The Stacks project

64.5 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

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.