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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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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 $...
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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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Residue division ring of a division ring over complete discrete valuation field

Let $K$ be a complete discrete valuation field with valuation ring $O_K$ residue field $k$. For a finite dimensional division ring $D$ over $K$ with center $K$, we can extend the valuation of $K$ to $...
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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 will ...
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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 ...
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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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Brauer groups, Milnor k theory and group cohomology

Can anyone suggest some basic material for learning connections between Brauer groups, Milnor k theory and group cohomology. I am an undergraduate. So, I find most of the sources available very hard.
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Question on fields with trival Brauer group

A common proof that an algebraically closed field has trivial Brauer group goes something like this: Take $D$ a finite central division algebra over $K$, where $K$ is an algebraically closed field....
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Construction of the Brauer Group

I've proven that the $K$ tensor product of two central simple $K$ algebras is itself central simple, and I've proven Wedderburn's theorem, but I now need to construct the Brauer group. I've been told ...