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periodicity theorem

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Context

Cohomology

cohomology

Special and general types

Special notions

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Extra structure

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Stable Homotopy theory

Higher algebra

Contents

Idea

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

Statement

A v nv_n-self-map on a p-local finite spectrum XX, for n1n \geq 1 is a map

f:Σ kXX f \;\colon\; \Sigma^k X \longrightarrow X

such that

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

  2. for nln \neq l the induced map K(l) *XK(l) *XK(l)_\ast X \longrightarrow K(l)_\ast X is nilpotent.

The periodicity theorem says:

Any p-local finite spectrum XX admits a v nv_n-self-map. (Lurie 10, theorem 4)

It is a corollary of the theorem, that for any such space, there is a v nv_n-self-map, such that for nln \neq l the induced map K(l) *XK(l) *XK(l)_\ast X \longrightarrow K(l)_\ast X is 00, 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.