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18
votes
1answer
321 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
214 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
180 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
54 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
374 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Brauer groups coincide for separated quasicompact schemes with ample line bundle, it is mentioned that Gabber had an unpublished proof of the ...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
164 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
410 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
215 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
214 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
321 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
458 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
286 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 ...
16
votes
1answer
956 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
649 views

When is Br(X) = H^2(X,G_m)?

In Milne, √Čtale 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
420 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 ...