The brauer-groups tag has no wiki summary.
8
votes
1answer
169 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
214 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 ...
3
votes
0answers
94 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 ...
27
votes
1answer
805 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 ...
23
votes
1answer
655 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 ...
4
votes
1answer
233 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
185 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 ...
3
votes
1answer
61 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 ...
13
votes
1answer
387 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
212 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 ...
5
votes
1answer
439 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
229 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 ...
2
votes
1answer
217 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
344 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
468 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 ...
5
votes
1answer
290 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 ...
17
votes
1answer
1k 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 ...
9
votes
2answers
659 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
1answer
454 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 ...
