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.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
dimension theory concept
ⓘ
mathematical invariant ⓘ |
| appliesTo |
Grothendieck categories
NERFINISHED
ⓘ
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 NERFINISHED ⓘ |
| invariantOf |
abelian categories up to equivalence
ⓘ
module categories of rings ⓘ |
| measures |
complexity of module categories
ⓘ
complexity of subobject lattices ⓘ |
| namedAfter |
Pierre Gabriel
NERFINISHED
ⓘ
Wolfgang Krull NERFINISHED ⓘ |
| property | finite for many representation-finite algebras ⓘ |
| refines | Krull dimension NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.