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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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)...
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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 ...
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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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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}/...
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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 ...
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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-...
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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{...
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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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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{...
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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 ...
user705174
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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 ...
user685167
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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 ...
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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 ...
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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 $...
user365205
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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 ...
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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 ...
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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}((...
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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 ...
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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 ...
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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 \...
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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 ...
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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$?
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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 ...
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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 ...
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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-...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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^*...
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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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Version of Wedderburn's theorem on central simple algebras
Suppose that $A$ be a central simple algebra over a field $k$. Then by Wedderburn's theorem $A\cong M_n(D)$ for some division $k$-algbera $D$. But to define the 'Brauer equivalence' I need that $D$ is ...
user276115
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If $L$ splits $D=I\otimes N$ does $L$ split $I$ and $N$?
Let $D$ be a central simple division algebra over the field $F$. Let $D\sim I\otimes N$ where $I$ and $N$ are division algebras in Br$(F)$ and $\sim$ is equivalence in Br$(F)$. I am interested in this ...
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Equivalent definitions of the Brauer group
I am trying to figure out a good way to see the equivalence in definitions of a brauer group of a field. The two are usually offered: either the brauer group has as elements, central simple algebras ...
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