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

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 $$\mathcal{H}$$?

• I think so. I'm looking at Milne's Etale cohomology book, Chapter IV, Section 2. For X a scheme you have an exact sequence $1\rightarrow G_{m,X}\rightarrow GL_{n,X}\rightarrow PGL_{n,X}\rightarrow 1$ (of etale sheaves on X). The group of azumaya algebras includes into $H^2_{et}(X,G_m)$ by Theorem 2.5 of loc. cit., so if one is trivial then it comes from an element of $H^1_{et}(X,GL_n)$. I think this correspondence sends a locally free sheaf $\mathcal{F}$ of rank $n$ to $\mathcal{E}nd(\mathcal{F})$. – Eoin Oct 27 '19 at 5:33
• @Eoin Thanks! Furthermore, would there be a way to express this $\mathcal{H}$ in terms of $\mathcal{F},\mathcal{G}$? (The result seems to claim we can always pick some line bundle $\mathcal{L}$, so that we can write $\mathcal{G}\otimes\mathcal{L}$ as $\mathcal{F}\otimes\mathcal{H}$? I am a bit confused how to show a module has a specific "tensor summand") – Qixiao Oct 27 '19 at 6:55
• The usual way to recover $k^n$ from $M_n(k)$ is to take a minimal left ideal. Does that work? – Martin Bright Oct 27 '19 at 9:17
• @MartinBright Thanks for the nice suggestion! Let me check it.. – Qixiao Oct 27 '19 at 11:16

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.