Lemma 59.62.1. Let $S$ be a scheme. Let $\mathcal{F}$ and $\mathcal{G}$ be finite locally free sheaves of $\mathcal{O}_ S$-modules of positive rank. If there exists an isomorphism $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, \mathcal{F}) \cong \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{G}, \mathcal{G})$ of $\mathcal{O}_ S$-algebras, then there exists an invertible sheaf $\mathcal{L}$ on $S$ such that $\mathcal{F} \otimes _{\mathcal{O}_ S} \mathcal{L} \cong \mathcal{G}$ and such that this isomorphism induces the given isomorphism of endomorphism algebras.
59.62 The Brauer group of a scheme
Let $S$ be a scheme. An $\mathcal{O}_ S$-algebra $\mathcal{A}$ is called Azumaya if it is étale locally a matrix algebra, i.e., if there exists an étale covering $\mathcal{U} = \{ \varphi _ i : U_ i \to S\} _{i \in I}$ such that $\varphi _ i^*\mathcal{A} \cong \text{Mat}_{d_ i}(\mathcal{O}_{U_ i})$ for some $d_ i \geq 1$. Two such $\mathcal{A}$ and $\mathcal{B}$ are called equivalent if there exist finite locally free $\mathcal{O}_ S$-modules $\mathcal{F}$ and $\mathcal{G}$ which have positive rank at every $s \in S$ such that
\[ \mathcal{A} \otimes _{\mathcal{O}_ S} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, \mathcal{F}) \cong \mathcal{B} \otimes _{\mathcal{O}_ S} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{G}, \mathcal{G}) \]
as $\mathcal{O}_ S$-algebras. The Brauer group of $S$ is the set $\text{Br}(S)$ of equivalence classes of Azumaya $\mathcal{O}_ S$-algebras with the operation induced by tensor product (over $\mathcal{O}_ S$).
Proof. Fix an isomorphism $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, \mathcal{F}) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{G}, \mathcal{G})$. Consider the sheaf $\mathcal{L} \subset \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{F}, \mathcal{G})$ generated as an $\mathcal{O}_ S$-module by the local isomorphisms $\varphi : \mathcal{F} \to \mathcal{G}$ such that conjugation by $\varphi $ is the given isomorphism of endomorphism algebras. A local calculation (reducing to the case that $\mathcal{F}$ and $\mathcal{G}$ are finite free and $S$ is affine) shows that $\mathcal{L}$ is invertible. Another local calculation shows that the evaluation map
\[ \mathcal{F} \otimes _{\mathcal{O}_ S} \mathcal{L} \longrightarrow \mathcal{G} \]
is an isomorphism. $\square$
The argument given in the proof of the following lemma can be found in [Saltman-torsion].
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. $\square$
In this setting, the analogue of the isomorphism $\delta $ of Theorem 59.61.6 is a map
\[ \delta _ S: \text{Br}(S) \to H_{\acute{e}tale}^2(S, \mathbf{G}_ m). \]
It is true that $\delta _ S$ is injective. If $S$ is quasi-compact or connected, then $\text{Br}(S)$ is a torsion group, so in this case the image of $\delta _ S$ is contained in the cohomological Brauer group of $S$
\[ \text{Br}'(S) := H_{\acute{e}tale}^2(S, \mathbf{G}_ m)_\text {torsion}. \]
So if $S$ is quasi-compact or connected, there is an inclusion $\text{Br}(S) \subset \text{Br}'(S)$. This is not always an equality: there exists a nonseparated singular surface $S$ for which $\text{Br}(S) \subset \text{Br}'(S)$ is a strict inclusion. If $S$ is quasi-projective, then $\text{Br}(S) = \text{Br}'(S)$. However, it is not known whether this holds for a smooth proper variety over $\mathbf{C}$, say.
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