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


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.


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.)

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