Auslander–Buchsbaum formula

Not to be confused with Auslander–Buchsbaum theorem.

In commutative algebra, the Auslander–Buchsbaum 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:

${\mathrm{p}\mathrm{d}}_R(M)+\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}(M)=\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}(R).$

Here pd stands for the projective dimension of a module, and depth for the depth of a module.

Applications

The Auslander–Buchsbaum 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 Auslander–Buchsbaum formula implies that A is Cohen–Macaulay if, and only if, pd_RA = codim_RA.

References

• Chapter 19 of