  The Countable Telescope Conjecture arose in the framework of stable homotopy
theory, as a tool conceived to study the chromatic filtration. It turned out,
however, to trigger extremely fertile research within the framework of Module
Categories. The project aims at presenting an almost self-contained review of
the recent work of Saroch on the Countable Telescope Conjecture for Module
Categories. After recalling some preliminaries, we report various devices of
independent interest that will lead to a proof of the aforementioned result.
This will be the outcome of inductive refinements of families of particularly
well-behaved dense systems of modules, our witnessing-notion for localness. The
procedure will be reminiscent of Cantor diagonal argument in the implementation
of a variant of Shelah's Compactness Principle. Then, we briefly review the
main applications to Enochs Conjecture of the just developed theory, and we
will also state a weaker version of it. The project closely follows the work of
Saroch (https://link.springer.com/article/10.1007%2Fs11856-018-1710-4);
however, for the sake of completeness and conciseness, we slightly modified
some well-known proofs applying the newly developed tools.

Rings and Algebras;Representation Theory