Assume A is an Azumaya algebra of rank r2 on a smooth projective scheme Y over C. Let f:X→Y be the Brauer-Severi variety associated to A.
I read here in a comment that the category of modules over A is equivalent to the categroy of modules on X which restrict to every fiber as a multiplicity of O(1). I did not find a reference in the literature of this fact.
Here is what I think of the situation:
Let us start with a left A-module M, then f∗M is a left f∗A-module. But f∗A=EndX(I)op≅EndX(I∨) for a locally free OX-module I which restricts to O(−1)r over every closed point. So using Morita equivalence f∗M=I∨⊗N for a module N on X. So M gives me the module N on X, but I don’t see any reason why we should know how N restricts to the fibers of f. Or am I missing something?
What is the exact equivalence of categories in this situation? I am also happy if we could only say something about locally free A-modules and locally free OX-modules. References are also very welcome.