Gabriel–Popescu theorem

E884933

mathematical theorem

The Gabriel–Popescu theorem is a fundamental result in category theory that characterizes Grothendieck categories as exact reflective localizations of module categories.

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All labels observed (1)

Label	Occurrences
Gabriel–Popescu theorem canonical	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ
appliesTo	Grothendieck abelian categories NERFINISHED ⓘ
assumes	Grothendieck category has a generator NERFINISHED ⓘ Grothendieck category satisfies AB5 ⓘ
characterizes	Grothendieck categories NERFINISHED ⓘ
characterizesAs	exact reflective localizations of module categories ⓘ
consequence	Grothendieck categories can be studied via module-theoretic methods ⓘ
domain	abelian categories ⓘ module categories ⓘ
field	category theory ⓘ homological algebra ⓘ
hasGeneralization	results on localization of Grothendieck categories ⓘ
hasImpactOn	modern homological algebra ⓘ theory of abelian categories ⓘ
involvesConcept	AB5 condition ⓘ Grothendieck category NERFINISHED ⓘ Serre subcategory ⓘ adjoint functors ⓘ exact embedding ⓘ exact functor ⓘ filtered colimits ⓘ fully faithful functor ⓘ generator of an abelian category ⓘ left exact functor ⓘ localization of categories ⓘ localizing subcategory ⓘ quotient of an abelian category ⓘ reflective subcategory ⓘ
languageOfFormulation	category-theoretic language ⓘ
namedAfter	Nicolae Popescu NERFINISHED ⓘ Pierre Gabriel NERFINISHED ⓘ
provedBy	Nicolae Popescu NERFINISHED ⓘ Pierre Gabriel NERFINISHED ⓘ
provides	representation of Grothendieck categories as localizations of module categories ⓘ
publishedIn	Publications Mathématiques de l’IHÉS NERFINISHED ⓘ
relatedTo	Freyd–Mitchell embedding theorem NERFINISHED ⓘ Gabriel’s theorem on abelian categories NERFINISHED ⓘ localization of abelian categories ⓘ
statesThat	a Grothendieck category is equivalent to the category of modules over a ring localized at a localizing subcategory ⓘ every Grothendieck abelian category is a localization of a module category ⓘ every Grothendieck category is equivalent to a reflective localization of a module category ⓘ there exists a fully faithful exact functor from a Grothendieck category into a module category ⓘ
usedFor	reducing problems in Grothendieck categories to problems in module categories ⓘ
usedIn	algebraic geometry ⓘ homological algebra ⓘ representation theory ⓘ
yearProved	1964 ⓘ

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Input

Subject: Gabriel–Popescu theorem
Description of subject: The Gabriel–Popescu theorem is a fundamental result in category theory that characterizes Grothendieck categories as exact reflective localizations of module categories.

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Grothendieck category → isCharacterizedBy → Gabriel–Popescu theorem ⓘ