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.
102
questions
38
votes
1
answer
2k
views
Why is there no Brauer scheme?
Let $X$ be a proper scheme over a base field $k$ (one could consider more general settings, but I am primarly interested in a "geometric" situation with $k$ being algebraically closed).
Then the ...
29
votes
2
answers
2k
views
Making $\mathbb{Q}$-cohomology integral
Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$
with $i>0$.
Does there always exist a variety $Y$ and a ...
18
votes
1
answer
2k
views
Grothendieck's question on the Brauer group for groups
Let $G$ be a group, and let $M(G)=H^2(G,\mathbb{C}^*)$ be the Schur multiplier of $G$. There is a group $Br(G)$ of complex projective representations of $G$ modulo those that can be lifted to linear ...
15
votes
1
answer
734
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 ...
14
votes
1
answer
975
views
Gabber's proof of Br' = Br for quasiprojective schemes
In a note by deJong showing the cohomological and ordinary Brauer groups coincide for separated quasicompact schemes with ample line bundle, it is mentioned that Gabber had an unpublished proof of the ...
13
votes
1
answer
1k
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}...
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 ...
13
votes
1
answer
1k
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 ...
12
votes
3
answers
2k
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 ...
12
votes
1
answer
717
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 ...
12
votes
1
answer
400
views
Can Enriques Surfaces have non-trivial TWISTED Fourier-Mukai partners?
It is a well-known fact that for an Enriques surface $Y$, if $D^b(Y)\cong D^b(X)$ for some smooth projective variety $X$, then $X\cong Y$. In other words, Enriques surfaces have no non-trivial ...
10
votes
1
answer
772
views
How to find examples of non-trival kernel of maps between Brauer groups Br(R) -> Br(K)
Background/Motivation: The facts about the Brauer groups I will be using are mainly in Chapter IV of Milne's book on Etale cohomology (unfortunately it was not in his online note).
Let $R$ be a ...
9
votes
2
answers
838
views
When is Br(X) = H^2(X,G_m)?
In Milne, Étale cohomology, it is proved that $\mathrm{Br}(X) = H^2(X,\mathbf{G}_m)$ for $X$ regular of dimension $\leq 2$. Are there in the meantime further results for $X$ regular?
9
votes
2
answers
651
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)...
9
votes
1
answer
672
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 ...
9
votes
0
answers
238
views
Grothendieck purity for Brauer groups of stacks
Let $X$ be a smooth variety over a field $k$ (for the sake of simplicity of characteristic $0$) and $\operatorname{Br}(X) := H^2_{\text{ét}}(X, \mathbb{G}_m)$ its (cohomological) Brauer group (...
8
votes
1
answer
483
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 ...
8
votes
1
answer
645
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 ...
8
votes
1
answer
418
views
When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?
For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For $...
8
votes
1
answer
381
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 ...
8
votes
0
answers
661
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) ...
7
votes
1
answer
1k
views
Brauer groups of punctured affine lines over a base
Let $R$ be a torsion-free regular noetherian ring. The Brauer group $Br(R)$ of $R$, defined equivalently (by a theorem of Gabber) as the group of Morita equivalence classes of Azumaya $R$-algebras or ...
7
votes
0
answers
220
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 ...
7
votes
0
answers
719
views
Brauer group elements associated to conic bundles
Let $X$ be a non-singular projective variety over a field $k$ (perhaps not of characteristic $2$), and let $\pi:Y\to X$ be a conic bundle over $X$ i.e. a proper morphism all of whose fibres are ...
6
votes
4
answers
3k
views
Brauer group of projective space
I've read that $\text{Br} \mathbb{P}^n_k$ (here $\text{Br}$ is the cohomological Brauer group, i.e. $H^2_{ét}(-,\mathbb{G}_m)$) is just isomorphic to $\text{Br} k$. As proof of this fact seems to be ...
6
votes
2
answers
503
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?
6
votes
1
answer
342
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 $b$...
6
votes
1
answer
276
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 ...
6
votes
1
answer
466
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{...
6
votes
1
answer
380
views
Brauer group of a field of power series in two variables.
Let $k$ be the field $F_2((X,Y))$, where $F_2$ is the field with two elements and
$X$ and $Y$ are two indeterminates. Can we describe the Brauer group of $k$, or at least its $2$-torsion?
(My ...
6
votes
1
answer
334
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.
...
6
votes
1
answer
249
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 ...
6
votes
1
answer
276
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-...
6
votes
0
answers
669
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:
...
5
votes
1
answer
779
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 ...
5
votes
1
answer
256
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 ...
5
votes
1
answer
551
views
What is known about the Brauer group of an arithmetic surface?
Let $X$ be an arithmetic surface over $\mathbb{Z}$, that is we have $\pi: X\rightarrow Spec(\mathbb{Z})$, $X$ is integral, two-dimensional and regular and $\pi$ is projective and flat.
What is known ...
5
votes
2
answers
291
views
smooth affine surfaces over algebraically closed fields with trivial l-torsion of the Brauer group
I am looking for examples of smooth affine surfaces over algebraically closed fields with trivial $\ell$-torsion of the Brauer group.
Related questions: Schemes with trivial brauer group and Brauer ...
5
votes
1
answer
229
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?
5
votes
1
answer
241
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)...
5
votes
1
answer
615
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 ...
5
votes
1
answer
271
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 \...
5
votes
1
answer
353
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 ...
5
votes
0
answers
484
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{...
5
votes
0
answers
437
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 ...
5
votes
0
answers
202
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 ...
4
votes
2
answers
521
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 $p$)?
4
votes
2
answers
592
views
How do Brauer groups relate to zeta functions?
There are two approaches to class field theory that I was taught. The first, is the theory of $L$-functions, Dirichlet characters and so forth (which I described succintly in the question What are the ...
4
votes
1
answer
293
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 ...
4
votes
1
answer
696
views
is generically split Azumaya algebra locally split?
Let $A$ be an Azumaya algebra over a scheme $X$ (or maybe more specifically a scheme of finite type over a field). Suppose that the restriction of $A$ to $U=X\setminus Z$ (where $Z$ is a closed set) ...