Contents

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).

### Landweber exact functor theorem

There is an analogue of the Landweber exact functor theorem for even 2-periodic cohomology theories, with MU replaced by MP (Hovey-Strickland 99, theorem 2.8, Lurie lecture 18, prop. 11).

## References

The concept of even 2-periodic multiplicative cohomology theories originates with

The analogue of the Landweber exact functor theorem for even 2-periodic cohomology is discussed in

Review includes

• section 5.1 by Markus Land in these “TMF seminar” notes: pdf

• Akhil Mathew, Lennart Meier, section 2.1 of Affineness and chromatic homotopy theory, J. Topol. 8 (2015), no. 2, 476–528 (arXiv:1311.0514)

Last revised on March 8, 2017 at 12:24:42. See the history of this page for a list of all contributions to it.