Auslander–Buchsbaum formula

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

\( \mathrm{pd}_R(M) + \mathrm{depth}(M) = \mathrm{depth}(R). \)

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

The Auslander–Buchsbaum formula 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

Auslander, Maurice; Buchsbaum, David A. (1957), "Homological dimension in local rings", Transactions of the American Mathematical Society 85: 390–405, ISSN 0002-9947, JSTOR 1992937, MR 0086822
Chapter 19 of Eisenbud, David (1995), Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8, MR 1322960

Mathematics Encyclopedia