Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of sheaves of $\mathbf{Z}/\ell^{n+1}$-modules defined by $\mathcal{F}_n$, i.e., the complex with $\mathcal{F}_n$ concentrated in degree zero and zeroes everywhere else.

In the proof of the generalized Trace formula (see Deligne's *Rapport sur la formule des traces* or de Jong's *Stacks Project: Etale cohomology*) the following fact is used.

**Fact.** We have that $K_n$ is of finite Tor dimension.

Equivalently:

**Fact.** We have that $K_n$ is isomorphic in $D^-(X,\mathbf{Z}/\ell^{n+1})$ to a bounded complex of constructible sheaves of flat $\mathbf{Z}/\ell^{n+1} $-modules.

Unfortunately, I haven't been able to find a proof of this fact in the literature.

*Question.* How does one prove the above **Fact**?

**[Edit: Explanation of Torsten Ekedahl's answer]**
By an $\ell$-adic sheaf, one usually means a $\mathbf{Q}_\ell$-sheaf. My mistake was that I considered an arbitrary $\mathbf{Z}_\ell$-sheaf $\mathcal{F}$. In fact, given a $\mathbf{Q}_\ell$-sheaf $\mathcal{G}$, we have that $\mathcal{G}= \mathcal{F}\otimes \mathbf{Q}_\ell$ for some torsion-free $\mathbf{Z}_\ell$-sheaf $\mathcal{F}$. But then it's clear that the complex defined by $\mathcal{F}_n$ is of finite Tor-dimension.

This **Fact** is very strange to me for the following reason:

**Example.** Consider the complex of $\mathbf{Z}/\ell^{n+1}$-modules $$\ldots \stackrel{\cdot l^n}{\longrightarrow} \mathbf{Z}/\ell^{n+1} \stackrel{\cdot l}{\longrightarrow} \mathbf{Z}/\ell^{n+1} \stackrel{\cdot l^n}{\longrightarrow} \mathbf{Z}/\ell^{n+1} \stackrel{\cdot l}{\longrightarrow}\mathbf{Z}/\ell^{n+1} \longrightarrow \mathbf{Z}/\ell \rightarrow 0 . $$ From this it follows that Tor_i$(\mathbf{Z}/\ell, \mathbf{Z}/\ell) = \mathbf{Z}/\ell \neq 0$ if $i>0$. In particular, the $\mathbf{Z}/\ell^{n+1}$-module $\mathbf{Z}/\ell$ is NOT of finite Tor dimension. Thus, we see that $K_n = \mathbf{Z}/\ell$ does not give an $\ell$-adic sheaf. (This would contradict the **Fact**.)

This Example suggests that the proof of the above **Fact** relies on the compatibility between the K_n.