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Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$.

I read herehere in a comment that the category of modules over $\mathcal{A}$ 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 $\mathcal{A}$-module $M$, then $f^{*}M$ is a left $f^{*}\mathcal{A}$-module. But $f^{*}\mathcal{A}=\mathcal{E}nd_X(I)^{op}\cong \mathcal{E}nd_X(I^{\vee})$ for a locally free $O_X$-module $I$ which restricts to $O(-1)^r$ over every closed point. So using Morita equivalence $f^{*}M=I^{\vee}\otimes N$ 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 $\mathcal{A}$-modules and locally free $O_X$-modules. References are also very welcome.

Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$.

I read here in a comment that the category of modules over $\mathcal{A}$ 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 $\mathcal{A}$-module $M$, then $f^{*}M$ is a left $f^{*}\mathcal{A}$-module. But $f^{*}\mathcal{A}=\mathcal{E}nd_X(I)^{op}\cong \mathcal{E}nd_X(I^{\vee})$ for a locally free $O_X$-module $I$ which restricts to $O(-1)^r$ over every closed point. So using Morita equivalence $f^{*}M=I^{\vee}\otimes N$ 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 $\mathcal{A}$-modules and locally free $O_X$-modules. References are also very welcome.

Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$.

I read here in a comment that the category of modules over $\mathcal{A}$ 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 $\mathcal{A}$-module $M$, then $f^{*}M$ is a left $f^{*}\mathcal{A}$-module. But $f^{*}\mathcal{A}=\mathcal{E}nd_X(I)^{op}\cong \mathcal{E}nd_X(I^{\vee})$ for a locally free $O_X$-module $I$ which restricts to $O(-1)^r$ over every closed point. So using Morita equivalence $f^{*}M=I^{\vee}\otimes N$ 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 $\mathcal{A}$-modules and locally free $O_X$-modules. References are also very welcome.

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

Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$.

I read here in a comment that the category of modules over $\mathcal{A}$ 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 $\mathcal{A}$-module $M$, then $f^{*}M$ is a left $f^{*}\mathcal{A}$-module. But $f^{*}\mathcal{A}=\mathcal{E}nd_X(I)^{op}\cong \mathcal{E}nd_X(I^{\vee})$ for a locally free $O_X$-module $I$ which restricts to $O(-1)^r$ over every closed point. So using Morita equivalence $f^{*}M=I^{\vee}\otimes N$ 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 $\mathcal{A}$-modules and locally free $O_X$-modules. References are also very welcome.