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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6
votes
1answer
207 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 ...
3
votes
0answers
148 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 ...
6
votes
1answer
170 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 ...
4
votes
0answers
106 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{...
12
votes
0answers
387 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 ...
3
votes
1answer
224 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 ...
3
votes
1answer
194 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 ...
5
votes
1answer
134 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 \...
3
votes
1answer
205 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}\...
5
votes
1answer
300 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{...
4
votes
0answers
212 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 ...
3
votes
0answers
91 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 ...
1
vote
0answers
162 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 ...
2
votes
0answers
180 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 $A\otimes_R A^{\mathrm{op}}$-...
2
votes
0answers
219 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 $N:=\ker(\mathrm{Br}(k) \rightarrow \mathrm{Br}(k(X))$. ...
3
votes
0answers
49 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/...
2
votes
0answers
151 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?
9
votes
2answers
460 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)...
4
votes
0answers
121 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 $l$-...
8
votes
1answer
492 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 ...
3
votes
1answer
200 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 +...
12
votes
3answers
633 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 ...
4
votes
0answers
104 views

3-torsion part of Brauer group

I want to solve this problem: If in field $K$ we have sufficient n-th roots of unity then the 3-torsion part of Brauer group is generated by classes of cyclic algebras I know that every element in 3-...
4
votes
1answer
265 views

Elementary proof that a central simple algebra over a field having a maximal subfield is a cyclic algebra

I'm currently reading the book "Central Simple Algebras and Galois Cohomology" written by Philippe Gille and Tamas Szamuely. In the book, I don't understand a computational proof of the theorem that ...
3
votes
1answer
117 views

Brauer group classifying some splitting categories

Notation: $k$ - field. "$k$-category" = $k$-linear abelian category. $Vect_k$ - the $k$-category of $k$-vector spaces. For a field extension $K/k$ and a $k$-category $\mathcal{A}$, denote by $\mathcal{...
6
votes
1answer
233 views

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 ...
11
votes
1answer
525 views

Smooth projective models of Severi-Brauer varieties over a DVR are also Severi-Brauer varieties

Let $R$ be a DVR with uniformizer $\pi$, fraction field $K$ and residue field $k$. Let $X/K$ be a Severi-Brauer variety and $\mathscr X/R$ a smooth, projective model of it. Is it true that $\mathscr ...
2
votes
0answers
62 views

A question about abelian varieties

For an abelian variety $A$ over a global field $K$, $\mathcal{A}$ its Néron model on $C$, either a smooth projective geometrically irreducible curve over a finite field, or the spectrum of the ring of ...
13
votes
1answer
777 views

finiteness of the Brauer group for flat proper schemes over $\mathrm{Spec}\,\mathbf{Z}$

One fundamental conjecture on the Brauer group is that $\mathrm{Br}(X)$ is finite for $X/\mathrm{Spec}\,\mathbf{Z}$ proper. By class field theory (the theorem of Albert-Brauer-Hasse-Noether), this is ...
2
votes
1answer
293 views

Is any element in $H^2_{et}(X,\mathcal{O}_X^*)$ locally trivial in the Zariski topology?

Let $X$ be an algebraic variety over a field $k$ and we consider the cohomological Brauer group $H^2_{et}(X,\mathcal{O}_X^*)$. For any element $\alpha \in H^2_{et}(X,\mathcal{O}_X^*)$ and any closed ...
3
votes
0answers
182 views

Examples of varieties with every stable sheaf simple

Are there examples of projective varieties over a non-algebraically closed field such that every geometrically stable sheaf on the variety is simple? I see, for example in Huybrechts-Lehn and in some ...
4
votes
0answers
108 views

On the quadratic equivalence of fields

I have spent the past two years studying abstract Witt rings. These objects are a generalization of "The Witt ring of a field," an algebraic invariant of fields of characteristic not equal to 2. ...
3
votes
1answer
67 views

