Modules over an Azumaya algebra and modules over the associated Brauer-Severi variety

Assume A is an Azumaya algebra of rank r2 on a smooth projective scheme Y over C. Let f:XY 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 fM is a left fA-module. But fA=EndX(I)opEndX(I) for a locally free OX-module I which restricts to O(1)r over every closed point. So using Morita equivalence fM=IN 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.

Answer

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Source : Link , Question Author : Bernie , Answer Author : Community

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