The Stacks project

Argument taken from [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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A2L. Beware of the difference between the letter 'O' and the digit '0'.