Krull–Gabriel dimension
E621108
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Krull–Gabriel dimension canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6833915 — 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: Krull–Gabriel dimension Context triple: [Krull dimension, hasVariant, Krull–Gabriel dimension]
-
A.
Gelfand–Kirillov dimension
The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
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B.
Krull dimension
Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
-
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.
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E.
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.
- 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: Krull–Gabriel dimension Target entity description: 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.
-
A.
Gelfand–Kirillov dimension
The Gelfand–Kirillov dimension is an invariant in noncommutative algebra that measures the growth rate of algebras and modules, serving as an analogue of Krull dimension for noncommutative settings.
-
B.
Krull dimension
Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
-
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.
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.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
dimension theory concept
ⓘ
mathematical invariant ⓘ |
| appliesTo |
Grothendieck categories
ⓘ
linked to:
Grothendieck category
abelian categories ⓘ categories of modules over a ring ⓘ module categories ⓘ |
| associatedWith |
hierarchies of localizing subcategories
ⓘ
structure theory of noetherian abelian categories ⓘ |
| comparedWith |
global dimension
ⓘ
representation dimension ⓘ |
| context |
lattice of Serre subcategories of an abelian category
ⓘ
lattice of subobjects of an object in an abelian category ⓘ |
| definedFor | objects of an abelian category via Serre subcategories ⓘ |
| definedUsing |
filtrations by Serre subcategories
ⓘ
localization of abelian categories ⓘ |
| field |
abelian category theory
ⓘ
category theory ⓘ module theory ⓘ representation theory of algebras ⓘ representation theory of rings ⓘ |
| generalizes | Gabriel dimension for module categories ⓘ |
| introducedBy | Pierre Gabriel ⓘ |
| invariantOf |
abelian categories up to equivalence
ⓘ
module categories of rings ⓘ |
| measures |
complexity of module categories
ⓘ
complexity of subobject lattices ⓘ |
| namedAfter |
Pierre Gabriel
ⓘ
Wolfgang Krull ⓘ |
| property | finite for many representation-finite algebras ⓘ |
| refines | Krull dimension ⓘ |
| relatedTo |
Gabriel dimension
ⓘ
Krull dimension of lattices ⓘ |
| studiedIn |
representation theory of Artin algebras
ⓘ
representation theory of finite-dimensional algebras ⓘ |
| takesValuesIn | extended natural numbers ⓘ |
| toolFor |
analyzing composition series of objects in abelian categories
ⓘ
stratifying module categories by complexity ⓘ |
| usedIn |
classification of Grothendieck categories by length conditions
ⓘ
representation type classification ⓘ study of length categories ⓘ study of locally finite abelian categories ⓘ |
| usedToDistinguish | tame and wild representation types in some contexts ⓘ |
| value |
0 for artinian module categories
ⓘ
0 for length categories ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Krull–Gabriel dimension Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.