# Explicit examples of Azumaya algebras

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

• 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. Jan 3, 2019 at 10:56
• "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. Jan 3, 2019 at 12:23
• 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 $\mathrm{Br}(X)$. Jan 3, 2019 at 12:42
• 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! Jan 3, 2019 at 12:50
• 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. Jan 3, 2019 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.