The Stacks project

63.5 Why derived categories?

With this definition of the trace, let us now discuss another issue with the formula as stated. Let $C$ be a smooth projective curve over $k$. Then there is a correspondence between finite locally constant sheaves $\mathcal{F}$ on $C_{\acute{e}tale}$ whose stalks are isomorphic to ${(\mathbf{Z}/\ell ^ n\mathbf{Z})}^{\oplus m}$ on the one hand, and continuous representations $\rho : \pi _1 (C, \bar c) \to \text{GL}_ m(\mathbf{Z}/\ell ^ n\mathbf{Z}))$ (for some fixed choice of $\bar c$) on the other hand. We denote $\mathcal{F}_\rho $ the sheaf corresponding to $\rho $. Then $H^2 (C_{\bar k}, \mathcal{F}_\rho )$ is the group of coinvariants for the action of $\rho (\pi _1 (C, \bar c))$ on ${(\mathbf{Z}/\ell ^ n\mathbf{Z})}^{\oplus m}$, and there is a short exact sequence

\[ 0 \longrightarrow \pi _1 (C_{\bar k}, \bar c) \longrightarrow \pi _1 (C, \bar c) \longrightarrow G_ k \longrightarrow 0. \]

For instance, let $\mathbf{Z} = \mathbf{Z} \sigma $ act on $\mathbf{Z}/\ell ^2\mathbf{Z}$ via $\sigma (x) = (1+\ell ) x$. The coinvariants are $(\mathbf{Z}/\ell ^2\mathbf{Z})_{\sigma } = \mathbf{Z}/\ell \mathbf{Z}$, which is not a flat $\mathbf{Z}/\ell ^2\mathbf{Z}$-module. Hence we cannot take the trace of some action on $H^2(C_{\bar k}, \mathcal{F}_\rho )$, at least not in the sense of the previous section.

In fact, our goal is to consider a trace formula for $\ell $-adic coefficients. But $\mathbf{Q}_\ell = \mathbf{Z}_\ell [1/\ell ]$ and $\mathbf{Z}_\ell = \mathop{\mathrm{lim}}\nolimits \mathbf{Z}/\ell ^ n\mathbf{Z}$, and even for a flat $\mathbf{Z}/\ell ^ n\mathbf{Z}$ sheaf, the individual cohomology groups may not be flat, so we cannot compute traces. One possible remedy is consider the total derived complex $R\Gamma (C_{\bar k}, \mathcal{F}_\rho )$ in the derived category $D(\mathbf{Z}/\ell ^ n\mathbf{Z})$ and show that it is a perfect object, which means that it is quasi-isomorphic to a finite complex of finite free module. For such complexes, we can define the trace, but this will require an account of derived categories.


Comments (4)

Comment #1920 by Matthieu Romagny on

typo : 'whose stalks' instead of 'which stalks'

Comment #2166 by Alex on

typo? `which is not a flat -module' should say ?


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Your email address will not be published.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 03T2. Beware of the difference between the letter 'O' and the digit '0'.