Cohen–Macaulay ring

E790527

Noetherian ring algebraic structure commutative ring mathematical object

A Cohen–Macaulay ring is a commutative Noetherian ring whose depth equals its Krull dimension, giving it especially well-behaved homological and geometric properties.

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All labels observed (2)

Label	Occurrences
Cohen–Macaulay rings	2
Cohen–Macaulay ring canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T9297166 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Cohen–Macaulay ring
Context triple: [Auslander–Buchsbaum formula, relatedConcept, Cohen–Macaulay ring]

A. Castelnuovo–Mumford regularity
Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
B. Noetherian rings
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
C. Auslander–Buchsbaum formula
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
D. Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
E. Krull–Gabriel dimension
Krull–Gabriel dimension is a refinement of Krull dimension used in the representation theory of rings and abelian categories to measure the complexity of their subobject lattices and module categories.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Cohen–Macaulay ring
Target entity description: A Cohen–Macaulay ring is a commutative Noetherian ring whose depth equals its Krull dimension, giving it especially well-behaved homological and geometric properties.

A. Castelnuovo–Mumford regularity
Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
B. Noetherian rings
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
C. Auslander–Buchsbaum formula
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
D. Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
E. Krull–Gabriel dimension
Krull–Gabriel dimension is a refinement of Krull dimension used in the representation theory of rings and abelian categories to measure the complexity of their subobject lattices and module categories.
F. None of above. chosen

Statements (51)

Predicate	Object
instanceOf	Noetherian ring ⓘ algebraic structure ⓘ commutative ring ⓘ mathematical object ⓘ
characterizedBy	depth equals dimension at all localizations at prime ideals ⓘ existence of a full-length regular sequence in every maximal ideal ⓘ
definedBy	depth(R) = dim(R) ⓘ
equivalentCondition	depth of localizations at all prime ideals equals their Krull dimension ⓘ every system of parameters is an R-regular sequence ⓘ
fieldOfStudy	algebraic geometry ⓘ commutative algebra ⓘ
generalizes	regular local ring ⓘ
hasCondition	all systems of parameters are regular sequences ⓘ depth of every maximal ideal equals Krull dimension ⓘ
hasExample	complete intersection ring ⓘ coordinate ring of a nonsingular affine variety over a field ⓘ regular local ring ⓘ standard graded k-algebra that is Cohen–Macaulay ⓘ
hasGeometricInterpretation	coordinate ring of a Cohen–Macaulay scheme ⓘ
hasHomologicalProperty	finite projective dimension for canonical modules when they exist ⓘ vanishing of certain local cohomology modules below top dimension ⓘ
hasInvariant	Krull dimension ⓘ depth ⓘ
hasModule	maximal Cohen–Macaulay module ⓘ
hasOriginIn	study of Hilbert functions and syzygies ⓘ
hasProperty	Noetherian ⓘ depth equals Krull dimension ⓘ well-behaved geometric properties ⓘ well-behaved homological properties ⓘ
hasSpecialModule	canonical module ⓘ
implies	equidimensional localizations at maximal ideals ⓘ unmixed ring ⓘ
isLocalVersion	Cohen–Macaulay local ring ⓘ
isStableUnder	completion at an ideal ⓘ finite integral extension under suitable hypotheses ⓘ localization ⓘ polynomial extension over a Cohen–Macaulay base ⓘ
namedAfter	Francis Sowerby Macaulay ⓘ Irvin Cohen ⓘ
relatedConcept	Gorenstein ring ⓘ linked to: Gorenstein rings Krull dimension ⓘ Serre’s conditions (S_n) ⓘ complete intersection ring ⓘ depth of a module ⓘ local cohomology ⓘ regular sequence ⓘ
satisfies	Serre’s condition (S_dim R) ⓘ
usedIn	intersection theory ⓘ local algebra ⓘ singularity theory ⓘ theory of Hilbert functions ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Auslander–Buchsbaum formula → relatedConcept → Cohen–Macaulay ring ⓘ

Introduction to Commutative Algebra → hasSubject → Cohen–Macaulay rings ⓘ

subject linked to: “Introduction to Commutative Algebra” (with Ian G. Macdonald)

linked to: Cohen–Macaulay ring

Eisenbud’s Commutative Algebra → topic → Cohen–Macaulay rings ⓘ