cohomology

# Contents

## Definition

An even 2-periodic cohomology theory or just periodic cohomology theory for short is an even multiplicative cohomology theory $E$ with a Bott element $\beta \in E^2({*})$ which is invertible (under multiplication in the cohomology ring of the point) so that multiplication by it induces an isomorphism

$(-)\cdot \beta : E^*({*}) \simeq E^{*+2}({*}) \,.$

Via the Brown representability theorem this corresponds to a periodic ring spectrum.

Compare with the notion of weakly periodic cohomology theory.

More generally one considers $2n$-periodic cohomology theories

## Properties

### Periodicity of the $\infty$-category of $\infty$-modules

For $E$ an E-∞ ring representing a periodic cohomology (a periodic ring spectrum) double suspension/looping on any $E$-∞-module $N$ is equivalent to the identity

$\Omega^2 N \simeq N \simeq \Sigma^2 N \,.$

This equivalence ought to be coherent to yield a $\mathbb{Z}/2\mathbb{Z}$ ∞-action on the (∞,1)-category of (∞,1)-modules $E Mod$ (MO discussion).

## References

Lecture notes include

Revised on November 1, 2014 22:26:06 by Urs Schreiber (141.0.9.61)