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 (1)
| Label | Occurrences |
|---|---|
| Brauer group canonical | 2 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure
ⓘ
group ⓘ |
| appearsIn |
Grothendieck’s Tôhoku paper on derived functors and cohomology
NERFINISHED
ⓘ
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) NERFINISHED ⓘ |
| identityElement | class of the base field as a central simple algebra ⓘ |
| inverseOperation | taking opposite algebra ⓘ |
| namedAfter | Richard Brauer NERFINISHED ⓘ |
| notation |
Br(K)
NERFINISHED
ⓘ
Br(X) ⓘ |
| operation | tensor product of algebras ⓘ |
| relatedConcept |
Azumaya algebra
NERFINISHED
ⓘ
Brauer–Manin obstruction NERFINISHED ⓘ Galois cohomology group H^2(Gal(K^sep/K), (K^sep)^×) ⓘ Picard group NERFINISHED ⓘ Severi–Brauer variety NERFINISHED ⓘ central simple algebra ⓘ class field theory NERFINISHED ⓘ 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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.