Berlekamp–Massey algorithm - Revision history

173.38.209.10: /* Description of algorithm */

2014-09-10T12:43:26Z

Description of algorithm

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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.

	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.~~

	~~Noncommutative analogues of 0-dimensional Gorenstein rings are called [[Frobenius ring]]s.~~

	~~==Definitions==~~

	~~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]].~~

	~~The classical definition reads:~~

	~~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}}~~

	~~For a [[Noetherian ring\|Noetherian]] commutative [[local ring]] <math>(R, m, k)</math> of Krull dimension <math>n</math>, the following are equivalent:~~
	* <math>R</math> has finite [[injective dimension]] as an <math>R</math>-module;
	* <math>R</math> has injective dimension <math>n</math> as an <math>R</math>-module;
	* <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>;
	* <math>\operatorname{Ext}^i_R (k, R) = 0</math> for some <math>i > n</math>;
	* <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>;
	* <math>R</math> is an ~~<math>n</math>-dimensional Gorenstein ring~~.

	~~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.~~

	~~==Examples==~~

	* Every local [[complete intersection ring]], in particular every [[regular local ring]], is Gorenstein.

	*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.~~

	*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.~~

	~~== Properties ==~~
	~~A noetherian commutative local ring is Gorenstein if and only if its completion is Gorenstein.<ref>{{harvnb\|Matsumura\|1986\|nb=Theorem 18.3}}</ref>~~

	~~==References==~~
	~~{{reflist}}~~
	*{{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}}~~
	*{{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}}~~
	*{{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}}
	*{{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}}
	*{{eom\|id=Gorenstein_ring}}
	*{{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}}~~
	*Hideyuki Matsumura, ''Commutative Ring Theory'', Cambridge studies in advanced mathematics 8.
	*{{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}}


	~~[[Category:Commutative algebra]~~]

en>Mr. Granger: Repairing links to disambiguation pages - You can help!

2014-01-21T20:02:20Z

Repairing links to disambiguation pages - You can help!

← Older revision		Revision as of 22:02, 21 January 2014
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	~~Wilber Berryhill~~ is ~~what his wife enjoys~~ to ~~contact him~~ and ~~he completely loves this title~~. ~~To play lacross~~ is the ~~thing I adore most~~ of all. ~~For years she~~'~~s been living~~ in ~~Kentucky but her spouse desires them to move~~. ~~My working day occupation~~ is a ~~journey agent~~.<br><br>~~Feel free to surf to my web~~-~~site~~ :: [http://www.~~discendodiscimus~~.~~com~~/~~AndreasigLumholtzkv psychic readings online~~]
			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.

			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.

			Noncommutative analogues of 0-dimensional Gorenstein rings are called [[Frobenius ring]]s.

			==Definitions==

			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]].

			The classical definition reads:

			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}}

			For a [[Noetherian ring\|Noetherian]] commutative [[local ring]] <math>(R, m, k)</math> of Krull dimension <math>n</math>, the following are equivalent:
			* <math>R</math> has finite [[injective dimension]] as an <math>R</math>-module;
			* <math>R</math> has injective dimension <math>n</math> as an <math>R</math>-module;
			* <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>;
			* <math>\operatorname{Ext}^i_R (k, R) = 0</math> for some <math>i > n</math>;
			* <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>;
			* <math>R</math> is an <math>n</math>-dimensional Gorenstein ring.

			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.

			==Examples==

			* Every local [[complete intersection ring]], in particular every [[regular local ring]], is Gorenstein.

			*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.

			*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.

			== Properties ==
			A noetherian commutative local ring is Gorenstein if and only if its completion is Gorenstein.<ref>{{harvnb\|Matsumura\|1986\|nb=Theorem 18.3}}</ref>

			==References==
			{{reflist}}
			*{{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}}
			*{{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}}
			*{{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}}
			*{{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}}
			*{{eom\|id=Gorenstein_ring}}
			*{{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}}
			*Hideyuki Matsumura, ''Commutative Ring Theory'', Cambridge studies in advanced mathematics 8.
			*{{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}}


			[[Category:Commutative algebra]]

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2012-07-12T20:47:02Z

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