Argument taken from .

Lemma 59.62.2. Let $S$ be a scheme. Let $\mathcal{A}$ be an Azumaya algebra which is locally free of rank $d^2$ over $S$. Then the class of $\mathcal{A}$ in the Brauer group of $S$ is annihilated by $d$.

Proof. Choose an étale covering $\{ U_ i \to S\}$ and choose isomorphisms $\mathcal{A}|_{U_ i} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}_ i, \mathcal{F}_ i)$ for some locally free $\mathcal{O}_{U_ i}$-modules $\mathcal{F}_ i$ of rank $d$. (We may assume $\mathcal{F}_ i$ is free.) Consider the composition

$p_ i : \mathcal{F}_ i^{\otimes d} \to \wedge ^ d(\mathcal{F}_ i) \to \mathcal{F}_ i^{\otimes d}$

The first arrow is the usual projection and the second arrow is the isomorphism of the top exterior power of $\mathcal{F}_ i$ with the submodule of sections of $\mathcal{F}_ i^{\otimes d}$ which transform according to the sign character under the action of the symmetric group on $d$ letters. Then $p_ i^2 = d! p_ i$ and the rank of $p_ i$ is $1$. Using the given isomorphism $\mathcal{A}|_{U_ i} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}_ i, \mathcal{F}_ i)$ and the canonical isomorphism

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}_ i, \mathcal{F}_ i)^{\otimes d} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}_ i^{\otimes d}, \mathcal{F}_ i^{\otimes d})$

we may think of $p_ i$ as a section of $\mathcal{A}^{\otimes d}$ over $U_ i$. We claim that $p_ i|_{U_ i \times _ S U_ j} = p_ j|_{U_ i \times _ S U_ j}$ as sections of $\mathcal{A}^{\otimes d}$. Namely, applying Lemma 59.62.1 we obtain an invertible sheaf $\mathcal{L}_{ij}$ and a canonical isomorphism

$\mathcal{F}_ i|_{U_ i \times _ S U_ j} \otimes \mathcal{L}_{ij} \longrightarrow \mathcal{F}_ j|_{U_ i \times _ S U_ j}.$

Using this isomorphism we see that $p_ i$ maps to $p_ j$. Since $\mathcal{A}^{\otimes d}$ is a sheaf on $S_{\acute{e}tale}$ (Proposition 59.17.1) we find a canonical global section $p \in \Gamma (S, \mathcal{A}^{\otimes d})$. A local calculation shows that

$\mathcal{H} = \mathop{\mathrm{Im}}(\mathcal{A}^{\otimes d} \to \mathcal{A}^{\otimes d}, f \mapsto fp)$

is a locally free module of rank $d^ d$ and that (left) multiplication by $\mathcal{A}^{\otimes d}$ induces an isomorphism $\mathcal{A}^{\otimes d} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{H}, \mathcal{H})$. In other words, $\mathcal{A}^{\otimes d}$ is the trivial element of the Brauer group of $S$ as desired. $\square$

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