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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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Unitary involutions on a simple central algebra after a scalar extension

$\DeclareMathOperator{id}{id}$ Let $L/K$ be a quadratic separable extension of fields. Let $A$ be a central simple algebra over $L$ such that its norm $N_{L/K}(A)$ splits. Then we know that there ...
Haowen Zhang's user avatar
1 vote
0 answers
114 views

The Brauer group of the function field of a proper curve

Let $X$ be a smooth proper geometrically connected curve over a number field $k$, and let $k(X)$ denote its field of rational functions, i.e., its function field. Then the (cohomological) Brauer group ...
oleout's user avatar
  • 835
2 votes
0 answers
103 views

Results concerning surjectivity of Brauer groups

Are there known cases of a morphism of smooth geometrically connected curves $f: X \rightarrow Y$ over a number field $k$ (to be specific) that would give rise to a surjective restriction map $f^*:\...
oleout's user avatar
  • 835
5 votes
1 answer
184 views

Torus gerbes over curves

Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected curve over $k$. Let $K(C)$ be the function field of $C$. Tsen's Theorem implies that every $\mathbb{G}_m$-gerbe over $K(C)...
lzww's user avatar
  • 51
1 vote
0 answers
162 views

Vanishing of the local étale cohomology sheaf (?)

Let $X$ be a locally noetherian regular scheme, and let $Z$ be a closed subscheme of $X$ whose codimension $d > 0$ at every point. Let $U$ be the complement of $Z$ in $X$. For a sheaf $\mathscr{F}$ ...
zom's user avatar
  • 185
1 vote
1 answer
221 views

Counterexample to purity of Brauer group for curves

The purity of Brauer group states that for a smooth (quasi-)projective variety $X$ over a field $k$, removing a closed subscheme $Z \subset X$ of codimension at least $2$ ensures that the restriction ...
oleout's user avatar
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4 votes
0 answers
138 views

A map between Brauer groups

Let $R$ be a henselian dvr over $\mathbb{C}$ and $A$ be a flat $R$-algebra of finite type. Suppose $\hat{R}$ is the completion of $R$ and $\hat{A}:=A\otimes_R \hat{R}$. For an ideal $I\subseteq A$, ...
nariri's user avatar
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2 votes
0 answers
111 views

Why is Maycock's Brauer group of graded C*-algebras connected while Moutuou's is not?

In her thesis The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras (cf. Proposition 3.4), Ellen Maycock described the Brauer group of graded continuous trace $C^\ast$-algebras with spectrum a ...
Jonathan Beardsley's user avatar
5 votes
1 answer
655 views

Brauer group of rational numbers

Reading about the calculation of the Brauer group of rational numbers, the calculations of the group are extremely lengthy and technical. First of all, it will be very helpful to me if someone can ...
Siavosh Ossareh's user avatar
3 votes
0 answers
241 views

Can moonshine be explained by the $q$-expansion map $\mathrm{tmf} \to KU[[q]]$?

If we form an equivariant version of both sides of the $q$-expansion map $\mathrm{tmf} \to KU[[q]]$ (with everything being $p$-completed), should the equivariant $q$-expansion map should have ...
KingVon's user avatar
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2 votes
0 answers
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Field of definition for a representation and Brauer group

I wonder if the following logic is true. And if it is true, how to prove it? Let $G$ be a profinite group, $\rho:G\to GL_n(k)$ a continuous representation, where $k$ is an algebraically closed field ...
user837898's user avatar
1 vote
0 answers
101 views

Galois extension of $ C_{k} $ field and irreducible polynomial over $ C_{k} $ field

A field is called a $ C_{k}$- field if every form i.e. every homogeneous polynomial of degree $ d $ in $ n > d^{k} $ indeterminates has a nontrivial zero. We know that any finite field is a $ C_{...
Sky's user avatar
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5 votes
1 answer
220 views

Countably many isomorphism classes of reductive groups over a field with countable Brauer and Witt groups

Assume a field has a countable Brauer group and a countable Witt group. Are there countably many isomorphism classes of reductive groups over it?
guriz's user avatar
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9 votes
1 answer
659 views

Software for detecting Brauer-Manin obstructions?

In the context of another MO question, the following question arose: Does there exist any software for detecting Brauer–Manin obstructions to the existence of integer solutions to a single polynomial ...
Timothy Chow's user avatar
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3 votes
0 answers
150 views

The Brauer group and the second Galois cohomology group

I know that for any finite Galois extension K of $\mathbb{Q}$ the group $H^2(Gal(K/\mathbb{Q}),K^*)$ is isomorphic to the Brauer group $Br(K/\mathbb{Q})$. The isomorphism goes as follows: to a 2-...
Jacques's user avatar
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1 vote
0 answers
66 views

Would the iterated finite abelian descent obstruction equality hold for curves?

Let $X$ be a smooth projective geometrically integral variety over a number field $k$. We begin with some established notions in the theory of descent obstruction to the local-global principle, ...
oleout's user avatar
  • 835
8 votes
1 answer
589 views

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 ...
Sky's user avatar
  • 693
3 votes
0 answers
177 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 ...
Sky's user avatar
  • 693
2 votes
0 answers
141 views

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 ...
Marty's user avatar
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3 votes
1 answer
222 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 ...
Weiwei Z.'s user avatar
3 votes
0 answers
91 views

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 ...
Mikhail Borovoi's user avatar
3 votes
0 answers
89 views

Multiplication law in a division algebra of dimension 9 over a non-archimedean local field

Let $k$ be a non-archimedean local field, for example, a $p$-adic field (a finite extension of the filed ${\Bbb Q}_p$ of $p$-adic numbers). It is well known that there is a canonical isomorphism $${\...
Mikhail Borovoi's user avatar
2 votes
0 answers
147 views

