I'm reading a set of notes that has an interlude about derived $p$-completions of abelian groups.

My first question stems from the fact that this interlude, although interesting, is not very furnished, so I would like to know if you knew some more references about this derived $p$-completion thing.

My more important question is about the definition of derived $p$-completion and more generally derived limit in this case. The context is the following : we have an abelian group $A$, its $p$-completion is the limit of $A/p^k$; we wish to make it more "homotopy friendly" in a way, so instead we take $A//p^k$ which is the non-negatively graded (for simplicity, perhaps "bounded below" is a better idea here, if so please tell me) chain complex $\dots\to 0\to A\to^{p^k} A$ which is some derived version $A/p^k$, and then we wish to take the derived limit of these complexes.

My question comes from the definition of derived limit in this context : we have a category $Ch(\mathbf{Ab})_{\geq 0}$ which has a wide subcategory $W$ of weak equivalences (which are the quasi-isomorphisms), and if I'm not mistaken this suffices to define what "homotopy limits" are (which I assume are what we mean by "derived limit"). It would mean a homotopical functor $\mathbb{R}\lim : Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}\to Ch(\mathbf{Ab})_{\geq 0}$ with a morphism $\lim\to \mathbb{R}\lim$ satisfying certain properties (I am using Riehl's Categorical homotopy theory for the definitions); and according to Riehl's book, such a functor is given by $\lim\circ F$ if $F: Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}\to Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}$ is a right deformation onto a subcategory on which $\lim$ is homotopical (we also have a transformation $f:id\to F$ as part of the data)

Now the notes I mentioned say that to compute $\mathbb{R}\lim_k C(k)$ you take a functor $D\in Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}$ which is injective in this category of diagrams together with a (pointwise) quasi-isomorphism $C\to D$ and compute $\lim_k D(k)$. My question is now :

why does this work ? Why is $\lim$ homotopical on the subcategory of injectives ? On the other hand, is it sufficient to take $D$ such that all $D(k)$'s have injective chain groups [I know that this doesn't imply that $D$ is injective, it's another question] ?

EDIT: I realized that maybe what was meant was to consider chain complexes of injective diagrams, not injective objects in $Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}$ ! That is, use the fact that $Ch(\mathbf{Ab})_{\geq 0}^\mathbb{N}\simeq Ch(\mathbf{Ab}^\mathbb{N})_{\geq 0}$ and take chain complexes of injective objects in $\mathbf{Ab}^\mathbb{N}$, which would definitely make more sense. Is that indeed what was meant ?


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