“Auslander–Buchsbaum formula”的意思、由来-开放百科全书

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Auslander–Buchsbaum formula

释义

Applications
References

In commutative algebra, the Auslander–Buchsbaum formula, introduced by {{harvs|txt|last=Auslander|author1-link=Maurice Auslander|last2=Buchsbaum|author2-link=David Buchsbaum|year=1957|loc=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_RA = codim_RA.

References

{{Citation | last1=Auslander | first1=Maurice | last2=Buchsbaum | first2=David A. | title=Homological dimension in local rings | jstor=1992937 | mr=0086822 | year=1957 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=85 | pages=390–405 | doi=10.2307/1992937}}
Chapter 19 of {{Citation

| last=Eisenbud
| first=David
| author-link=David Eisenbud
| title=Commutative algebra with a view toward algebraic geometry
| publisher=Springer-Verlag
| location=Berlin, New York
| series=Graduate Texts in Mathematics
| isbn=978-0-387-94269-8
| mr=1322960
| year=1995
| volume=150
}}{{DEFAULTSORT:Auslander-Buchsbaum formula}}{{abstract-algebra-stub}}

2 : Commutative algebra|Theorems in abstract algebra

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