# A Mini-Course on Morava Stabilizer Groups and Their Cohomology

## TL;DR

This paper provides an overview of Morava stabilizer groups and their cohomology, emphasizing their crucial role in understanding chromatic stable homotopy theory at various primes and levels.

## Contribution

It offers a comprehensive mini-course on the structure and cohomology of Morava stabilizer groups, highlighting their importance in chromatic homotopy theory.

## Key findings

- Morava stabilizer groups are central to chromatic stable homotopy theory.
- The cohomology of these groups controls the chromatic type of spectra.
- The paper clarifies the relationship between p-adic Lie groups and stable homotopy types.

## Abstract

The Morava stabilizer groups play a dominating role in chromatic stable ho-motopy theory. In fact, for suitable spectra X, for example all finite spectra, thechromatic homotopy type of X at chromatic level n \textgreater{} 0 and a given prime p islargely controlled by the continuous cohomology of a certain p-adic Lie group Gn,in stable homotopy theory known under the name of Morava stabilizer group oflevel n at p, with coefficients in the corresponding Morava module (En)$\star$X.