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# 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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### Unitary involutions on a simple central algebra after a scalar extension

$\DeclareMathOperator{id}{id}$ Let $L/K$ be a quadratic separable extension of fields. Let $A$ be a central simple algebra over $L$ such that its norm $N_{L/K}(A)$ splits. Then we know that there ...
1 vote
114 views

### The Brauer group of the function field of a proper curve

Let $X$ be a smooth proper geometrically connected curve over a number field $k$, and let $k(X)$ denote its field of rational functions, i.e., its function field. Then the (cohomological) Brauer group ...
103 views

1 vote
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### Vanishing of the local étale cohomology sheaf (?)

Let $X$ be a locally noetherian regular scheme, and let $Z$ be a closed subscheme of $X$ whose codimension $d > 0$ at every point. Let $U$ be the complement of $Z$ in $X$. For a sheaf $\mathscr{F}$ ...
1 vote
221 views

### Counterexample to purity of Brauer group for curves

The purity of Brauer group states that for a smooth (quasi-)projective variety $X$ over a field $k$, removing a closed subscheme $Z \subset X$ of codimension at least $2$ ensures that the restriction ...
138 views

### A map between Brauer groups

Let $R$ be a henselian dvr over $\mathbb{C}$ and $A$ be a flat $R$-algebra of finite type. Suppose $\hat{R}$ is the completion of $R$ and $\hat{A}:=A\otimes_R \hat{R}$. For an ideal $I\subseteq A$, ...
111 views

### Why is Maycock's Brauer group of graded C*-algebras connected while Moutuou's is not?

In her thesis The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras (cf. Proposition 3.4), Ellen Maycock described the Brauer group of graded continuous trace $C^\ast$-algebras with spectrum a ...
655 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 ...
241 views

### Can moonshine be explained by the $q$-expansion map $\mathrm{tmf} \to KU[[q]]$?

If we form an equivariant version of both sides of the $q$-expansion map $\mathrm{tmf} \to KU[[q]]$ (with everything being $p$-completed), should the equivariant $q$-expansion map should have ...
125 views

### Field of definition for a representation and Brauer group

I wonder if the following logic is true. And if it is true, how to prove it? Let $G$ be a profinite group, $\rho:G\to GL_n(k)$ a continuous representation, where $k$ is an algebraically closed field ...
1 vote
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### 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 ...
276 views

### What is known about lower etale cohomology of unirational varieties?

Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
269 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 ...
257 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 \...
283 views

### Are Azumaya algebras of trivial Brauer class isomorphic to $\mathcal{E}nd(\mathcal{H})$?

Let $X$ be a scheme, let $\mathcal{A}$ be a sheaf of locally free algebras on $X$. We say $\mathcal{A}$ is an azumaya algebra, if the natural map \mathcal{A}\otimes_{\mathcal{O}_X}\mathcal{A}^{opp}\... 404 views

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### 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?