# 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. – pbelmans Jan 3 '19 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. – Jason Starr Jan 3 '19 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 $\mathrm{Br}(X)$. – Uriya First Jan 3 '19 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! – Uriya First Jan 3 '19 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. – 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.