The Stacks project

Proof. We explain how to deduce this from Theorem 64.20.4. We first use some étale cohomology arguments to reduce the proof to an algebraic statement which we subsequently prove.

Let $\mathcal{F}$ be as in the theorem. We can write $\mathcal{F}$ as $\mathcal{F}'\otimes \mathbf{Q}_\ell $ where $\mathcal{F}' = \left\{ \mathcal{F}'_ n\right\} $ is a $\mathbf{Z}_\ell $-sheaf without torsion, i.e., $\ell : \mathcal{F}'\to \mathcal{F}'$ has trivial kernel in the category of $\mathbf{Z}_\ell $-sheaves. Then each $\mathcal{F}_ n'$ is a flat constructible $\mathbf{Z}/\ell ^ n\mathbf{Z}$-module on $X_{\acute{e}tale}$, so $\mathcal{F}'_ n \in D_{ctf}(X, \mathbf{Z}/\ell ^ n\mathbf{Z})$ and $\mathcal{F}_{n+1}' \otimes ^{\mathbf{L}}_{\mathbf{Z}/\ell ^{n+1}\mathbf{Z}} \mathbf{Z}/\ell ^ n\mathbf{Z} = \mathcal{F}_ n'$. Note that the last equality holds also for standard (non-derived) tensor product, since $\mathcal{F}'_ n$ is flat (it is the same equality). Therefore,

1. the complex $K_ n = R\Gamma _ c\left(X_{\bar k}, \mathcal{F}_ n'\right)$ is perfect, and it is endowed with an endomorphism $\pi _ n : K_ n\to K_ n$ in $D(\mathbf{Z}/\ell ^ n\mathbf{Z})$,

2. there are identifications

   \[ K_{n+1} \otimes ^{\mathbf{L}}_{\mathbf{Z}/\ell ^{n+1}\mathbf{Z}} \mathbf{Z}/\ell ^ n\mathbf{Z} = K_ n \]

   in $D_{perf}(\mathbf{Z}/\ell ^ n\mathbf{Z})$, compatible with the endomorphisms $\pi _{n+1}$ and $\pi _ n$ (see [Rapport 4.12, SGA4.5]),

3. the equality $\text{Tr}\left(\pi _ X^* |_{K_ n}\right) = \sum _{x\in X(k)} \text{Tr}\left(\pi _ x |_{(\mathcal{F}'_ n)_{\bar x}}\right)$ holds, and

4. for each $x\in X(k)$, the elements $\text{Tr}(\pi _ x |_{\mathcal{F}'_{n, \bar x}}) \in \mathbf{Z}/\ell ^ n\mathbf{Z}$ form an element of $\mathbf{Z}_\ell $ which is equal to $\text{Tr}(\pi _ x |_{\mathcal{F}_{\bar x}}) \in \mathbf{Q}_\ell $.

It thus suffices to prove the following algebra lemma. $\square$