nLab
periodic cohomology theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

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

()β:E *(*)E *+2(*). (-)\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 2n2n-periodic cohomology theories

Properties

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

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

Ω 2NNΣ 2N. \Omega^2 N \simeq N \simeq \Sigma^2 N \,.

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

Examples

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