# A short introduction to the telescope and chromatic splitting conjectures

## TL;DR

This paper provides an overview of the telescope and chromatic splitting conjectures in stable homotopy theory, proving the equivalence of the telescope conjecture across all heights with the generalized telescope conjecture, and discusses connections to modular representation theory.

## Contribution

It offers a proof of the folklore result linking the telescope conjecture for all heights to the generalized telescope conjecture, clarifying their relationship.

## Key findings

- Proves the equivalence of the telescope conjecture for all heights with the generalized telescope conjecture.
- Highlights similarities between stable homotopy theory and modular representation theory.
- Provides an overview of the telescope and chromatic splitting conjectures.

## Abstract

In this note, we give a brief overview of the telescope conjecture and the chromatic splitting conjecture in stable homotopy theory. In particular, we provide a proof of the folklore result that Ravenel's telescope conjecture for all heights combined is equivalent to the generalized telescope conjecture for the stable homotopy category, and explain some similarities with modular representation theory.