**Disclaimer** : I asked this question on Maths.StackExchange 20 days ago (and started a bounty) [here](https://math.stackexchange.com/questions/3723218/perfect-mathbb-z-ell-modules) but got no answer, so I'm asking it here now (with no modification).


$\newcommand{\l}{\ell} \newcommand{\Z}{\mathbb Z}$
I'm trying to understand a remark in *Weil's conjectures for function fields I*, by Gaitsgory-Lurie.


Specifically it's remark 2.3.4.3., which essentially states the following : an object $M$ in $\mathrm{Mod}\_{\mathbb Z\_\l}$ is perfect if and only if it's $\l$-complete and $M\otimes\_{\Z\_\l}\Z/\l$ is perfect in $\mathrm{Mod}\_{\Z/\l}$ (this latter condition implies that this is also true for $M\otimes\_{\Z\_\l}\Z/\l^d$, for any $d\geq 0$)


They state it without proof, and it looks like there is no proof elsewhere in the book (as far as I can see, although I haven't checked the whole book).


What is clear to me is the forward direction : indeed $\Z\_\l\otimes\_{\Z\_\l}\Z/\l$ is perfect over $\Z/\l$ and $\Z\_\l$ is $\l$-complete, and these conditions are closed under retracts, so the sub-$\infty$-category of those $M$'s that satisfy the latter conditions is a stable subcategory, contains $\Z\_\l$ and is closed under retracts so it contains all perfect $\Z\_\l$-modules.


The reverse direction, however, does not seem so clear. I think one ought to use the fact that the $\infty$-category of $\l$-complete modules is equivalent to $\underset{d}{\varprojlim} \mathrm{Mod}\_{\Z/\l^d}$ but that doesn't seem to be enough since we want $M$ to be compact in the whole $\mathrm{Mod}\_{\Z\_\l}$, not just in $\l$-complete modules.



Perfect $\mathbb Z_\ell$-modules
Over a PID, you can check perfectness on homology. So the claim is that the homology of an $\ell$-complete $\mathbb{Z}\_\ell$-module is finitely generated if and only if it is so mod $\ell$. The homology groups are $\ell$-complete, and so this follows from the long exact sequence of mod $\ell$-reduction.


