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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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About $(\frac{a,1-a}{k})\cong\mathrm{M}_2(k).$

$k$ is a field. A quaternion algebra over $k$ is a $4$-dimensional $k$-algebra with a basis $1,i,j,ij$ with the following multiplicative relations: $i^2\in k^\times, j\in k^\times, ij=-ji$ and every $...
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Representations with traces in a finite field

Let $F$ be a finite field. Let $\rho:G\to GL_2(\overline F)$ be a continuous representation from a profinite group $G$, where $\overline F$ is an algebraic closure of $F$. Suppose that for all $g\in G$...
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How does Wedderburns's little Theorem imply that central simple Algebras are isomorphic to matrix Algebras over a divison Algebra?

For a thesis I need the statement that every central simple algebra $A$ is isomorphic to $M_n(D)$ for some divison algebra $D$ and integer $n$. In a reference I am reading it states that this is a ...
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Cohomology of local fields in positive characteristic

It is well-known from local class field theory that the Brauer group $\text{Br}(k)$ of a local field $k$ gets killed as you pass to sufficiently large extensions of $k$. In particular, $\text{Br}(L)(p)...
aspear's user avatar
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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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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 ...
Ulysse Keller's user avatar
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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 ...
oleout's user avatar
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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 ...
Adjoint Functor's user avatar
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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: ...
Justin Desrochers's user avatar
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Where can I find a proof of the Hasse norm theorem (in english)?

I am interested in determining the Brauer group of $\mathbb{Q}$. A while ago I started reading about central simple algebras and Brauer groups. Till now I have proved up to the $B(K^{unr})=\mathbb{Z}/...
Suchir Kaustav's user avatar
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When is a two-dimensional representation of a finite group over a finite field defined over a subfield?

I'm reading the proof of the Deligne-Serre theorem attaching Galois representations to newforms of weight one, and there's a representation-theoretic argument that I don't understand at all. The setup ...
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Simple question about the Brauer group of a variety

If $X$ is a variety over a field $k$ and $\overline{k}$ is an algebraic closure of $k$, then there is a homomorphism $\text{Br}(X) \rightarrow \text{Br}(X \times_k \overline{k})$. Since $\text{Br}$ is ...
Mehta's user avatar
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Quaternion algebras over a non-Archimedean local field $K$, up to isomorphism

I want to know the number of non-isomorphic quaternion algebras over a non-Archimedean local field $K$. What is the number of non-isomorphic central simple algebras of dimension $n^2$ over a non-...
Tireless and hardworking's user avatar
2 votes
1 answer
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Relationship between ideals in an algebra to ideals after extending scalars

Fix a field $K$. The following is a basic fact about central simple algebras. Theorem. Let $A$ be a central simple $K$-algebra, and let $B$ be an arbitrary $K$-algebra. Any two-sided ideal $\mathfrak{...
Joshua Ruiter's user avatar
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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 & ...
Lord Haydari's user avatar
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2 answers
312 views

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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Reduction step for studying cyclic algebras over local fields

I am trying to study cyclic algebras $(\chi,a)$ over a local field $K$ (specifically of characteristic 0). Given a cyclic Galois extension $L/K$ with isomorphism $\chi\colon Gal(L/K)\rightarrow\mathbb{...
InvisibleMango's user avatar
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1 answer
198 views

Brauer group of a field and norms

For $a,b \in K^\times$ the symbol $(a,b)$ denotes the element of the Brauer group of $K$ represented by the $2$-cocycle on the absolute Galois group $G_K$ of $K$ sending $(g_1,g_2)$ to $$ \sqrt{a}^{\...
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Brauer group of $k$-rational scheme

let $X$ be a smooth, projective and geometrically integral $k$-scheme. the Brauer group of $X$ is defined by $Br(X)=H^2_{ét}(X, \mathbb{G}_m)$. I'm searching for a proof of this Theorem: assume that ...
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smooth curve of genus $0$ and Brauer group

In this blog of Alex Youcis, I see a sentence in the proof of theorem 4 which says that "since $C$ has genus 0 that it defines a class $[C]\in\mathrm{Br}(\mathbb{Q})$ which is trivial if and only ...
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154 views

Are all central simple algebras unital?

The definition I'm using for a CSA over a field $k$ is the following: A CSA over $k$ is a finite-dimensional associative $k$-algebra which is simple and has center precisely $k$. My question ...
Charalambos Kioulos's user avatar
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213 views

Brauer Group for a Global Field with $l$-roots of unity $l\neq \text{char}(F)$

Let $F$ be global field that contains the $l$-roots of unity with $l$ a prime number different with the characteristic of $F$ and $\text{Br}F$ the Brauer Group of $F$. How can i proof that all ...
Elvis Torres Pérez's user avatar
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165 views

Field with vanishing Brauer group which is not $C_1$

In Serre's Local Fields he gives several examples of fields with trivial Brauer group. However, all of these examples are $C_1$ or conjectured to be $C_1$. Is there an example of a field which is not $...
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The local invariant map is a group homomorphism

Let $ (K, \nu) $ be a nonarchimedian local field. I have read that the Brauer group, $ \text{Br}(K) $ (which for me, is defined by the similarity classes of CSAs with group operation as tensor product)...
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Serre's proof of $ \operatorname{Br}(K/k) = H^2(K/k)$

