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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Observed surface forms (1)

Surface form	Occurrences
Cohen–Macaulay rings	1

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 ⓘ

Referenced by (2)

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” (with Ian G. Macdonald) → hasSubject → Cohen–Macaulay ring ⓘ

subject surface form: Introduction to Commutative Algebra

this entity surface form: Cohen–Macaulay rings