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13
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1answer
350 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Brauer groups coincide for separated quasicompact schemes with ample line bundle, it is mentioned that Gabber had an unpublished proof of the ...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
130 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
381 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 ...
0
votes
2answers
186 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
207 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 $Br(X) \to Br(\mathrm{Spec}\ K)$.
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4
votes
1answer
302 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
447 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
279 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
909 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
638 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
397 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 ...
