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| In [[commutative algebra]], a '''Gorenstein local ring''' is a [[Noetherian ring|Noetherian]] commutative [[local ring]] ''R'' with finite [[injective dimension]], as an ''R''-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition.
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| Gorenstein rings were introduced by Grothendieck, who named them because of their relation to a duality property of singular plane curves studied by {{harvs|txt|last=Gorenstein|authorlink=Daniel Gorenstein|year=1952}} (who was fond of claiming that he did not understand the definition of a Gorenstein ring). The zero dimensional case had been studied by {{harvtxt|Macaulay|1934}}. {{harvtxt|Serre|1961}} and {{harvtxt|Bass|1963}} publicized the concept of Gorenstein rings.
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| Noncommutative analogues of 0-dimensional Gorenstein rings are called [[Frobenius ring]]s.
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| ==Definitions==
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| A '''Gorenstein ring''' is a commutative ring such that each [[localization of a ring|localization]] at a [[prime ideal]] is a Gorenstein local ring. The Gorenstein ring concept is a special case of the more general [[Cohen–Macaulay ring]].
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| The classical definition reads:
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| A local [[Cohen–Macaulay ring]] ''R'' is called '''Gorenstein''' if there is a maximal [[regular sequence (algebra)|''R''-regular sequence]] in the maximal ideal generating an [[irreducible ideal]].{{Citation needed|date=May 2012}}
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| For a [[Noetherian ring|Noetherian]] commutative [[local ring]] <math>(R, m, k)</math> of Krull dimension <math>n</math>, the following are equivalent:
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| * <math>R</math> has finite [[injective dimension]] as an <math>R</math>-module;
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| * <math>R</math> has injective dimension <math>n</math> as an <math>R</math>-module;
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| * <math>\operatorname{Ext}^i_R (k, R) = 0</math> for <math>i \neq n</math> and <math>\operatorname{Ext}^n_R (k, R)</math> is isomorphic to <math>k</math>;
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| * <math>\operatorname{Ext}^i_R (k, R) = 0</math> for some <math>i > n</math>;
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| * <math>\operatorname{Ext}^i_R (k, R) = 0</math> for all <math>i < n</math> and <math>\operatorname{Ext}^n_R (k, R)</math> is isomorphic to <math>k</math>;
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| * <math>R</math> is an <math>n</math>-dimensional Gorenstein ring.
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| A (not necessarily commutative) ring ''R'' is called Gorenstein if ''R'' has finite injective dimension both as a left ''R''-module and as a right ''R''-module. If ''R'' is a local ring, we say ''R'' is a local Gorenstein ring.
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| ==Examples==
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| * Every local [[complete intersection ring]], in particular every [[regular local ring]], is Gorenstein.
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| *The ring ''k''[''x'',''y'',''z'']/(''x''<sup>2</sup>, ''y''<sup>2</sup>, ''xz'', ''yz'', ''z''<sup>2</sup>–''xy'') is a 0-dimensional Gorenstein ring that is not a complete intersection ring.
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| *The ring ''k''[''x'',''y'']/(''x''<sup>2</sup>, ''y''<sup>2</sup>, ''xy'') is a 0-dimensional Cohen–Macaulay ring that is not a Gorenstein ring.
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| == Properties ==
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| A noetherian commutative local ring is Gorenstein if and only if its completion is Gorenstein.<ref>{{harvnb|Matsumura|1986|nb=Theorem 18.3}}</ref>
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| ==References==
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| {{reflist}}
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| *{{Citation | last1=Bass | first1=Hyman | title=On the ubiquity of Gorenstein rings | doi=10.1007/BF01112819 | id={{MR|0153708}} | year=1963 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=82 | pages=8–28}}
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| *{{Citation | last1=Bruns | first1=Winfried | last2=Herzog | first2=Jürgen | title=Cohen-Macaulay rings | url=http://books.google.com/books?id=LF6CbQk9uScC | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-41068-7 | id={{MR|1251956}} | year=1993 | volume=39}}
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| *{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=An arithmetic theory of adjoint plane curves | url= http://www.jstor.org/stable/1990710 | id={{MR|0049591}} | year=1952 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=72 | pages=414–436}}
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| *{{Citation | last1=Grothendieck | first1=Alexandre | author1-link=Alexandre Grothendieck | title=Séminaire Bourbaki, Vol. 4 | url=http://www.numdam.org/item?id=SB_1956-1958__4__169_0 | publisher=[[Société Mathématique de France]] | location=Paris | id={{MR|1610898}} | year=1957 | chapter=Théorèmes de dualité pour les faisceaux algébriques cohérents | pages=169–193}}
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| *{{eom|id=Gorenstein_ring}}
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| *{{Citation | last1=Macaulay | first1=F. S. | title= Modern algebra and polynomial ideals | doi=10.1017/S0305004100012354 | year=1934 | journal=Mathematical Proceedings of the Cambridge Philosophical Society | issn=0305-0041 | volume=30 | issue=1 | pages=27–46}}
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| *Hideyuki Matsumura, ''Commutative Ring Theory'', Cambridge studies in advanced mathematics 8.
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| *{{Citation |year=1961| last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Sur les modules projectifs | url=http://www.numdam.org/item?id=SD_1960-1961__14_1_A2_0 | series=Séminaire Dubreil. Algèbre et théorie des nombres | volume=14 | pages=1–16}}
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| [[Category:Commutative algebra]]
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