Cohen-Macaulay ring - Revision history

Vipul at 15:39, 18 July 2013

2013-07-18T15:39:44Z

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 {{further|[[Equivalence of definitions of Cohen-Macaulay]]}}
 ==Relation with other properties==
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 |}
 ==Spectrum==

Vipul: /* Metaproperties */

2013-07-18T15:37:36Z

Metaproperties

← Older revision		Revision as of 15:37, 18 July 2013
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	==Metaproperties==		==Metaproperties==

	{~~{poly~~-closed ~~curing~~ property}}		{\| class="sortable" border="1"
			! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
	A polynomial ~~ring over~~ a Cohen-Macaulay ring is Cohen-Macaulay. ~~{{proofat~~\|[[~~Cohen-Macaulay is polynomial-closed]]}}~~		\|-
	{{strongly local ~~curing property}}~~		\| [[satisfies metaproperty::polynomial-closed property of commutative unital rings]] \|\| Yes \|\| [[Cohen-Macaulay is polynomial-closed]] \|\| Suppose <math>R</math> is a Cohen-Macaulay ring. Then, the polynomial ring <math>R[x]</math> is also a Cohen-Macaulay ring.
			\|-
	~~The~~ property of ~~being a~~ Cohen-Macaulay ~~ring~~ is ''local~~'' in the sense that~~ a commutative unital ring is Cohen-Macaulay ~~if and only if its localizations~~ at ~~all maximal ideals are~~ Cohen-Macaulay, ~~if and only if its localizations~~ at ~~all prime ideals are~~ Cohen-Macaulay. ~~{{proofat~~\|[[Cohen-Macaulay is ~~strongly local]]}}~~		\| [[satisfies metaproperty::strongly local property of commutative unital rings]] \|\| Yes \|\| [[Cohen-Macaulay is strongly local]] \|\| The following are eqiuvalent for a commutative unital ring <math>R</math>: (i) <math>R</math> is Cohen-Macaulay, (ii) the [[localization at a prime ideal\|localization at any prime ideal]] of <math>R</math> is Cohen-Macaulay, (iii) the localization at any maximal ideal of <math>R</math> is Cohen-Macaulay.
			\|-
	{{not Q-closed ~~curing property}}~~		\| [[dissatisfies metaproperty::quotient-closed property of commutative unital rings]] \|\| No \|\| [[Cohen-Macaulay is not quotient-closed]] \|\| It is possible to have a Cohen-Macaulay ring <math>R</math> and an ideal <math>I</math> in <math>R</math> such that <math>R/I</math> is not Cohen-Macaulay. However, if <math>I</math> is generated by a regular sequence, then the quotient is a Cohen-Macaulay ring. Thus, for instance, the quotient by a [[principal ideal]] generated by an element which is not a zero divisor is again a Cohen-Macaulay ring.
			\|-
	~~A quotient of~~ a Cohen-Macaulay ring by an ideal ~~need~~ not be Cohen-Macaulay. However, if ~~the ideal~~ is generated by a regular sequence, then the quotient is a Cohen-Macaulay ring. Thus, for instance, the quotient by a [[principal ideal]] generated by an element which is not a zero divisor, is again a Cohen-Macaulay ring.		\| [[satisfies metaproperty::finite direct product-closed property of commutative unital rings]] \|\| Yes \|\| [[Cohen-Macaulay is finite direct product-closed]] \|\| Suppose <math>R_1,R_2,\dots,R_n</math> are all Cohen-Macaulay rings. Then, the external direct product <math>R_1 \times R_2 \times \dots \times R_n</math> is also a Cohen-Macaulay ring.
			\|}
	{{finite DP-closed ~~curing~~ property}}

	~~A finite direct product~~ of ~~Cohen-Macaulay~~ rings ~~is Cohen Macaulay. {{proofat~~\|[[Cohen-Macaulay is finite direct product-closed]]}}

	==Spectrum==		==Spectrum==

Vipul: /* Conjunction with other properties */

2013-07-18T15:05:56Z

Conjunction with other properties

← Older revision		Revision as of 15:05, 18 July 2013
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	===Conjunction with other properties===		===Conjunction with other properties===

