Disclaimer : I asked this question on Maths.StackExchange 20 days ago (and started a bounty) here 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.
