Definition of morphism between pro-A-modules?

Question

In Bhatt's paper On the Direct Summand Conjecture and its Derived Invariant, he makes the following "almost" definition for pro-A-modules:

Definition A pro-$A$-module $\{M_n\}_{n\ge1}$ is said to be almost-pro-zero if for any $k\ge0$ and any $n\ge1$, there exists some $m = m(n,k) ≥ n$ such that $im(M_m → M_n)$ is killed by $t^{1/{p^k}}$; equivalently, for each $k ≥ 0$, the map $\{M_n[t^{1/p^k}]\}_{n≥1} \to \{M_n\}_{n≥1}$ is a pro-isomorphism in the usual sense. A map of pro-objects in $D^b(A)$ is said to be an almost-pro-isomorphism if the cohomology groups of cones form an almost-pro-zero system.

My problem seems to be that I'm ignorant on the definition of a pro-$A$-module and morphisms of these objects in general. The only seemingly relevant reference I can find is the following page on nlab

https://ncatlab.org/nlab/show/pro-object

which defines for any category $C$ a "pro-category" $\text{pro-}C$ where the objects are diagrams of the form $F:D\to C$ for $D$ a small cofiltered category, and morphisms are defined by

$$\text{pro-}C(F,G)=\varprojlim_{e\in E}\varinjlim_{d\in D} C(Fd,Ge)$$

for $F:D\to C$ and $G:E\to C$. This definition isn't very enlightening but I assume pro-$A$-modules are basically just this construction with $D$ taken to be $\Bbb N^{\rm{op}}$ (considered as a category) for all objects, meaning that objects of pro-$A$-mod are just sequences

$$\cdots\to M_{n+1}\to M_n\to M_{n-1}\to\cdots\to M_1$$

of morphisms of $A$-modules. I still feel unenlightened as to what the (iso)morphisms of these objects are, and why the remark Bhatt makes in the second statement of the quoted definition holds true.

Answer 1

Where to direct you to learn about pro-objects depends on how general a form you need and, to some extent, your background. If you look at inverse systems/sequences of modules (as in the case you cite), then look at the nLab entry on towers, and also on profinite groups. Pro-objects provide a sense of approximation to a limit object that may not be in the category you are looking at, e.g. a pro-finite group is an inverse system of finite groups. If you take its limit the result will often not be a finite group. The finite quotients of a group form a profinite group in this sense. There is also a topological approach which comes in in your pro-A-module setting as well. Nice systems of cofinite submodules will form a pro-module in an analogous way.

I wrote an introduction to pro-objects (in general) in a set of notes available via my nLab page: https://ncatlab.org/timporter/files/ProfAlgHomotopy.pdf You only need the first few pages of the main text, although you may find some useful ideas later on in the first chapter.

The definition of morphism in a procategory can possibly be best understood by looking at pro-representable functors. Left exact functors will be pro-representable (check again on the nlab for the definition) and natural transformations between pro-representable functors give exactly that definition of pro-morphism. An approach that will perhaps help you understand Bhatt's second statement is to work when a pro-module (in general) will be isomorphic to a zero pro-module.(Some ideas relating to this can be found in a book by Cordier and myself that you may find answers some of the basic points, see the nLab page on pro-objects for the details of the book. There are also some old notes of Duskin based on ideas of Verdier, who wrote a note J.-L. Verdier, Équivalence essentielle des syst`emes projectifs, C. R. Acad. Sci. Paris, 261, (1965), 4950–4953.)