Grothendieck group
E254131
The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic geometry.
All labels observed (8)
How this entity was disambiguated
This entity first appeared as the object of triple T2290651 — 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: Grothendieck group Context triple: [Alexander Grothendieck, notableConcept, Grothendieck group]
-
A.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
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B.
Abelian groups
Abelian groups are algebraic structures in which the group operation is commutative, meaning the order of combining elements does not affect the result.
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C.
Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
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D.
Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
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E.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
- 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: Grothendieck group Target entity description: The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic geometry.
-
A.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
-
B.
Abelian groups
Abelian groups are algebraic structures in which the group operation is commutative, meaning the order of combining elements does not affect the result.
-
C.
Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
-
D.
Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
-
E.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
- F. None of above. chosen
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
abelian group
ⓘ
algebraic construction ⓘ group completion ⓘ |
| alsoKnownAs | group completion of a commutative monoid ⓘ |
| appliedTo |
isomorphism classes of coherent sheaves
ⓘ
isomorphism classes of modules ⓘ isomorphism classes of objects in an abelian category ⓘ isomorphism classes of objects in an exact category ⓘ isomorphism classes of representations ⓘ isomorphism classes of vector bundles ⓘ |
| constructionMethod |
formal differences of monoid elements
ⓘ
quotient of free abelian group on the monoid ⓘ |
| definesElement | formal difference [a] - [b] of monoid elements ⓘ |
| generalizes |
construction of K0 in K-theory
ⓘ
construction of Z from N ⓘ difference of integers from natural numbers ⓘ |
| hasCanonicalMapFrom | underlying commutative monoid ⓘ |
| hasCanonicalMapType | monoid homomorphism ⓘ |
| hasDomain |
K-theory
ⓘ
algebra ⓘ algebraic geometry ⓘ category theory ⓘ |
| hasHistoricalContext | introduced in the development of Grothendieck’s K-theory ⓘ |
| hasKeyOperation | addition induced by monoid operation ⓘ |
| hasKeyRelation | [a] + [b] = [a + b] in the group ⓘ |
| hasProperty | universal for monoid homomorphisms into abelian groups ⓘ |
| inputStructure | commutative monoid ⓘ |
| mathematicsSubjectClassification |
14-XX algebraic geometry
ⓘ
18-XX category theory ⓘ 19-XX K-theory ⓘ |
| namedAfter | Alexander Grothendieck ⓘ |
| outputStructure | abelian group ⓘ |
| relatedConcept |
Grothendieck K0-group
ⓘ
linked to:
Grothendieck group
Grothendieck group of a category ⓘ Grothendieck group of a scheme ⓘ Grothendieck group of an abelian category ⓘ Grothendieck group of an exact category ⓘ Grothendieck group of coherent sheaves ⓘ
linked to:
Grothendieck group
Grothendieck group of vector bundles ⓘ
linked to:
Grothendieck group
Grothendieck ring ⓘ |
| satisfies | universal property of group completion ⓘ |
| universalProperty | every monoid homomorphism to an abelian group factors uniquely through it ⓘ |
| usedIn |
algebraic K-theory
ⓘ
linked to:
K-theory
algebraic geometry ⓘ homological algebra ⓘ number theory ⓘ operator algebras ⓘ representation theory ⓘ topological K-theory ⓘ
linked to:
K-theory
|
| usedToDefine |
Grothendieck group of varieties
ⓘ
linked to:
Grothendieck group
K0 of a ring ⓘ K0 of a scheme ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Grothendieck group Description of subject: The Grothendieck group is an algebraic construction that formally turns a commutative monoid (often arising from isomorphism classes of objects like vector bundles or modules) into an abelian group, playing a central role in K-theory and modern algebraic geometry.
Referenced by (11)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Grothendieck group
linked to: Grothendieck group
linked to: Grothendieck group
linked to: Grothendieck group
linked to: Grothendieck group
linked to: Grothendieck group
linked to: Grothendieck group
linked to: Grothendieck group