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, pdRA = codimRA.
auslander, buchsbaum, formula, confused, with, auslander, buchsbaum, theorem, commutative, algebra, introduced, auslander, buchsbaum, 1957, theorem, states, that, commutative, noetherian, local, ring, zero, finitely, generated, module, finite, projective, dime. 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 p d R M d e p t h M d e p t h R displaystyle 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 EditThe 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 pdRA codimRA References EditAuslander Maurice Buchsbaum David A 1957 Homological dimension in local rings Transactions of the American Mathematical Society 85 2 390 405 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 vol 150 Berlin New York Springer Verlag ISBN 978 0 387 94269 8 MR 1322960 This abstract algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Auslander Buchsbaum formula amp oldid 1150576002, wikipedia, wiki, book, books, library,