	* [[Weaker than::Cohen-Macaulay domain]]~~: A Cohen-Macaulay ring that is also an~~ [[integral domain]].		{\| class="sortable" border="1"
	* [[Weaker than::~~Local~~ Cohen-Macaulay ring]]~~: A Cohen-Macaulay ring that is also a~~ [[local ring]].		! Conjunction !! Other component of conjunction !! Comments
			\|-
			\| [[Weaker than::Cohen-Macaulay domain]] \|\| [[integral domain]] \|\|
			\|-
			\| [[Weaker than::local Cohen-Macaulay ring]] \|\| [[local ring]] \|\|
			\|}

	===Stronger properties===		===Stronger properties===

Vipul: /* Relation with other properties */

2013-07-18T04:48:24Z

Relation with other properties

← Older revision		Revision as of 04:48, 18 July 2013
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	===Weaker properties===		===Weaker properties===
	* [[Stronger than::Noetherian ring]]
	* [[Stronger than::~~Universally~~ catenary ring]]~~: {{proofofstrictimplicationat~~\|[[Cohen-Macaulay implies universally catenary]]\|[[universally catenary not implies Cohen-Macaulay]]}}		{\| class="sortable" border="1"
	* [[Stronger than::~~Catenary~~ ring]]~~: {{proofofstrictimplicationat~~\|[[Cohen-Macaulay implies catenary]]\|[[~~Catenary~~ not implies Cohen-Macaulay]]}}		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::Noetherian ring]] \|\| every ideal is [[finitely generated ideal\|finitely generated]] \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|Cohen-Macaulay ring}}
			\|-
			\| [[Stronger than::universally catenary ring]] \|\| \|\| [[Cohen-Macaulay implies universally catenary]]\|\|[[universally catenary not implies Cohen-Macaulay]] \|\| {{intermediate notions short\|universally catenary ring\|Cohen-Macaulay ring}}
			\|-
			\| [[Stronger than::catenary ring]] \|\| \|\| [[Cohen-Macaulay implies catenary]]\|\| [[catenary not implies Cohen-Macaulay]] \|\| {{intermediate notions short\|catenary ring\|Cohen-Macaulay ring}}
			\|}

	==Metaproperties==		==Metaproperties==

Vipul: /* Relation with other properties */

2013-07-18T04:43:18Z

Relation with other properties

← Older revision		Revision as of 04:43, 18 July 2013
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	===Stronger properties===		===Stronger properties===
	* [[Weaker than::Regular local ring]]~~: {{proofofstrictimplicationat~~\|[[~~Regular~~ local implies Cohen-Macaulay]]\|[[Cohen-Macaulay not implies regular local]]}}
	* [[Weaker than::~~Regular~~ ring]]~~: {{proofat~~\|[[Regular implies Cohen-Macaulay]]\|[[Cohen-Macaulay not implies regular]]}}		{\| class="sortable" border="1"
	* [[Weaker than::~~Polynomial~~ ring over a field]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Weaker than::~~Multivariate~~ polynomial ring over a field]]		\|-
	* [[Weaker than::~~Principal~~ ideal domain]]		\| [[Weaker than::Regular local ring]] \|\| \|\| [[regular local implies Cohen-Macaulay]]\|\| [[Cohen-Macaulay not implies regular local]] \|\| {{intermediate notions short\|Cohen-Macaulay ring\|regular local ring}}
	* [[Weaker than::Dedekind domain]]		\|-
	* [[Weaker than::Finite-dimensional algebra over a field]]		\| [[Weaker than::regular ring]] \|\| its [[localization at a prime\|localization]] at any [[prime ideal]] is a [[regular local ring]] \|\| [[Regular implies Cohen-Macaulay]]\|\|[[Cohen-Macaulay not implies regular]] \|\| {{intermediate notions short\|Cohen-Macaulay ring\|regular ring}}
	* [[Weaker than::Artinian ring]]~~: {{proofat~~\|[[Artinian implies Cohen-Macaulay]]}}		\|-
	* [[Weaker than::~~One~~-dimensional Noetherian domain]]~~: {{proofat~~\|[[One-dimensional Noetherian domain implies Cohen-Macaulay]]}}		\| [[Weaker than::polynomial ring over a field]] \|\| has the form <math>K[x]</math> where <math>K</math> is a [[field]] \|\| \|\| \|\| {{intermediate notions short\|Cohen-Macaulay ring\|polynomial ring over a field}}
			\|-
			\| [[Weaker than::multivariate polynomial ring over a field]] \|\| has the form <math>K[x_1,x_2,\dots,x_n]</math> where <math>K</matH> is a [[field]] \|\| \|\| \|\| {{intermediate notions short\|Cohen-Macaulay ring\|multivariate polynomial ring over a field}}
			\|-
			\| [[Weaker than::principal ideal domain]] \|\| every [[ideal]] is a [[principal ideal]], i.e., it is generated by one element \|\| \|\| \|\| {{intermediate notions short\|Cohen-Macaulay ring\|principal ideal domain}}
			\|-
			\| [[Weaker than::Dedekind domain]] \|\| [[one-dimensional ring\|one-dimensional]] [[Noetherian ring\|Noetherian]] [[normal domain]]\|\| \|\| \|\| {{intermediate notions short\|Cohen-Macaulay ring\|Dedekind domain}}
			\|-
			\| [[Weaker than::Finite-dimensional algebra over a field]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Cohen-Macaulay ring\|finite-dimensional algebra over a field}}
			\|-
			\| [[Weaker than::Artinian ring]] \|\| every descending chain of ideals stabilizes at a finite length \|\|\|[[Artinian implies Cohen-Macaulay]] \|\| \|\| {{intermediate notions short\|Cohen-Macaulay ring\|Artinian ring}}
			\|-
			\| [[Weaker than::one-dimensional Noetherian domain]] \|\| [[one-dimensional ring\|one-dimensional]] [[Noetherian ring\|Noetherian]] [[integral domain]] \|\| [[One-dimensional Noetherian domain implies Cohen-Macaulay]] \|\| \|\| {{intermediate notions short\|Cohen-Macaulay ring\|one-dimensional Noetherian domain}}
			\|}

