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Brauer group: Products, Construction & The Brauer group of a scheme

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.

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Brauer group topic overview

The analysis highlights Products, Construction and The Brauer group of a scheme as prominent areas in the source structure around Brauer group.

Related topics
132
Source areas
11
Connected nodes
143
Extracted relationships
70
Concept neighborhoods
74
Bridge connections
143

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Construction · 23 topics
The Brauer group of a scheme · 20 topics
Examples · 18 topics
Overview · 13 topics
Class field theory · 12 topics
Cyclic algebras · 12 topics
Relation to the Tate conjecture · 12 topics
Severi–Brauer varieties · 9 topics
Galois cohomology · 7 topics
The Brauer–Manin obstruction · 4 topics
The period-index problem · 2 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Construction

Examples

Severi–Brauer varieties

Cyclic algebras

The period-index problem

Class field theory

Galois cohomology

The Brauer group of a scheme

Relation to the Tate conjecture

The Brauer–Manin obstruction

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Brauer group connects Entity context

The extracted context around Brauer group shows recurring relationship patterns in the source. For example, Brauer group → Archimedean, Br, Brauer, BrR, C1, Equivalently, In, Let, More, Q/Z, Since, The Brauer, Then Br, There, Tsen's, Wedderburn's Another extracted example is Brauer group → Archimedean, Br, Br Kv, Brauer, Hasse, If, If Kv, Kv, Q/Z, The, The Brauer, This. Use these groups to spot repeated connection types before inspecting the individual relationships.

Brauer group

Top relations

related to Examples · 16
Brauer group → Archimedean, Br, Brauer, BrR, C1, Equivalently, In, Let, More, Q/Z, Since, The Brauer, Then Br, There, Tsen's, Wedderburn's
related to Class field theory · 12
Brauer group → Archimedean, Br, Br Kv, Brauer, Hasse, If, If Kv, Kv, Q/Z, The, The Brauer, This
related to The Brauer group of a scheme · 10
Brauer group → Auslander, Azumaya, Brauer, Goldman, Grothendieck, In, The, The Brauer, There, Zariski
related to Galois cohomology · 9
Brauer group → Brauer, For, Gal, Galois, Gm, H2, Ks, Ks/K, More
related to Relation to the Tate conjecture · 9
Brauer group → Artin, Brauer, For, Indeed, Jacobian, Shafarevich, Tate, The, This
related to The Brauer–Manin obstruction · 7
Brauer group → Brauer, K-points, K-rational, Kv, Let, Manin, The Hasse
related to Severi–Brauer varieties · 6
Brauer group → Another, Brauer, For, Severi, Such, The
is a · 1
Brauer group → important birational invariant

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

brauer group algebra field central simple algebras division degree projective scheme cyclic defined cohomology isomorphic hasse finite algebraic classes ring

Brauer group relationships Subject–Predicate–Object triples

TTTA extracted 70 structured relationships around Brauer group. Examples in this analysis include Brauer group → is a → important birational invariant and Brauer group → related to Class field theory → The Brauer. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Brauer groupis aimportant birational invariant0.90text
Brauer grouprelated to Class field theoryThe Brauer0.60section
Brauer grouprelated to Class field theoryIf Kv0.60section
Brauer grouprelated to Class field theoryArchimedean0.60section
Brauer grouprelated to Class field theoryBr Kv0.60section
Brauer grouprelated to Class field theoryQ/Z0.60section
Brauer grouprelated to Class field theoryHasse0.60section
Brauer grouprelated to Class field theoryThe0.60section
Brauer grouprelated to Class field theoryIf0.60section
Brauer grouprelated to Class field theoryKv0.60section
Brauer grouprelated to Class field theoryThis0.60section
Brauer grouprelated to Class field theoryBrauer0.60section

Related concept clusters Concept neighborhoods

The concept neighborhoods around Brauer group bring nearby vocabulary together. In this analysis, examples include Group, Field and Scheme. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Brauer group
    • Group
    • Field
    • Scheme
    • Algebras
    • Algebra
    • Projective
    • Example
    • Cyclic
    • Central
    • Degree
    • Defined
    • Varieties
  • brauer group
    • Group
    • Field
    • Scheme
    • Algebras
    • Projective
    • Cohomology
    • Algebra
    • Example
    • Cyclic
    • Algebraic
    • Central
    • Degree
  • field
    • Group
    • Algebra
    • Algebraic
    • Central
    • Division
    • Every
    • Local
    • Finite
    • Simple
    • Theory
    • Class
    • Ring
  • abelian group
    • Scheme
    • Projective
    • Cohomology
    • Algebraic
    • Defined
    • Cyclic
    • Algebra
    • Simple
    • Galois
    • Theory
    • Class
    • Numbers
  • central simple algebras
    • Simple
    • Algebra
    • Cyclic
    • Degree
    • Division
    • Tensor
    • Product
    • Central
    • Brauer
    • Center
    • Field
    • Group
  • tensor product of algebras
    • Product
    • Tensor
    • Cyclic
    • Degree
    • Division
    • Algebras
    • Equivalence
    • Simple
    • Central
    • Algebra
    • Classes
    • Period
  • richard brauer
    • Group
    • Field
    • Scheme
    • Algebras
    • Algebra
    • Projective
    • Example
    • Cyclic
    • Central
    • Degree
    • Defined
    • Varieties
  • division algebras
    • Algebra
    • Every
    • Cyclic
    • Degree
    • Division
    • Tensor
    • Product
    • Simple
    • Theorem
    • Finite
    • Central
    • Brauer

Connections between topic areas Semantic bridges

For Brauer group, one of the stronger structural bridges in this analysis connects Brauer group with Construction. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Brauer group — Construction · splits 120 ⟂ 24
Brauer group — The Brauer group of a scheme · splits 123 ⟂ 21
Brauer group — Examples · splits 125 ⟂ 19
Brauer group — Overview · splits 130 ⟂ 14
Brauer group — Cyclic algebras · splits 131 ⟂ 13
Brauer group — Class field theory · splits 131 ⟂ 13
Brauer group — Relation to the Tate conjecture · splits 131 ⟂ 13
Brauer group — Severi–Brauer varieties · splits 134 ⟂ 10
Brauer group — Galois cohomology · splits 136 ⟂ 8
Brauer group — The Brauer–Manin obstruction · splits 139 ⟂ 5
Brauer group — The period-index problem · splits 141 ⟂ 3

Map overview Semantic statistics

Brauer group

Nodes144
Edges143
Triples70
Avg. degree1.99
Density0.013889
Components1

Source & methodology

TTTA analyzes the structure around Brauer group to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Products, Construction & The Brauer group of a scheme, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Brauer group · EN edition · Analysis: TopicsToTalkAbout

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