Cohen-Macaulay ring: Difference between revisions

Latest revision as of 15:39, 18 July 2013

This article is about a definition in group theory that is standard among the commutative algebra community (or sub-community that dabbles in such things) but is not very basic or common for people outside

View a list of other standard non-basic definitions

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

Symbol-free definition

A commutative unital ring is termed Cohen-Macaulay if it is Noetherian and satisfies the following equivalent conditions:

For any maximal ideal, the depth equals the codimension
For any prime ideal, the depth equals the codimension
For any ideal, the depth equals the codimension

Equivalence of definitions

Further information: Equivalence of definitions of Cohen-Macaulay

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
polynomial-closed property of commutative unital rings	Yes	Cohen-Macaulay is polynomial-closed	Suppose $R$ is a Cohen-Macaulay ring. Then, the polynomial ring $R[x]$ is also a Cohen-Macaulay ring.
strongly local property of commutative unital rings	Yes	Cohen-Macaulay is strongly local	The following are eqiuvalent for a commutative unital ring $R$ : (i) $R$ is Cohen-Macaulay, (ii) the localization at any prime ideal of $R$ is Cohen-Macaulay, (iii) the localization at any maximal ideal of $R$ is Cohen-Macaulay.
quotient-closed property of commutative unital rings	No	Cohen-Macaulay is not quotient-closed	It is possible to have a Cohen-Macaulay ring $R$ and an ideal $I$ in $R$ such that $R/I$ is not Cohen-Macaulay. However, if $I$ 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.
finite direct product-closed property of commutative unital rings	Yes	Cohen-Macaulay is finite direct product-closed	Suppose $R_{1},R_{2},\dots ,R_{n}$ are all Cohen-Macaulay rings. Then, the external direct product $R_{1}\times R_{2}\times \dots \times R_{n}$ is also a Cohen-Macaulay ring.

Relation with other properties

Conjunction with other properties

Conjunction	Other component of conjunction	Comments
Cohen-Macaulay domain	integral domain
local Cohen-Macaulay ring	local ring

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Regular local ring		regular local implies Cohen-Macaulay	Cohen-Macaulay not implies regular local	click here
regular ring	its localization at any prime ideal is a regular local ring	Regular implies Cohen-Macaulay	Cohen-Macaulay not implies regular	click here
polynomial ring over a field	has the form $K[x]$ where $K$ is a field			click here
multivariate polynomial ring over a field	has the form $K[x_{1},x_{2},\dots ,x_{n}]$ where $K$ is a field			click here
principal ideal domain	every ideal is a principal ideal, i.e., it is generated by one element			click here
Dedekind domain	one-dimensional Noetherian normal domain			click here
Finite-dimensional algebra over a field				click here
Artinian ring	every descending chain of ideals stabilizes at a finite length	Artinian implies Cohen-Macaulay		click here
one-dimensional Noetherian domain	one-dimensional Noetherian integral domain	One-dimensional Noetherian domain implies Cohen-Macaulay		click here

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Noetherian ring	every ideal is finitely generated			click here
universally catenary ring		Cohen-Macaulay implies universally catenary	universally catenary not implies Cohen-Macaulay	click here
catenary ring		Cohen-Macaulay implies catenary	catenary not implies Cohen-Macaulay	click here

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 $k[x,y,z]/(xz,yz)$ .

However:

The spectrum need not be irreducible: The ring $k[x,y]/(xy)$ 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,

@@ Line 6: / Line 6: @@
 ===Symbol-free definition===
-A [[commutative unital ring]] is termed '''Cohen-Macaulay''' if it satisfies the following equivalent conditions:
+A [[commutative unital ring]] is termed '''Cohen-Macaulay''' if it is [[Noetherian ring|Noetherian]] and satisfies the following equivalent conditions:
 * For any [[maximal ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]]
 * For any [[prime ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]]
 * For any [[ideal]], the [[depth of an ideal|depth]] equals the [[codimension of an ideal|codimension]]
+===Equivalence of definitions===
+{{further|[[Equivalence of definitions of Cohen-Macaulay]]}}
+==Metaproperties==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[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.
+|-
+| [[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.
+|-
+| [[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.
+|-
+| [[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.
+|}
 ==Relation with other properties==
-==Metaproperties==
+===Conjunction with other properties===
+{| class="sortable" border="1"
+! Conjunction !! Other component of conjunction !! Comments
+|-
+| [[Weaker than::Cohen-Macaulay domain]] || [[integral domain]] ||
+|-
+| [[Weaker than::local Cohen-Macaulay ring]] || [[local ring]] ||
+|}
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[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::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::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===
+{| class="sortable" border="1"
+! 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}}
+|}
+==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.
-{{poly-closed curing property}}
+This rules out, for instance, rings like <math>k[x,y,z]/(xz,yz)</math>.
-{{strongly local curing property}}
+However:
-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.
+* 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,