# Questions tagged [brauer-groups]

Questions concerning Brauer groups of fields, rings, varieties, schemes or more general ringed spaces, invariants associated to Brauer classes such as index and period.

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1answer
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### Subspaces of $A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible

Let $A_{n}(\mathbb{Q})$ denote the $n$ times $n$ skew symmetric matrices over the rational number field. Let $N$ be a subspace of $A_{n}(\mathbb{Q})$. If all the non-zero matrices in $N$ are ...
0answers
124 views

### Central division algebras over $\mathbb{Q}$

Quaternions over $\mathbb{Q}$ are an example of a Central Division algebra over $\mathbb{Q}$ for which the basis elements $\{ i,j,ij \}$ other than $1$ are represented by skew-symmetric matrices ...
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### Understanding Sha through $K_2$

Consider the following setup, to keep things easy: let $F$ be a number field with ring of integers $A$. Let $E$ be an elliptic curve over $F$ with Neron model $N$ over $A$. Let $Sha(E)$ be the ...
1answer
183 views

### A local-to global principle for splitting of Azumaya algebras

Let $S$ be a finitely generated domain with the field of fractions $F.$ Let X be a smooth, geometrically connected affine variety over $S.$ Let $A$ be an Azumaya algebra over $X.$ Assume that for all ...
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### Multiplication law in a central simple algebra of dimension 9 over a global field

Let $k$ be a global field, for example $k=\Bbb Q$. Let $D$ denote the central simple algebra of dimension 9 over $k$ with given local invariants $i_v$. Here $v$ runs over the set $\Omega_f(k)$ of ...
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### Brauer group of $\mathbb{Z}_{(p)}$

This may be a well known result but I could not find it in the standard references. What is the Brauer group of the local ring $\mathbb{Z}_{(p)}$ (the ring of integers localized at $p$)?
0answers
177 views

### Brauer group of the Henselization

Let $R$ be a Noetherian local ring and let $R^h$ be its Henselization. What can we say about the kernel and range of the map $$\operatorname{Br}(R) \rightarrow \operatorname{Br}(R^h)?$$ Are there ...
1answer
89 views

### $0$-th Galois cohomology with topological Milnor K-groups coefficients

In local class field theory, the reciprocity map is constructed by using the isomorphism ${\rm Br}(F)\simeq \mathbb{Q/Z}$, where $F$ is a local field and ${\rm Br}(F)$ is its Brauer group. The ...
1answer
305 views

### Cohomological Brauer group vs classical

Let $X$ be a smooth scheme over $\mathbb{C}$. A $O_X$-algebra $A$ is called Azumaya algebra on $X$ if locally it's ismorphic to matrix algebra: ie for every $p \in X$ there exist open $U \subset X$ ...
1answer
403 views

### Category of modules over an Azumaya algebra and the Brauer group

Let $k$ be a field, and let $\alpha \in \mathrm{Br}(k)$. Let $A$ be an Azumaya algebra representing $\alpha$. Then the category $A$–$\mathrm{mod}$ depends only on $\alpha$. I would like to know ...
0answers
169 views

### Why doesn't the Manin obstruction work for quadratic forms?

The Manin obstruction explains why the Hasse principle doesn't work for non-quadratic forms. Let's write the notation first; $V(\mathbb{Q})$ is variety for rational numbers. $V(A_\mathbb{Q})$ is ...
1answer
197 views

### Involution action on Brauer group of an abelian variety

Let $k$ be an algebraically closed field of characteristic $p>2$, let $A/k$ be an abelian variety. Let $\iota\colon A\to A, a\mapsto -a$ be the natural involution. Let $x\in\mathrm{Br}(A)[p]$ be a ...
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### Topological Hochschild homology of Azumaya algebra

Let $R$ be a commutative ring, let $A$ be an Azumaya algebra over $R$, does its topological Hochschild homology coincide with that of $R$? For example, let $\mathbb{H}$ be the quaternion algebra over ...
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97 views

### Topological Brauer group and sheaf of $C^{\infty}$-functions

Given a smooth manifold $X$; by definition the topological Brauer group $B(X)$ is the group of classes of Azumaya algebras over the sheaf $\mathcal{O}_X$ of $\mathbb{C}$-valued functions on $X$. If ...
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166 views

### Killing a Brauer class by a flat projective morphism

Let $X$ be a Noetherian scheme and $\beta \in H^2(X, \mathbb{G}_m)$ that is the image of some $\alpha \in H^1(X, \mathrm{PGL}_{n + 1})$ for some $n \ge 0$, so $\beta$ is a class in the Brauer group of ...
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211 views

### Azumaya algebra and central separable algebra over local rings

let be $R$ a commutative ring Auslander and Goldman define a central separabl $R$-algebra $A$ as a faithful $R$-module with $Z(A) = R$ , and such that $A$ is a projective $A\otimes_R A^{\mathrm{op}}$-...
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224 views

### Brauer groups and del Pezzo surfaces

Let $k$ be a field of characteristic $0$ und let $X$ be a del Pezzo surface over $k$. Note that $X$ may not have points. Let us consider $N:=\ker(\mathrm{Br}(k) \rightarrow \mathrm{Br}(k(X))$. ...
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### Brauer groups over local fields

Let X be a smooth projective variety over a local field of characteristic $(0,p)$. The Brauer group of X is a torsion group whose $l$-part is of cofinite type of some corank. Is it know that the $l$-...
1answer
647 views

### Explicit examples of Azumaya algebras

I'm trying to understand the Brauer group of a scheme better. I know how to compute $\text{Br}(X)$ as an abstract group in some cases, but don't have a good idea of what the individual Azumaya ...
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### Calculating topological index

Consider the space $X=BSL(8,\mathbb{C})/(\mathbb{Z}/2)$. The topological Brauer group of $X$ is given by $Br_{top}(X)=Tor(H^{3}(X;\mathbb{Z}))=\mathbb{Z}/2$. I'm studying concepts of topological ...
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1answer
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### Some questions on division algebras

Given a field $K$, is there a finite dimensional quiver algebra, such that any finite dimensional division algebra is isomorphic to End(M)/rad(End(M)) for some indecomposable finite dimensional module ...
1answer
274 views

### Brauer groups and field extensions

Let $k$ be a field and $\mathrm{Br}(k)$ the Brauer group of $k$. Let $k \subset L$ be a field extension. Let $b \in \mathrm{Br}(k)$ and denote by $b \otimes L \in \mathrm{Br}(L)$ the base-change of $b$...
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244 views

### Interaction between the Brauer group and abelian extensions

If $k$ is a field of characteristic zero and $k$ has no (non-trivial) abelian extensions (e.g. the composite of all solvable extensions of $\mathbb{Q}$), then $\text{Br(k)} = 0$ by the norm residue ...
1answer
296 views

### What is Mumford's example of a normal complex algebraic surface $X$ with non-torsion elements in $H^2_{et}(X,\mathbb{G}_m)$?

I have heard that Mumford has constructed an example of a normal complex algebraic surface $X$ such that $H^2_{et}(X,\mathbb{G}_m)$ contains a non-torsion element. But I cannot find the reference. ...
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415 views

### Algebraization of Brauer classes in a paper of Lieblich

I am reading a paper of Lieblich on the unirationality of K3 surfaces and am trying to understand the result of Proposition 4.1: Proposition 4.1: Let $k$ be an algebraically closed field of ...