A Proof of the Countable Telescope Conjecture for Module Categories

TL;DR

This paper reviews and proves the Countable Telescope Conjecture for Module Categories, using inductive refinements and dense systems of modules, with applications to the Enochs Conjecture.

Contribution

It provides a proof of the Countable Telescope Conjecture for Module Categories and introduces new techniques involving dense systems and localness concepts.

Findings

Proof of the Countable Telescope Conjecture for Module Categories

Development of new tools involving dense systems of modules

Applications to the Enochs Conjecture

Abstract

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.