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.



For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension? 
I think you have (slightly) misread your sources. If you take *Rapport*
for instance (which is the one I am familiar with) Deligne never make this claim
(as far as I can see). As a typical example consider 4.11 (proof of 2.30) where
he assumes that $(K\_n)$ is a torsion free $\mathbb Z\_\ell$-complex in which case
$K\_n$ is indeed of finite Tor-dimension. Alternatively, he could have considerd
(even when $K$ is just a sheaf) the derived reduction modulo $\ell^n$,
$K\bigotimes^{\mathbb L}\_{\mathbb Z\_\ell}\mathbb Z/\ell^n$ which always will
have finite Tor-dimension. Your example is a proof that not taking the derived
reduction but just the reduction will not necessarily give something of finite
Tor-dimension.


