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 algebras actually look like (in terms of an explicit trivialisation on an etale open cover). For instance:

  • What do the Azumaya algebras on $\mathbb{P}^n_k$ look like? (Since $\text{Br}(\mathbb{P}^n_k)=\text{Br}(k)$, I imagine there's an explicit way to get an Azumaya algbera on $\mathbb{P}^n$ from a central simple algebra).
  • What do the Azumaya algebras on a smooth curve in $\mathbb{P}^2$ given by $f=0$ look like? (For some $k$ not algberaically closed or finite, like a number field, since otherwise $\text{Br}(X)=0$).
  • Are there any other examples like this?

I'm mainly interested in this to compute examples of the Brauer-Manin obstruction, i.e. taking Azumaya algebra $A$ and point $p\in X(k_\nu)$ and computing $$\text{inv}_\nu(A_{p})$$ where $A_{p}$ is the image of Azumaya algebra $A$ under $\text{Br}(X)\to \text{Br}(k_\nu)$ arising from the inclusion of $p\hookrightarrow X$, and $\text{inv}_\nu$ is the Hasse invariants map. I'd be interested in seeing this computed in the examples above.

My confusion might just arise from not knowing how to explicitly compute the map $\text{inv}_\nu:\text{Br}(k_\nu)\to \mathbb{Q}/\mathbb{Z}$.

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    $\begingroup$ On $\mathbb{P}_k^n$ the Azumaya algebras are of the form $\mathcal{O}_{\mathbb{P}_k^n}\otimes_k A$ where $A$ is a central simple algebra over $k$. On smooth projective curves over finite fields the Brauer group is trivial, (just like it is over an algebraically closed field), so nothing happens there. $\endgroup$ – pbelmans Jan 3 '19 at 10:56
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    $\begingroup$ "On $\mathbb{P}^n_k$ the Azumaya algebras are of the form $\mathcal{O}_{\mathbb{P}^n_k}\otimes_k A$ ..." There are also Azumaya algebras on $\mathbb{P}^n_k$ of the form $\textit{End}_{\mathcal{O}_{\mathbb{P}^n}}(F)\otimes_{k} A$, where $F$ is a locally free $\mathcal{O}_{\mathbb{P}^n}$-module. $\endgroup$ – Jason Starr Jan 3 '19 at 12:23
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    $\begingroup$ If $X$ is a regular integral scheme with fraction field $K$, then results of Grothendieck (or Auslander-Goldman in the affine case) show that the natural map $\mathrm{Br}(X)\to \mathrm{Br}(K)$ is injective and its image is the unramified Brauer group of $K$, namely, the intersection of the images of $\mathrm{Br}(\mathcal{O}_{X,x})$ in $\mathrm{Br}(K)$ as $x$ ranges over the codimension $1$ points of $X$. This means that one can describe Brauer classes over $X$ as Brauer classes over $K$ which extend to every codimension $1$ point of $X$. This is one approach to describing $\mathrm{Br}(X)$. $\endgroup$ – Uriya First Jan 3 '19 at 12:42
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    $\begingroup$ Relating to @JasonStarr 's comment, describing the Brauer group using Azumaya algebra representatives is usually not the same as describing all Azumaya algebras. The latter problem is usually much more difficult! $\endgroup$ – Uriya First Jan 3 '19 at 12:50
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    $\begingroup$ The easiest non-trivial case is quaternion algebras, which are completely explicit. These give rise to $2$-torsion elements in the Brauer group. Other cases, which generalise these, are cyclic algebras. I would recommend looking at Poonen's book "Rational points on varieties" which has explicit examples of the Brauer-Manin obstruction. Note that constant algebras arn't useful for the Brauer-Manin obstruction as all local invariants sum to $0$ here. $\endgroup$ – Daniel Loughran Jan 3 '19 at 14:12

Another example I found: if $X=G$ is a linear algebraic group over an algebraically closed field $k$, then every Azumaya algebra is given by a projective representation $$\pi_1G \ \longrightarrow \text{PGL}_n(k)$$ which is trivial iff it lifts to an honest representation $G\to\text{GL}_n(k)$. See ``Brauer Group of a Linear Algebraic Group'' by Bircher Iversen for a proof.

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