Cohen–Macaulay ring
E790527
A Cohen–Macaulay ring is a commutative Noetherian ring whose depth equals its Krull dimension, giving it especially well-behaved homological and geometric properties.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cohen–Macaulay ring canonical | 1 |
| Cohen–Macaulay rings | 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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Irvin Cohen NERFINISHED ⓘ |
| relatedConcept |
Gorenstein ring
NERFINISHED
ⓘ
Krull dimension NERFINISHED ⓘ 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Cohen–Macaulay ring Description of subject: A Cohen–Macaulay ring is a commutative Noetherian ring whose depth equals its Krull dimension, giving it especially well-behaved homological and geometric properties.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Introduction to Commutative Algebra
this entity surface form:
Cohen–Macaulay rings