Interpretation of some maps involving cohomology groups

I've asked some questions on Math Stackexchange regarding areas around my research but I received very little success with responses, so I thought I might try to share some of my other problems here ...
oleout's user avatar
  • 835
8 votes
0 answers
625 views

An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich

Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$. In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map $$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
Joshua Mundinger's user avatar
4 votes
2 answers
458 views

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 $p$)?
user123's user avatar
  • 81
7 votes
0 answers
213 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 ...
user123's user avatar
  • 81
1 vote
1 answer
118 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 ...
M masa's user avatar
  • 479
3 votes
1 answer
632 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 $U \subset X$ ...
user267839's user avatar
  • 4,966
7 votes
1 answer
452 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 ...
David Corwin's user avatar
  • 14.7k
3 votes
0 answers
173 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 ...
Lord Haydari's user avatar
6 votes
1 answer
232 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 ...
user avatar
5 votes
0 answers
411 views

Brauer groups of a local ring and of its residue field

This is a question of DeMeyer (see the last paragraph of [1]): What's an example of a local ring $A$ with residue field $k$ such that the restriction map on Brauer groups $\varphi : \operatorname{...
Minseon Shin's user avatar
  • 1,967
15 votes
1 answer
705 views

Postnikov invariants of the Brauer 3-group

Given a commutative ring $k$ there is a bicategory with algebras over $k$ as objects, bimodules as morphisms, bimodule homomorphisms as 2-morphisms. This is a monoidal bicategory, since we can ...
John Baez's user avatar
  • 20.6k
3 votes
1 answer
276 views

What is known about lower etale cohomology of unirational varieties?

Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
Mikhail Bondarko's user avatar
4 votes
1 answer
269 views

Can base-change be non-surjective on Brauer groups?

Is there a finite-degree separable field extension $\mathbb{K} \subset \mathbb{L}$ such that the induced map on Brauer groups $\operatorname{Br}(\mathbb{K}) \to \operatorname{Br}(\mathbb{L})$ is not a ...
Theo Johnson-Freyd's user avatar
5 votes
1 answer
257 views

Relation in Brauer group coming from trace form

Let $L/K$ be a cubic (or, more generally, odd-order) extension of fields of characteristic $0$. To every element $a \in L^\times$ we can associate the quadratic form \begin{align*} q_a : L &\to K \...
Evan O'Dorney's user avatar
3 votes
1 answer
283 views

Are Azumaya algebras of trivial Brauer class isomorphic to $\mathcal{E}nd(\mathcal{H})$?

Let $X$ be a scheme, let $\mathcal{A}$ be a sheaf of locally free algebras on $X$. We say $\mathcal{A}$ is an azumaya algebra, if the natural map $$\mathcal{A}\otimes_{\mathcal{O}_X}\mathcal{A}^{opp}\...
user avatar
6 votes
1 answer
404 views

Purity of Brauer group for stacks

Let $k$ be a field, let $X$ be a smooth quasi-projective $k$-variety, let $Z\subset X$ be a closed subscheme of codimension at least $2$, it is shown that the restriction map $\mathrm{H}^2(X,\mathbb{...
user avatar
5 votes
1 answer
330 views

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 ...
user avatar
3 votes
0 answers
116 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 ...
Moutand Mohammed's user avatar
1 vote
0 answers
180 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 ...
Typo Detective's user avatar
2 votes
0 answers
276 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 $A\otimes_R A^{\mathrm{op}}$-...
Moutand Mohammed's user avatar
2 votes
0 answers
246 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 $N:=\ker(\mathrm{Br}(k) \rightarrow \mathrm{Br}(k(X))$. ...
nxir's user avatar
  • 1,387
3 votes
0 answers
62 views

Splitting of central simple algebras in the Schur subgroup over residue fields of places

​Recall that a valuation domain of ​a field extension ​$K/k$ is a $k$-subalgebra $V$ of $K$ ​not equal to $K$ ​such that for every $a\in K$ at least one of $a$ and $a^{-1}$ is in $V$. A​ ​place of $K/...
Fabian Meumertzheim's user avatar
2 votes
0 answers
222 views

Brauer Group of a nodal curve

What is known about the Brauer Group of a Nodal curve (complete integral curve) over $k$ with singularity as ordinary double point? Is it trivial if $k$ algebraically closed?
PSUN's user avatar
  • 137
9 votes
2 answers
599 views

On a morphism from the Brauer group to the Picard group

Suppose that $k$ is a commutative ring and that $A$ is an Azumaya $k$-algebra. Then there is a well-known morphism from $Aut_{Alg_k}(A)$, the group of algebra automorphisms, to the Picard group $Pic(k)...
Tyler Lawson's user avatar
  • 49.6k
4 votes
0 answers
207 views

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 $l$-...
Thomas Geisser's user avatar
13 votes
1 answer
1k 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 ...
Pulcinella's user avatar
  • 5,112
3 votes
1 answer
241 views

Are local fields $C_{2}$?

We say that a field $K$ is $C_{m}$ if it satisfies the following property: for every positive integer $n$ and every sequence of positive integers $(d_{1},\dotsc,d_{r})$ satisfying $d_{1}^{m} + \dotsb +...
Minseon Shin's user avatar
  • 1,967
12 votes
3 answers
1k views

Brauer group of a curve over non-algebraically closed field

It is a famous consequence of Tsen's theorem that a smooth curve over an algebraically closed field has trivial Brauer group. But what about curves over non algebraically closed fields? Let us fix a ...
Giulio Bresciani's user avatar