# nLab periodicity theorem

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A statement in chromatic homotopy theory about periodicity of p-local spectra.

## Statement

A $v_n$-self-map on a p-local finite spectrum $X$, for $n \geq 1$ is a map

$f \;\colon\; \Sigma^k X \longrightarrow X$

such that

1. it induces an isomorphism $K(n)_\ast X \longrightarrow K(n)_\ast X$

2. for $n \neq l$ the induced map $K(l)_\ast X \longrightarrow K(l)_\ast X$ is nilpotent.

The periodicity theorem says:

Any p-local finite spectrum $X$ admits a $v_n$-self-map. (Lurie 10, theorem 4)

It is a corollary of the theorem, that for any such space, there is a $v_n$-self-map, such that for $n \neq l$ the induced map $K(l)_\ast X \longrightarrow K(l)_\ast X$ is $0$, and not just nilpotent. (Hopkins, Smith, Corollary 3.3)

## References

The periodicity theorem is due to

A quick review is in

Lecture notes are in

Quick lecture notes are in

Last revised on March 24, 2018 at 08:28:33. See the history of this page for a list of all contributions to it.