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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votes
0answers
134 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
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2answers
361 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
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0answers
96 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
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1answer
340 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
179 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
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3answers
449 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
89 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-...
3
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1answer
224 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
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1answer
108 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
224 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
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1answer
470 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
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0answers
61 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
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1answer
704 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
286 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
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0answers
174 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
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0answers
103 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
65 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
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1answer
197 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
1answer
228 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
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0answers
196 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
278 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
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0answers
390 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
321 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
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1answer
238 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
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0answers
193 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
429 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
305 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
367 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
184 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
182 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 ...
11
votes
1answer
694 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
471 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
171 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
460 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?
12
votes
1answer
335 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 ...
8
votes
1answer
361 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 $...
4
votes
0answers
138 views

Zeta functions with Brauer class

In algebraic geometry there are examples when a variety $X$ is somehow related (I call it double-mirror) to another variety $Y$, together with a 2-torsion Brauer class. By "related" I mean statements ...
34
votes
1answer
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 ...
28
votes
2answers
1k 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 ...
5
votes
1answer
411 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
2answers
240 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 ...
4
votes
1answer
117 views

Does every equivalence class in a Brauer-Wall group have a (graded) division algebra?

It is known that each equivalence class in a Brauer group has a division algebra (or, in other words, every central simple algebra is isomorphic to $\mathrm{Mat}(D)$ for some division algebra $D$). Is ...
14
votes
1answer
666 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 ...
0
votes
1answer
725 views

Explicit bijection between Azumaya algebras and Brauer-Severi schemes

This is kind of the relative version of this question. Even though I made extensive enquiries, I couldn't find good references for this and it seems to me that these questions are pretty well ...
6
votes
1answer
708 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 ...
1
vote
2answers
331 views

a question on Brauer groups and trivialised Azumaya algebras

If an Azumaya algebra $A$ on a scheme $X$ is trivialised by $A = \mathcal{End}(\mathcal{O})$, $\mathcal{O}$ locally free on $X$, why is $\mathcal{O}$ uniquely determined up to tensoring with a line ...
7
votes
0answers
540 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 ...
2
votes
1answer
270 views

when is a map of analytic Brauer groups induced by inclusion injective?

A theorem of Auslander and Goldman states that for a regular integral scheme $X$ the inclusion of the generic point $\mathrm{Spec}\ K \to X$ induces an injective map $Br(X) \to Br(\mathrm{Spec}\ K)$. ...
4
votes
1answer
562 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) ...
4
votes
2answers
548 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 ...