TITLE: Explicit examples of Azumaya algebras
QUESTION [9 upvotes]: 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 $\text{inv}_\nu:\text{Br}(k_\nu)\to \mathbb{Q}/\mathbb{Z}$.

REPLY [2 votes]: 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.