in seminar today we got puzzled about a basic definition, beginning of §2 of Mathew-Naumann-Noel "Nilpotence and descent in ESHT"
namely the categories of $p$-torsion and $p$-complete $\mathbb Z$-modules, and in particular the claim that $p$-completion gives, by Dwyer-Greenlees theory, an equivalence of categories between the two
(where category means $\infty$-category and $\mathbb Z$-module means $H\mathbb Z$-module spectrum)
I feel like $\mathbb Z_p$ lives in $p$-complete $\mathbb Z$-modules, but $\mathbb Z$ doesn't live in $p$-torsion modules, so idk how you can hit $\mathbb Z_p$ with $p$-completion
@YuriSulyma The point is that the ∞-categories of p-complete and p-torsion Z-modules are abstractly isomorphic, but they are different subcategories of D(Z)
There is a "completion" functor, that has both a left and a right adjoint. If one of the two functors is fully faithful, so is the other one and so they give you an identification of two different subcategories of your "big" category
@Twistediso I sent you the Harvard ones too, didn't you see them in my email? Or are you looking for other notes?
@YuriSulyma Also, the inverse functor of the equivalence is given by smashing with ΩZ/p^∞. If I'm not mistaken Z_p is indeed the p-completion of ΩZ/p^∞ (which is p-torsion), because of the fiber sequence ΩZ/p^∞→Z→Z[1/p]