	===Weaker properties===		===Weaker properties===

Vipul: /* Relation with other properties */

2009-01-17T17:15:27Z

Relation with other properties

← Older revision		Revision as of 17:15, 17 January 2009
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	* [[Weaker than::Finite-dimensional algebra over a field]]		* [[Weaker than::Finite-dimensional algebra over a field]]
	* [[Weaker than::Artinian ring]]: {{proofat\|[[Artinian implies Cohen-Macaulay]]}}		* [[Weaker than::Artinian ring]]: {{proofat\|[[Artinian implies Cohen-Macaulay]]}}
	* [[Weaker than::One-dimensional domain]]: {{proofat\|[[One-dimensional domain implies Cohen-Macaulay]]}}		* [[Weaker than::One-dimensional Noetherian domain]]: {{proofat\|[[One-dimensional Noetherian domain implies Cohen-Macaulay]]}}

	===Weaker properties===		===Weaker properties===

Vipul: /* Relation with other properties */

2009-01-17T17:08:51Z

Relation with other properties

← Older revision		Revision as of 17:08, 17 January 2009
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	==Relation with other properties==		==Relation with other properties==

			===Conjunction with other properties===

			* [[Weaker than::Cohen-Macaulay domain]]: A Cohen-Macaulay ring that is also an [[integral domain]].
			* [[Weaker than::Local Cohen-Macaulay ring]]: A Cohen-Macaulay ring that is also a [[local ring]].

	===Stronger properties===		===Stronger properties===
	* [[Regular local ring]]: {{proofofstrictimplicationat\|[[Regular local implies Cohen-Macaulay]]\|[[Cohen-Macaulay not implies regular local]]}}		* [[Weaker than::Regular local ring]]: {{proofofstrictimplicationat\|[[Regular local implies Cohen-Macaulay]]\|[[Cohen-Macaulay not implies regular local]]}}
	* [[Regular ring]]: {{proofat\|[[Regular implies Cohen-Macaulay]]\|[[Cohen-Macaulay not implies regular]]}}		* [[Weaker than::Regular ring]]: {{proofat\|[[Regular implies Cohen-Macaulay]]\|[[Cohen-Macaulay not implies regular]]}}
	* [[Polynomial ring over a field]]		* [[Weaker than::Polynomial ring over a field]]
	* [[Multivariate polynomial ring over a field]]		* [[Weaker than::Multivariate polynomial ring over a field]]
	* [[Principal ideal domain]]		* [[Weaker than::Principal ideal domain]]
	* [[Dedekind domain]]		* [[Weaker than::Dedekind domain]]
	* [[Finite-dimensional algebra over a field]]		* [[Weaker than::Finite-dimensional algebra over a field]]
	* [[Artinian ring]]: {{proofat\|[[Artinian implies Cohen-Macaulay]]}}		* [[Weaker than::Artinian ring]]: {{proofat\|[[Artinian implies Cohen-Macaulay]]}}
	* [[One-dimensional domain]]: {{proofat\|[[One-dimensional domain implies Cohen-Macaulay]]}}		* [[Weaker than::One-dimensional domain]]: {{proofat\|[[One-dimensional domain implies Cohen-Macaulay]]}}

