Are Azumaya algebras of trivial Brauer class isomorphic to $\mathcal{E}nd(\mathcal{H})$?

Question

Let $X$ be a scheme, let $\mathcal{A}$ be a sheaf of locally free algebras on $X$. We say $\mathcal{A}$ is an azumaya algebra, if the natural map $$\mathcal{A}\otimes_{\mathcal{O}_X}\mathcal{A}^{opp}\to\mathcal{E}nd_{\mathcal{O}_X}(\mathcal{A}), $$ $$a\otimes b\mapsto (x\mapsto axb)$$ is an isomorphism.

Two azumaya algebras $\mathcal{A},\mathcal{B}$ are called Morita equivalent, if there exists locally free sheaves $\mathcal{F},\mathcal{G}$, such that $$\mathcal{A}\otimes\mathcal{E}nd_{\mathcal{O}_X}(\mathcal{F})\cong\mathcal{B}\otimes\mathcal{E}nd_{\mathcal{O}_X}(\mathcal{G}).$$

Let $\mathcal{A}$ be an Azumaya algebra, which is Morita equivalent to $\mathcal{B}:=\mathcal{O}_X$, does $\mathcal{A}$ necessarily have the form $\mathcal{E}nd_{\mathcal{O}_X}(\mathcal{H})$ for some locally free sheaf $\mathcal{H}$ on $X$?(I think this is true by the comment of Eoin). Is there a way to express this $\mathcal{H}$?

Answer 1

The answer is yes and one can take $$\mathcal{H}:=\mathcal{Hom}_{\mathcal{End}(\mathcal G)}(\mathcal F\otimes \mathcal A, \mathcal G).$$ Here, $\mathcal F\otimes \mathcal A$ is viewed as a left $\mathcal{End}(\mathcal G)$-module via the isomorphism $\mathcal{End}(\mathcal G)\cong \mathcal{End}(\mathcal F)\otimes \mathcal A$.

More precisely, the right action of $\mathcal{A}$ on $\mathcal F\otimes\mathcal A$ determines a left $\mathcal{A}$-module structure on $\mathcal H=\mathcal{Hom}_{\mathcal{End}(\mathcal G)}(\mathcal F\otimes \mathcal A, \mathcal G)$, which in trun determines a morphism of $\mathcal O_X$-algebras $\psi:\mathcal A\to \mathcal{End}(\mathcal H)$. This morphism is an isomorphism.

Since both the source and target of $\psi$ are locally free $\mathcal O_X$-modules, it is enough to check that $\psi$ is an isomorphism after specializing to geometric points. Then the claim follows by noting that the source and target are central simple algebras of the same dimension.

---

I do feel that something should be said about the use of Morita theory lying in the background, so let me elaborate on this. I will consider consider the affine case $X=\mathrm{Spec} R$ for simplicity, and write $A=\Gamma(X,\mathcal{A})$, etc.

Our goal is to construct an $(A,R)$-progenerator $H$, i.e., an $(A,R)$-bimodule $H$ such that $H_R$ is projective (i.e. locally free), finitely generated and the natural map $A\to\mathrm{End}_R(H_R)$ is an isomorphism. (In this case, Morita theory tells us that ${}_AH$ is f.g. projective and $R=\mathrm{End}_A({}_AH)$ if $A$-endomorphism are written on the right.)

Write $F'=F\otimes A$ and view it as a right $A$-module. Then, since $F$ is f.g. projective, there is an $R$-algebra isomorphism $$ \mathrm{End}_A(F')\cong \mathrm{End}_R(F)\otimes A\cong \mathrm{End}_R(G). $$ Thus, both $F'$ and $G$ can be regarded as left $\mathrm{End}_R(G)$-modules.

Since $G$ is f.g. projective over $R$ and $F'$ is f.g. projective over $A$, we see that $G$ is an $(\mathrm{End}(G),R)$-progenerator and $G$ is an $(\mathrm{End}_A(F'),A)$-progenerator, which we view as a $(\mathrm{End}_R(G),A)$-progenerator. Now, Morita theory tells us that $H=\mathrm{Hom}_{\mathrm{End}_R(G)}(F',G)$, which is naturally an $(A,R)$-bimodule, is an $(A,R)$-progenerator, which is exactly what we want.