# 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

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_{R}*A* = codim_{R}*A*.

## References

- Auslander, Maurice; Buchsbaum, David A. (1957), "Homological dimension in local rings",
*Transactions of the American Mathematical Society*,**85**: 390–405, doi:10.2307/1992937, 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

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