Brauer group
E753153
The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Brauer group canonical | 8 |
| Azumaya algebra | 1 |
| Brauer group of a variety | 1 |
| cohomological Brauer group Br'(X) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8733411 — 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: Brauer group Context triple: [Hasse invariant, relatedTo, Brauer group]
-
A.
Grothendieck group
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.
-
B.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
-
C.
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.
-
D.
Milnor K-theory
Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
-
E.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
- 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: Brauer group Target entity description: The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
-
A.
Grothendieck group
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.
-
B.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
-
C.
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.
-
D.
Milnor K-theory
Milnor K-theory is an algebraic K-theory constructed from fields using tensor powers of their multiplicative groups modulo Steinberg relations, playing a central role in modern algebraic geometry and number theory.
-
E.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
group ⓘ |
| appearsIn |
Grothendieck’s Tôhoku paper on derived functors and cohomology
ⓘ
Grothendieck’s theory of Brauer groups of schemes ⓘ |
| classifies |
Azumaya algebras over a scheme
ⓘ
equivalence classes of central simple algebras over a field ⓘ |
| definedOver |
field
ⓘ
scheme ⓘ |
| dependsOn |
base field
ⓘ
scheme structure ⓘ |
| elementType |
Brauer class
ⓘ
class of a central simple algebra ⓘ |
| equivalenceRelation | similarity of central simple algebras ⓘ |
| fieldOfStudy |
Galois cohomology
ⓘ
algebra ⓘ algebraic geometry ⓘ homological algebra ⓘ number theory ⓘ |
| generalizationOf | Brauer group of a field to Brauer group of a scheme ⓘ |
| hasIsomorphism | H^2(Gal(K^sep/K), (K^sep)^×) for a field K with separable closure K^sep ⓘ |
| hasProperty |
abelian group
ⓘ
functorial in the base field or scheme ⓘ torsion group for fields ⓘ |
| hasVariant |
Azumaya Brauer group Br_Az(X)
ⓘ
cohomological Brauer group Br'(X) ⓘ
linked to:
Brauer group
|
| identityElement | class of the base field as a central simple algebra ⓘ |
| inverseOperation | taking opposite algebra ⓘ |
| namedAfter | Richard Brauer ⓘ |
| notation |
Br(K)
ⓘ
Br(X) ⓘ |
| operation | tensor product of algebras ⓘ |
| relatedConcept |
Azumaya algebra
ⓘ
linked to:
Brauer group
Brauer–Manin obstruction ⓘ Galois cohomology group H^2(Gal(K^sep/K), (K^sep)^×) ⓘ Picard group ⓘ Severi–Brauer variety ⓘ central simple algebra ⓘ class field theory ⓘ
linked to:
global class field theory
cohomological Brauer group ⓘ crossed product algebra ⓘ division algebra ⓘ period-index problem ⓘ étale cohomology ⓘ |
| usedIn |
classification of division algebras over local and global fields
ⓘ
descent theory in algebraic geometry ⓘ moduli problems in algebraic geometry ⓘ obstructions to the Hasse principle ⓘ obstructions to weak approximation ⓘ study of rational points on varieties ⓘ |
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: Brauer group Description of subject: The Brauer group is an algebraic structure that classifies equivalence classes of central simple algebras over a field (or more general schemes), playing a key role in number theory, algebraic geometry, and cohomology.
Referenced by (11)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Brauer group
linked to: Brauer group
linked to: Brauer group