	===Weaker properties===		===Weaker properties===
	* [[Noetherian ring]]		* [[Stronger than::Noetherian ring]]
	* [[Universally catenary ring]]: {{proofofstrictimplicationat\|[[Cohen-Macaulay implies universally catenary]]\|[[universally catenary not implies Cohen-Macaulay]]}}		* [[Stronger than::Universally catenary ring]]: {{proofofstrictimplicationat\|[[Cohen-Macaulay implies universally catenary]]\|[[universally catenary not implies Cohen-Macaulay]]}}
	* [[Catenary ring]]: {{proofofstrictimplicationat\|[[Cohen-Macaulay implies catenary]]\|[[Catenary not implies Cohen-Macaulay]]}}		* [[Stronger than::Catenary ring]]: {{proofofstrictimplicationat\|[[Cohen-Macaulay implies catenary]]\|[[Catenary not implies Cohen-Macaulay]]}}

	==Metaproperties==		==Metaproperties==

Vipul: 15 revisions

2008-05-12T16:19:22Z

15 revisions

← Older revision	Revision as of 16:19, 12 May 2008
(No difference)

Vipul at 14:07, 11 March 2008

2008-03-11T14:07:47Z

← Older revision		Revision as of 14:07, 11 March 2008
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	==Relation with other properties==		==Relation with other properties==
	===Stronger properties===		===Stronger properties===
	* [[Regular local ring]]		* [[Regular local ring]]: {{proofofstrictimplicationat\|[[Regular local implies Cohen-Macaulay]]\|[[Cohen-Macaulay not implies regular local]]}}
	* [[Regular ring]]		* [[Regular ring]]: {{proofat\|[[Regular implies Cohen-Macaulay]]\|[[Cohen-Macaulay not implies regular]]}}
	* [[Polynomial ring over a field]]		* [[Polynomial ring over a field]]
	* [[Multivariate polynomial ring over a field]]		* [[Multivariate polynomial ring over a field]]
	* [[Principal ideal domain]]		* [[Principal ideal domain]]
	* [[Dedekind domain]]		* [[Dedekind domain]]
	* [[~~Zero~~-dimensional ring]]		* [[Finite-dimensional algebra over a field]]
	* [[One-dimensional domain]]		* [[Artinian ring]]: {{proofat\|[[Artinian implies Cohen-Macaulay]]}}
			* [[One-dimensional domain]]: {{proofat\|[[One-dimensional domain implies Cohen-Macaulay]]}}

	===Weaker properties===		===Weaker properties===

Vipul at 02:09, 10 March 2008

2008-03-10T02:09:37Z

← Older revision		Revision as of 02:09, 10 March 2008
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	A finite direct product of Cohen-Macaulay rings is Cohen Macaulay. {{proofat\|[[Cohen-Macaulay is finite direct product-closed]]}}		A finite direct product of Cohen-Macaulay rings is Cohen Macaulay. {{proofat\|[[Cohen-Macaulay is finite direct product-closed]]}}

			==Spectrum==

			The spectrum of a Cohen-Macaulay ring has the following important geometric property: ''If two irreducible components intersect, they must have the same dimension''. Here, by the ''dimension'' of an irreducible component, we mean the dimension of the corresponding [[minimal prime ideal]], or more explicitly, the [[Krull dimension]] of the quotient ring by that minimal prime ideal.

			This rules out, for instance, rings like <math>k[x,y,z]/(xz,yz)</math>.

			However:

			* The spectrum need not be irreducible: The ring <math>k[x,y]/(xy)</math> is Cohen-Macaulay, although its spectrum has two irreducible components.
			* All components of the spectrum need not have the same dimension: For instance, we could have a ring that is a disjoint union of irreducible subsets of different dimensions
			* Not every irreducible space is the spectrum of a Cohen-Macaulay ring. In other words, not every integral domain, or even every [[affine domain]], is Cohen-Macaulay,