This is kind of the relative version of [this question](https://mathoverflow.net/q/93812/15782). Even though I made extensive enquiries, I couldn't find good references for this and it seems to me that these questions are pretty well understood by experts.


Given an Azumaya algebra on a scheme $X$, adopting the construction of the Brauer-Severi variety for a central simple algebra, (I think) I'm able to define a scheme over $X$ which is a twisted form of projective space, representing the same class in $\check H^1(X\_{et},PGL\_n)$ than the Azumaya algebra. (The sub scheme of the relative Grassmanian parametrising right ideals of appropriate dimension.)


Does someone know how to get the other way around, i.e. how to read the Azumaya algebra from an Brauer-Severi scheme explicitly? Perhaps one could turn the construction in Bhargavs comments to the answer in the question linked above to a global one, but I'm uncertain concerning the details.



Explicit bijection between Azumaya algebras and Brauer-Severi schemes
A straightforward way is to find an etale covering $\tilde{X} \to X$ such that the pullback of the Severi--Brauer variety has a rational section and so is isomorphic to a projectiviation of a vector bundle $E$, then take $\mathcal{E}nd(E)$ and descend it to $X$. The covering $\tilde{X}$ can be obtained from (a collection of) multisections of the Severi--Brauer variety.


If you want a more explicit way, you can find a vector bundle $F$ on the Severi-Brauer variety which restricts to any its fiber over $X$ as $\mathcal{O}(1)^{\oplus n}$. Then the descent of $\mathcal{E}nd(F)$ is the required Azumaya algebra.


