nLab periodicity theorem

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)

Related concepts

• telescopic localization

References

The periodicity theorem is due to

• Ethan Devinatz, Michael Hopkins, Jeff Smith, 1980s, and appears in Nilpotence and Stable Homotopy Theory II

A quick review is in

• Haynes Miller, section 4 of “Chromatic” Homotopy theory, 2001 (pdf)

Lecture notes are in

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 27 The periodicity theorem (pdf)

Quick lecture notes are in

• Robb Legg, The periodicity theorem, talk at 2013 Pre-Talbot Seminar (pdf)