1
Central Simple Algebras
▶
1.1
Basic Theory
1.2
Subfields of Central Simple Algebras
2
Morita Equivalence
▶
2.1
Construction of the equivalence
2.2
Stacks 074E
3
Results in Noncommutative Algebra
▶
3.1
A Collection of Useful Lemmas
▶
3.1.1
Tensor Product
3.1.2
Centralizer and Center
3.1.3
Some Isomorphisms
3.2
Wedderburn-Artin Theorem for Simple Rings
▶
3.2.1
Classification of Simple Rings
3.2.2
Uniqueness of the Classification
3.3
Skolem-Noether Theorem
3.4
Double Centralizer Theorem
4
Brauer Group
▶
4.1
Construction of Brauer Group
4.2
Base Change
4.3
Good Representative Lemma
▶
4.3.1
Basic Properties
4.3.2
Conjugation Factors and Conjugation Sequences
4.4
The Second Galois Cohomology
▶
4.4.1
From \(\operatorname{Br}(K/F)\) to \(\operatorname{H}^{2}\left(\operatorname{Gal}(K/F),K^{\star }\right)\)
4.4.2
Cross Product as a Central Simple Algebra
4.4.3
From \(\operatorname{H}^{2}\left(\operatorname{Gal}(K/F), K^{\star }\right)\) to \(\operatorname{Br}(K/F)\)
4.4.4
\(\operatorname{H}^{2} \circ \operatorname{\mathfrak {C}}\) and \(\operatorname{\mathfrak {C}}\circ \operatorname{H}^{2}\)
4.4.5
Group Homomorphism
Dependency graph
Brauer Group and Galois Cohomology
Jujian Zhang Yunzhou Xie
1
Central Simple Algebras
1.1
Basic Theory
1.2
Subfields of Central Simple Algebras
2
Morita Equivalence
2.1
Construction of the equivalence
2.2
Stacks 074E
3
Results in Noncommutative Algebra
3.1
A Collection of Useful Lemmas
3.1.1
Tensor Product
3.1.2
Centralizer and Center
3.1.3
Some Isomorphisms
3.2
Wedderburn-Artin Theorem for Simple Rings
3.2.1
Classification of Simple Rings
3.2.2
Uniqueness of the Classification
3.3
Skolem-Noether Theorem
3.4
Double Centralizer Theorem
4
Brauer Group
4.1
Construction of Brauer Group
4.2
Base Change
4.3
Good Representative Lemma
4.3.1
Basic Properties
4.3.2
Conjugation Factors and Conjugation Sequences
4.4
The Second Galois Cohomology
4.4.1
From \(\operatorname{Br}(K/F)\) to \(\operatorname{H}^{2}\left(\operatorname{Gal}(K/F),K^{\star }\right)\)
4.4.2
Cross Product as a Central Simple Algebra
4.4.3
From \(\operatorname{H}^{2}\left(\operatorname{Gal}(K/F), K^{\star }\right)\) to \(\operatorname{Br}(K/F)\)
4.4.4
\(\operatorname{H}^{2} \circ \operatorname{\mathfrak {C}}\) and \(\operatorname{\mathfrak {C}}\circ \operatorname{H}^{2}\)
4.4.5
Group Homomorphism