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| {{distinguish|Auslander–Buchsbaum theorem}}
| | I'm Crystle (22) from Vitry-Sur-Seine, France. <br>I'm learning Portuguese literature at a local high school and I'm just about to graduate.<br>I have a part time job in a the office.<br><br>Feel free to surf to my site - [http://Odsrenovations.Beep.com home remodeling contractor new orleans] |
| 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 ring|Noetherian]] [[local ring]] and ''M'' is a [[finitely generated module|finitely generated]] ''R''-module of finite [[projective dimension]], then
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| : <math> \mathrm{pd}_R(M) + \mathrm{depth}(M) = \mathrm{depth}(R).</math>
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| Here pd stands for the projective dimension of a module, and depth for the [[depth (ring theory)|depth]] of a module.
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| ==Applications==
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| The Auslander–Buchsbaum formula implies that a Noetherian local ring is [[regular local ring|regular]] if, and only if, it has finite [[global dimension]]. In turn this implies that the [[localization of a ring|localization]] of a regular local ring is regular.
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| If ''A'' is a local finitely generated [[algebra over a ring|''R''-algebra]] (over a regular local ring ''R''), then the Auslander–Buchsbaum formula implies that ''A'' is [[Cohen–Macaulay ring|Cohen–Macaulay]] if, and only if, pd<sub>''R''</sub>''A'' = codim<sub>''R''</sub>''A''.
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| ==References==
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| *{{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}}
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| *Chapter 19 of {{Citation
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| | last=Eisenbud
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| | first=David
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| | author-link=David Eisenbud
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| | title=Commutative algebra with a view toward algebraic geometry
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| | publisher=[[Springer-Verlag]]
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| | location=Berlin, New York
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| | series=[[Graduate Texts in Mathematics]]
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| | isbn=978-0-387-94269-8
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| | mr=1322960
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| | year=1995
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| | volume=150
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| }}
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| {{DEFAULTSORT:Auslander-Buchsbaum formula}}
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| [[Category:Commutative algebra]]
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I'm Crystle (22) from Vitry-Sur-Seine, France.
I'm learning Portuguese literature at a local high school and I'm just about to graduate.
I have a part time job in a the office.
Feel free to surf to my site - home remodeling contractor new orleans