I am reading Serre's proof on the Galois cohomological interpretation of the Brauer group, i.e. on the isomorphism $ \operatorname{Br}(K/k) \to H^2(K/k)$ (Serre, Local Fields, X, § 5). In Proposition ...
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Direct product of group with itself mod diagonal subgroup

Let $G$ be any abelian group, and let $\triangle_G = \{(g,g)\mid g\in G\}\subset G\times G.$ Is there any significance in studying the quotient group ${(G\times G)}\left/{\triangle_G}\right.?$ If ...
Chickenmancer's user avatar
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1 answer
117 views

Cohomological proof of Wedderburn's theorem on the finite division algebra

Wedderburn proved that any finite division algebra is a field. I saw a beautiful proof which uses class equation and some basic analysis in the book Proofs from the Book. However, I want to know ...
Seewoo Lee's user avatar
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Brauer group of the field of Laurent series with coefficients in a finite field

In a course I attended at university, we calculated the Brauer group of $\mathbb{F}_q((t))$ with $q=p^n$ , $p$ prime number and we proved it was $\dfrac {\mathbb{Q}}{\mathbb{Z}}=Br(\bar{\mathbb{F}_q}((...
Tommaso Scognamiglio's user avatar
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Degree $n$ extension of local field splits degree $n$ division algebra

I am trying to write an article which is pretty self-contained on the number theory side, and would like to use the following result: Let $K$ be a local field, $n > 1$ a natural number, $D$ a ...
Bib-lost's user avatar
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4 votes
2 answers
600 views

Brauer group of global fields

Is the Brauer group $\text{Br}(K)$ of a global field $K$ an $\ell$-divisible group for some prime $\ell$? If so, what $\ell$? Is $\text{Br}(K)[n]$ finite, for $n$ integer? I know from class field ...
Julie's user avatar
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Showing that a Severi-Brauer Variety with a point is trivial

Let $X/k$ be a variety over a field such that $X_{\overline k} \cong \mathbb P^n$ over $\overline k$ for some $n$. Suppose moreover that $X$ has a rational $k$-point $P$. Then, I know that $X \cong \...
Asvin's user avatar
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What is an example of a formal group law that does not come from an abelian variety?

I am curious to find an example of a 1-d formal group law that does not come from a splitting of the formal group law of an abelian variety. I am aware that we can craft logarithms from the formal ...
Catherine Ray's user avatar
3 votes
1 answer
105 views

Realising finite abelian groups as Brauer groups of a field

Which finite abelian groups appear as Brauer groups of a field? Given a finite abelian group $G$, what are the (easiest) examples of fields with Brauer group equal to $G$?
Mare's user avatar
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On group graded algebras and Brauer groups

I was reading the paper "Algebras graded by groups" by Knus. I want to test and further my understanding of the paper by asking several questions. Since the paper is not readily available I ...
Mathematician 42's user avatar
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On Brauer groups

The Brauer group of a braided monoidal category $\mathcal{C}$ is defined in general in this paper. Essentially it's defined as the equivalence classes of Azumaya algebras in $\mathcal{C}$ (see the ...
Mathematician 42's user avatar
1 vote
1 answer
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Maschke's theorem for $G$-graded algebras

I am reading the paper Algebras graded by a group of Knus. Immediately I run into problems, which I will now detail: Let $G$ be a group and $K$ a field. A $G$-graded algebra $A$ is a finite-...
Mathematician 42's user avatar
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Field with trivial Brauer group that is not of dimension $\leq 1$

In Serre's book Galois cohomology he describes an example of a field with trivial Brauer group that is not of dimension $\leq 1$, as follows: Exercise II.3.1.1. Let $k_0$ be a field of ...
Gnuk's user avatar
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1 answer
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Geometric fibers $\mathbb P^n$ + Vanishing of Brauer group implies projective bundle

Let $\pi: X\to Y$ be a projective flat morphism over a Noetherian integral scheme all of whose geometric fibers are isomorphic to $\mathbb P^n$ over the geometric point. If there is a line bundle $\...
Asvin's user avatar
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5 votes
1 answer
382 views

Hasse invariants under extension of scalars

Let $K\subset L$ be finite extensions of $\Bbb{Q}$. Background. Let $D$ be a finite dimensional division algebra with center $K$. Its class in the Brauer group $Br(K)$ then maps injectively into the ...
Jyrki Lahtonen's user avatar
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Proof of $\operatorname{Br}(K) = H^2(G,K^*)$

The proofs I know of the fact $\operatorname{Br}(K) = H^2(G,K^*)$ ($G= \operatorname{Gal}(K^s/K))$ involve non-abelian group cohomology of $H^1(G,PGL_n(K))$. Are there any nice conceptual proofs which ...
grok's user avatar
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2 votes
1 answer
84 views

Cyclic algebras of degree $4$ and period $2$

Recall that if a field $k$ has a primitive $n$-root of unity $\omega$, then the cyclic $k$-alegbras of degree $n$ (ie of dimension $n^2$) have the following familiar presentation : they are generated ...
Captain Lama's user avatar
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4 votes
1 answer
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Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb Q^*/N(L^*...
Rasmus's user avatar
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4 votes
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Crossed products and division algebras

I am currently reading some introductory material on Brauer groups ("Noncommutative Algebra", by Farb and Dennis) and the following two questions came to my mind: 1) Are all crossed products algebras,...
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