Gabriel localization theory
E884934
Gabriel localization theory is a framework in homological algebra and category theory that studies how to construct and analyze localizations of Grothendieck categories via torsion theories and exact functors.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
localization theory
ⓘ
mathematical theory ⓘ theory in category theory ⓘ theory in homological algebra ⓘ |
| aimsTo |
classify localizing subcategories
ⓘ
describe localizations via torsion theories ⓘ |
| appliesTo |
Grothendieck categories
NERFINISHED
ⓘ
categories of quasi-coherent sheaves ⓘ module categories ⓘ sheaf categories ⓘ |
| characterizes |
exact localizations of Grothendieck categories
ⓘ
localizing subcategories of Grothendieck categories ⓘ |
| concerns |
exactness properties of localization functors
ⓘ
structure of abelian categories ⓘ |
| developedBy | Pierre Gabriel NERFINISHED ⓘ |
| field |
category theory
ⓘ
homological algebra ⓘ |
| frameworkFor |
analyzing localization in abelian categories
ⓘ
constructing quotient categories ⓘ |
| generalizes |
localization of abelian groups
ⓘ
localization of module categories ⓘ |
| influenced |
modern theory of abelian categories
ⓘ
noncommutative algebraic geometry ⓘ |
| involves |
exact reflective subcategories
ⓘ
kernels of localization functors ⓘ local objects ⓘ torsion objects ⓘ |
| provides |
classification of localizations of Grothendieck categories
ⓘ
correspondence between localizing subcategories and localization functors ⓘ |
| relatedTo |
Gabriel–Popescu theorem
NERFINISHED
ⓘ
Grothendieck abelian categories ⓘ Serre quotient ⓘ derived functors ⓘ exact sequences ⓘ torsion theory in abelian categories ⓘ |
| studies | localizations of Grothendieck categories ⓘ |
| usedIn |
module theory
ⓘ
representation theory of algebras ⓘ sheaf theory ⓘ |
| usesConcept |
Gabriel topologies
NERFINISHED
ⓘ
Serre subcategories ⓘ adjoint functors ⓘ exact functors ⓘ localization functors ⓘ quotient categories ⓘ torsion pairs ⓘ torsion theories ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.