Splitting variety for bicyclic algebras

Let $F$ be a field contains a primitive root of unity of order $p$, where $p$ is a prime number. Let $a,b \in F^\times$, then one can look at the cyclic algebra $(a,b)_p \in {_p}Br(F)$ where ${_p}Br(F)...
5
votes
1answer
221 views

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 ...
6
votes
1answer
259 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 $b$...
3
votes
0answers
221 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 ...
6
votes
1answer
283 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. ...
5
votes
0answers
401 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 ...
1
vote
2answers
351 views

Twisted line bundles Brauer class

This question is mainly a reference request about the order of a Brauer class on a smooth projective variety over $\mathbb{C}$. Namely, let $X$ be a smooth complex projective variety and $\alpha$ be a ...
2
votes
1answer
275 views

Do $PGL_n$-torsors induce elements of the Brauer group

Let $K$ be a field and let $n\geq 2$. If $n=2$, then the set of $K$-isomorphism classes of $PGL_n$-torsors is in bijection with the $n$-torsion of the Brauer group of $K$. Is this only for $n=2$? Is ...
4
votes
0answers
215 views

Is a central simple algebra necessarily cyclic if it splits after a cyclic Galois extension?

Let $A$ be a central simple algebra of degree $n$ over $k$, $\dim_kA=n^2$, let $K/k$ be a cyclic galois extension of degree $n$. Suppose $A\times_kK\cong M_n(K)$, does this imply that $A$ is a cyclic ...
5
votes
1answer
467 views

Brauer group of a product of curves

By a famous theorem of Tate, we know that the Tate conjecture holds for a product of curves over a finite field. But this implies that the Brauer group of a product of curves (over finite field) is ...
8
votes
1answer
333 views

Gerbes on the multiplicative group

Let $k$ be an arbitrary field with absolute Galois group $\Gamma$. The group $\text{Hom}(\Gamma,\mathbb{Q}/\mathbb{Z})$ injects into $H^2(\mathbb{A}^1 \setminus \{ 0 \},\mathbb{G}_m)$, as one can see ...
1
vote
2answers
398 views

Central division algebras and splitting fields

Let $K$ be a field and $D$ be a central division algebra over $K$ of degree $n$. Suppose that $L\subset D$ is a maximal subfield, so that $[L:K]=n$. Then we know that $L$ is a splitting field, so ...
5
votes
0answers
186 views

Extension of sheaf of Azumaya algebras and derived equivalence

Suppose there is a smooth variety $X$ and a sheaf of algebra $\mathcal{B}$. Let $Z\subseteq X$ be a closed subvariety, whose codimension is large (say $\geq 2$). If the restriction of $\mathcal{B}$ to ...
2
votes
0answers
192 views

Can one control the ramification of a Brauer class under birational morphisms?

Assume we are given a Brauer class $\xi\in Br(k(\mathbb{P}^n))$ ramified at some divisor $D\subset \mathbb{P}^n$, here $k=\mathbb{C}$. If $f: \mathbb{P}^n\mathrel{-\,}\rightarrow \mathbb{P}^n$ is a ...
12
votes
1answer
755 views

Is the Brauer group functor a Zariski sheaf?

For any scheme $X$, let $\operatorname{Br}X$ denote the (Azumaya) Brauer group of $X$, namely the Morita equivalence classes of Azumaya $\mathcal{O}_{X}$-algebras. Is the functor $$\operatorname{Br}...
6
votes
0answers
515 views

Brauer group of a rational variety

This is a follow-up question to this question. There and here $X$ is a normal projective rational surface over $\mathbb{C}$ with finitely generated divisor class group $\text{Cl}(X)$. My question is: ...
2
votes
0answers
193 views

What is the relation (if any) between Clifford algebras and Azumaya algebras?

Suppose the base field is $\mathbb{C}$ and the Clifford algebra is the classical one (i.e. associated to a quadratic form in $n$ variables). It seems that there are relations between Clifford algebras ...
6
votes
2answers
470 views

Obstruction and rational points on curves

Is etale-Brauer the only obstruction to the existence of rational points on projective plane curves over number fields?