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 $\text{inv}_\nu:\text{Br}(k_\nu)\to \mathbb{Q}/\mathbb{Z}$.

unramified Brauer groupof $K$, 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