# Explicit bijection between Azumaya algebras and Brauer-Severi schemes

This is kind of the relative version of this question. 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.

• You can construct the underlying locally free sheaf of the Azumaya algebra "explicitly" (i.e., without explicit mention of descent). This is Bhatt's suggestion for the other question. Constructing the algebra structure is a bit trickier. You can construct the associated Lie bracket, and then you can recover the multiplication from that. Inside the associated vector bundle of the locally free sheaf, you can construct the open subset parameterizing invertible elements and the group law on that. None of the constructions I know is particularly direct. Feb 21, 2014 at 14:09
• But the fact is that you don't usually need to know the algebra structure explicitly. What you may need is a description of the category of modules over the algebra. And the latter is equivalent to the category of sheaves on the Severy-Brauer variety which restrict to any fiber as a multiplicity of $O(1)$. Feb 21, 2014 at 14:14

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.