaum formula, introduced by Auslander and Buchsbaum ( 1957, theorem 3.7), states that if R is a commutative Noetherian local ring and M is a non-zero finitely generated R-module of finite projective dimension, then:
Here pd stands for the projective dimension of a module, and depth for the depth of a module.
The AuslanderâBuchsbaum theorem implies that a Noetherian local ring is regular if, and only if, it has finite global dimension. In turn this implies that the localization of a regular local ring is regular.
If A is a local finitely generated R-algebra (over a regular local ring R), then the AuslanderâBuchsbaum formula implies that A is CohenâMacaulay if, and only if, pdRA = codimRA.