# Questions tagged [brauer-group]

For questions about Brauer groups; 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.

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### Wedderburn Artin's theorem for algebras over a field

I learned about the Wedderburn Artin's theorem for simple left artinian ring, says that if $R$ is simple left Artinian ring then $R\cong\mathrm{M}_n(\Delta)$, for some division ring $\Delta$. I want ...
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### On central simple algebras and the Wedderburn Artin's theorem

I learned about the Wedderburn Artin's theorem, says that if $R$ is simple left Artinian ring then $R\cong\mathrm{M}_n(\Delta)$, for some division ring $\Delta$. Now, I was studying the Brauer group. ...
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### What is the map $f^*: Br(Y)\to Br(X)$?

Let $f:X \to Y$ be a morphism of schemes. Let $Br(X)$ be Brauer group of $X$. I heard $Br(-)$ is functrial, that is, there is induced map $f^*: Br(Y)\to Br(X)$. How $f$ induces $f^*$, in other words, ...
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### Why $Br(Spec(K))=Br(K)$ holds?

This question may be very elementary, sorry but I would like to ask this question. Brauer group of local ring $(R,m)$ is defined to be a group of equivalent classes of Azumaya algebra over $R$. Local ...
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1 vote
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### Quaternion algebras Brauer equivalent iff isomorphic

Let $k$ be a field, and for any $a,b\in k^\times$ let $(a,b)$ be the quaternion algebra over $k$ with parameters $a,b$. Is the following correct? I haven't been able to find a place where this is ...
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### Problems with the exact formal definition of the Brauer group of a field [duplicate]

My question is not about the algebra of this thing, but related to formal set theory. I think that usually one will read that the Brauer group of the field K is made of equivalence classes of algebras ...
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### Local invariant map in the case of closed points of a curve

In class field theory, we have the well-known local invariant map $\mathrm{inv}_v: \mathrm{Br}(k_v) \rightarrow \mathbb{Q}/\mathbb{Z}$, where $k$ is a field and $v$ is a place of $k$. Similarly, we ...
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### Brauer group of a subfield

Let $A$ be a central simple algebra with a finite dimension over the field $F$. Let $A \supset K \supset F$ be a subfield. Show that $C_A(K)$ and $A \otimes_F K$ are both central simple algebras over ...
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### Isomorphism of Brauer groups

A recent big result proved by $\mathrm{\check{C}}$esnavi$\mathrm{\check{c}}$ius states that For a regular, integral, noetherian scheme $X$ and an open subset $U \subset X$ whose complement is of ...
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### Is the $F$-rank of the multiplicity group of central simple algebra $M_m(D)$ over $F$ equal to $m$?

Suppose $F$ is a field, and $A=M_m(D)$ is a central simple algebra over $F$, where $m$ is a natural number and $D$ is a central division algebra over $F$ of rank $d$. Let $G=GL(m,D)$ the multiplicity ...
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### Relationship between trivial Brauer group and commutative division algebras

Let $k$ be an algebraically closed field and $F$ a finite field extension of the field of rational functions $k(t)$. I've heard two different statements of Tsen's theorem under these conditions: ...
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### Proving that central simple algebra $B$ over a field $k$ splits in Brauer group by a field $L$ contained in $B$, then $[B:k]=[L:k]^2$

I want to prove the following statement (I read it here - https://www.jmilne.org/math/CourseNotes/CFT.pdf#X.4.3.6) The similarity in the below statement comes from the equivalence relation from the ...
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### What is the central algebra?

I' m studying over Brauer group. But I have a problem in starting point. What is the central algebra? Defination of center of $\mathbb{k}$-algebra S as following ( Noncommutative Algebra, Farb & ...
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### Why are central simple algebras classified by cohomology?

In their article on the Brauer group Wikipedia writes: Since all central simple algebras over a field $K$ become isomorphic to the matrix algebra over a separable closure of $K$, the set of ...
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